THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES The RALPH D. REED LIBRARY DEPARTMENT OF GROLOGY UNIVERSITY of CALIFORNIA LOS ANGBLES. CALIF. Cambridge Natural defence GEOLOGICAL SERIES. CKYSTALLOGEAPHY. EonDon: C. J. CLAY AND SONS, CAMBRIDGE UNIVEKSITY PRESS WAREHOUSE, AVE MARIA LANE, AND H. K. LEWIS, 136, GOWER STREET, W.C. 263, ABGYLB STREET. ILeipjis: F. A. BROCKHAUS. }Sorfe: THE MACMILLAN COMPANY. Bombap: E. SEYMOUR HALE. A TEEATISE ON CRYSTALLOGRAPHY BY W. J. LEWIS, M.A. PROFESSOR OF MINERALOGY IN THE UNIVERSITY OF CAMBRIDGE, FELLOW OF ORIEL COLLEGE, OXFORD. CAMBRIDGE : AT THE UNIVERSITY PRESS. 1899 [All Rights reserved.] CamimUge: PRINTED BY J. AND 0. F. CLAY, AT THE UNIVERSITY PRESS. Library PREFACE. MY purpose in this text-book is to set before the student the views held at the present time as to the classification of crystals and the principles of symmetry on which the classification is based ; to describe the ' forms ' which are a consequence of the symmetry; to determine the geometrical relations of the forms ; and, finally, to explain the methods by which the crystals are drawn and their forms represented graphically. The treatment has been as far as possible geo- metrical, and, with the exception of the formulae of plane and spherical trigonometry necessary for the solution of triangles, very little analysis has been introduced into the main discussion. Seeing that good figures are important aids to the understanding of the geometrical relations of crystals, and that practice in drawing crystals develops the student's power of solving crystal- lographic problems, I have throughout given prominence to the methods used in making diagrams. The principles of pro- jection followed in the construction of such figures are there- fore explained at an early stage. Hence it is necessary in certain parts of Chapters VI and VII to presuppose some knowledge of the crystals of the different systems : and, accordingly, those articles which refer to a given system should VI PREFACE. be read with the corresponding chapter, and with the examples which it contains. A brief sketch of the methods depending on analytical geometry suitable for the treatment of crystallo- graphic problems has been given in Chapter XIX. It is now generally held that each of the thirty-two possible classes of crystals is a definite group, the forms in which are a direct consequence of the 'elements of symmetry' (p. 21) present in it. Some of these classes have certain geo- metrical and physical relations in common, and form larger groups called systems. In discussing the classes seriatim, I have taken first the crystals which have no symmetry; and, in passing from one class to another, I have, in general, pro- ceeded to that which involves the least addition to the sym- metry of its predecessor, or follows from it on the addition of a centre of symmetry. When no further classes can be obtained by this process, a fresh start is made with a class which has the least symmetry of a new kind. Thus, for instance, the acleistous tetragonal class, which has only a single tetrad axis, comes third in Chapter XIV, whilst the two preceding classes of the tetragonal system have been derived from particular combinations of dyad axes. The geometrical characters of rhombohedral and hexagonal crystals require special treat- ment, and, accordingly, their systems have been placed last. The general order of discussion has involved some repetition ; and, unfortunately, in almost every system it entails discussing at the outset a class in which the geometrical relations do not lend themselves readily to elementary treatment. The beginner will therefore do well to read through the general discussion of all the classes of a system before endeavouring to master the geometrical relations between the face-symbols and the interfacial angles. It is not easy to find distinctive and appropriate names PREFACE. Vll for the several classes. Those which I have adopted are, with one or two exceptions, intended to indicate the shape of the 'general form' of the class. Names based on physical characters fail to bring clearly before the mind the shape of the crystal, or to indicate the elements of symmetry present ; and those based on particular substances may, with greater know- ledge, have to be changed, if the name-substance has to be transferred from one class to another. The simple notation for the crystal-forms, and the elegant method of geometrical treatment by the stereographic pro- jection and the anharmoiiic ratio of four tautozonal faces, with which Miller's name is indissolubly associated, have been adopted throughout. In the scalenohedral class of the rhom- bohedral system Naumann's symbols are so expressive of the geometrical relations of the various rhombohedra and scaleno- hedra, that I have used his notation as well as Miller's in the representation of these forms. Free use has been made of the works of previous writers Naumann, Haidinger, Miller, Story-Maskelyne, Groth, and others. From the late Professor W. H. Miller and Professor M. H. N. Story-Maskelyne I received my training as a crystallo- grapher, so that my debt to them is of a very special and personal nature. I have drawn largely on the stores of informa- tion in Professor E. S. Dana's System of Mineralogy, 1892 ; and I am indebted to him and to his publishers, Messrs. Wiley of New York, for cliches of several of the figures in that work. I have to thank Professor Groth and his publisher, Herr Engel- mann, for cliches of several of the figures in the Physikalische Krystallographie ; Mr. Hilary Bauerman for the loan of the blocks of several of the figures in Miller's works ; and Dr. J. H. Pratt, of the Geological Survey of North Carolina, for several figures, the cliches of which were kindly supplied by Professor Vlll PREFACE. Dana, the editor of the American Journal of Science. Professor Miers of Oxford has permitted the use of the figure of the student's goniometer designed by him ; and Herr Fuess of Berlin generously placed at my disposal cliches of his cele- brated instruments. I have had valuable help from numerous friends, to each and all of whom I give my hearty thanks. I am specially indebted to Mr. L. Fletcher, Keeper of the Minerals of the British Museum ; to Mr. L. J. Spencer, also of the Mineral Department of the British Museum ; to my colleague, Mr. A. Hutchinson, Fellow of Pembroke College, Cambridge ; and to my sister, Mrs. G. T. Pilcher; who have all been good enough to read the proof-sheets, and some parts of the manuscript. To, their care and the valuable suggestions made by them the book owes much ; and it is hoped that few errors of any kind have escaped detection. The very full index is the work of Mr. Spencer. I have much pleasure also in expressing my obligation to Mr. Edwin Wilson for the care exercised in the preparation of the diagrams needed to illustrate the text. W. J. LEWIS. CAMBRIDGE, 28th September, 1899. CONTENTS. CHAPTER I. PAGE CRYSTALS AND THEIR FORMATION 1 CHAPTER II. THE LAW OP CONSTANCY OF ANGLE 8 CHAPTER III. SYMMETRY 15 CHAPTER IV. AXIAL REPRESENTATION 23 The law of rational indices ....... 26 The equations of the normal 29 CHAPTER V. ZONE-INDICES AND RELATIONS OF ZONES . , . . . 33 Weiss's zone-law . . .. ..-... . 39 Face common to two zones .... 43 CHAPTER VI. CRYSTAL-DRAWINGS . . . ,,.... ... 48 Plans and elevations . . . ;;v : . , vv , . 49 Orthographic drawings .'.'* . 55 Clinographic drawings 65 X CONTENTS. CHAPTER VII. PAGE LINEAR AND STEREOGRAPHIC PROJECTIONS . . . 70 CHAPTER VIII. THE ANHARMONIC RATIO OP FOUR TAUTOZONAL FACES . . 87 Transformation of axes 104 CHAPTER IX. CONDITIONS FOR PLANES AND AXES OF SYMMETRY, AND RELA- TIONS BETWEEN THE ELEMENTS OF SYMMETRY . . 108 CHAPTER X. THE SYSTEMS ; AND SOME OF THE PHYSICAL CHARACTERS ASSOCIATED WITH THEM 138 Optical characters 140 CHAPTER XI. THE ANORTHIC SYSTEM 148 I. Pediad class; a{hkl} . . . 148 II. Pinakoidal class ; {hkl} 154 Formulae connecting crystal-elements, indices and angles 161 CHAPTER XII. THE OBLIQUE SYSTEM . . 172 I. Hemimorphic class ; a {hkl} 173 II. Gonioid class ; K {hkl} . ... . . . . 177 III. Plinthoid class ; {hkl} 178 Formulae and methods of calculation . . . .181 CHAPTER XIII. THE PRISMATIC SYSTEM 197 I. Sphenoidal class ; a {hkl} . . . . . . ' . 198 II. Bipyramidal class ; {hkl} . , . -. ' Y . . 205 III. Acleistous pyramidal class ; p. {hkl} 209 Formulae and methods of calculation 211 CONTENTS. xi CHAPTER XIV. PAGE THE TETRAGONAL SYSTEM . . .-. ... . . 224 I. Scalenohedral class ; K {hkl} ...:.. . . .225 II. Diplohedral ditetragonal class ; {hkl} ... 243 III. Acleistous (polar) tetragonal class; r{hkl] . . . 254 IV. Diplohedral tetragonal class ; ir{hkl} .... 260 V. Trapezohedral class : a{hkl} 263 VI. Acleistous ditetragonal class ; p. {hkl} .... 268 VII. Sphenoidal class ; T a {hkl} 270 CHAPTER XV. THE CUBIC SYSTEM 275 I. Plagihedral class ; a {hkl} 284 II. Hexakis-octahedral class ; {hkl} 303 III. Tetrahedral class; r{hkl} 311 IV. Dyakis-dodecahedral class; Tr{hkl} . . . .328 V. Hexakis-tetrahedral class; fi{hkl} 333 To draw the general forms 340 CHAPTER XVI. THE RHOMBOHEDRAL SYSTEM 344 I. Acleistous trigonal class ; T {hkl} 350 II. Diplohedral trigonal class ; IT {hkl} .... 360 III. Scalenohedral class ; {hkl} 365 IV. Trapezohedral class ; a {hkl} . . . . . .408 V. Acleistous ditrigonal class; ft. {hkl} . . . .418 VI. Trigonal bipyramidal class; r{hkl,pqr} '. . . 422 VII. Ditrigonal bipyramidal class ; K {hkl, pqr} . . . 424 CHAPTER XVII. THE HEXAGONAL SYSTEM 426 Rhombohedral axes 427 Hexagonal axes . 430 I. Acleistous hexagonal class; r{hkl, pqr}=r{bkil} . . 443 II. Diplohedral hexagonal class; n{hkl,pqr} = ir $h\til} . 448 III. Acleistous dihexagonal class ; ft. {hkl, pqr}=p {hkil} . 451 IV. Diplohedral dihexagonal class; {hkl, pqr} = {hkil} . . 455 V. Trapezohedral class; a {hkl, pqr] = a {hkil} . . .457 Xli CONTENTS. CHAPTER XVIII. PAGE TWIN-CRYSTALS AND OTHER COMPOSITE CRYSTALS . . .461 General introduction .461 i. Twins of the cubic system 466 ii. tetragonal system 482 iii. prismatic system 497 iv. rhombohedral system . . . .513 v. hexagonal system 527 vi. oblique system 528 vii. anorthic system 543 Twin-axis a line in a face perpendicular to one of its edges 554 On the method of determining the position of the twin- axis 556 CHAPTER XIX. ANALYTICAL METHODS AND DIVERS NOTATIONS .... 557 Some propositions relating to the rhombohedral system . 573 Grassmann's method of axial representation . . . .581 Weiss's, Naumann's and Levy's notations .... 585 CHAPTER XX. ON GONIOMETERS * 589 The vertical-circle goniometer 592 The horizontal-circle goniometer 597 Theodolite goniometers .601 Three-circle goniometer . 604 INDEX 605 CRYSTALLOGRAPHY. CHAPTER I. CRYSTALS AND THEIR FORMATION. 1. CRYSTALS are homogeneous solid bodies bounded by plane surfaces arranged according to definite laws of symmetry : their physical properties, such as cohesion, elasticity, optical and thermal characters, are intimately connected with the symmetry of the external form. The plane surfaces are called the faces of the crystal. 2. Examination of a few crystals of common substances, such as calcite (CaCO 3 ), quartz (SiO 2 ), gypsum (CaSO 4 . 2H 2 O), and soda (Na 2 CO 3 . 10H 2 O), brings to our notice two important characteristics of the external form. They are: (1) the parallelism of the planes in pairs, and (2) the arrangement of them in zones, i.e. in sets, the planes of each of which intersect one another in parallel edges. A straight line drawn through some fixed point, called the origin, parallel to the edges of a set is called a zone-axis. In Fig. 1, the planes marked b, m, m f and those parallel to them form a zone with a vertical zone- axis; the faces marked b, I and I' form a second zone. 3. The connection of the physical properties with external form is strikingly manifested in the case of the cleavage, i.e. the property, characteristic j? IO> of many crystals, of splitting along plane surfaces having certain directions. It is well shown in calcite, fluor and 2 CHARACTERISTICS OF CRYSTALS. FIG. 2. gypsum. Thus in the latter there is a good cleavage parallel to the plane marked b in Fig. 1. The cleavage-planes, shortly called cleavages, are always parallel to actual or possible 1 (Chap. iv. Art. 9) faces of the crystal. They often enable us to get fragments, com- pletely bounded by cleavages, which differ from the original crystals in no respect except in their mode of formation. Such cleavage- fragments as, for instance, those of calcite and rock-salt are sometimes hard to distinguish from true crystals. Experience in the criticism of the character of the crystal-faces will in most cases enable us to distinguish between them and those resulting from cleavage. For the natural faces often show coarse or fine markings striae, pittings, fec. which accord with the symmetry of the face, and are an important characteristic of it. Thus the cubes of pyrites, Fig. 2, are frequently striated in the manner shown in the diagram. A perfect cleavage should be smooth ; and markings, perceived on a cleavage-face of a simple crystal, arise from the accidental interruptions in the continuity of the cleavage. Imitations of crystals in glass, or by the cutting of artificial faces on fragments of crystals or metal, such as gold, can generally be easily distinguished from true crystals. False faces, however, sometimes occur in Nature, and may be due to several causes : as, for instance, to the growth of the crystal being hampered by the surface of some other body, just as that of a crystal growing in a vessel is hindered by the sides and bottom ; or to accidental cleavage followed by corrosion, as may have occurred in some crystals of calcite. True crystal-forms are also found in which the internal structure is not correlated with the external form. In such cases the originally formed crystal has undergone a change in its internal structure with, or without, change of substance. Such altered bodies, in which the substance, or the internal structure, or both, no longer correspond with the external form, are called pseudomorphs ; and are not to be included amongst crystals : as instances of such transformations we may mention the brownish translucent crystals of sulphur 1 By a possible face or edge of a crystal is meant one which satisfies the law of development of the crystal. A face possible on one crystal may actually exist on another crystal of the same substance. FORMATION OF CRYSTALS. 3 formed by fusion, which after the lapse of a day or two change into aggregates of yellow sulphur, and the brown pseudoraorphs after pyrites (FeS 2 ) resulting from the conversion of the iron disul- phide into a hydroxide. 4. Crystallization is a property common to almost every sub- stance of which the structure is not the result of organic growth. Instances of crystallization are familiar to everyone : e.g. salt and sugar. The crystalline form is characteristic of the substance, and conversely, each chemical compound has a distinctive and charac- teristic crystalline form, subject however to the modifying principles of isomorphism and polymorphism. Crystallography is the science which treats of the distribution of the faces on a crystal, of the geometrical relations to which their positions are subject, and of the classification of crystals in groups depending on the distribution of the faces. It applies equally to the natural bodies known as minerals and to the crystals manu- factured in the laboratory. Formation of Crystals. 5. Crystals are formed during the passage of their substance, under favourable conditions, from a fluid to a solid state. Thus, snow is the result of the rapid cooling of aqueous vapour, and consists of numerous minute crystals of ice aggregated together. Crystals of phosphorus have been formed by volatilizing the substance in a hermetically closed tube heated at one end and cooled at the other. Liquid bismuth, on cooling, gives six-faced crystals of bismuth. In the larger number of cases crystals are formed from solutions, in which the liquid, which serves as solvent, contains one or several substances. The quantity of a substance which a given quantity of liquid can, under ordinary circumstances, hold in solution at a definite temperature is limited. At this limit the solution is said to be saturated. The higher the temperature the greater the quantity which can as a rule be held in solution ; consequently a solution saturated at a high temperature will deposit crystals as the temperature falls. Thus a solution of nitre (KNO 3 ) saturated at 30 C. deposits crystals when it is allowed to cool freely. Crystals are also obtained by the evaporation of a solution. They are likewise formed by the mixture of two non-saturated solutions of different substances which react on one another chemically; thus, 12 4 GROWTH OF CRYSTALS. for instance, if dilute solutions of Epsom salts (MgSO 4 . 7H 2 O) and calcium chloride (CaCl 2 ) be mixed together crystals of gypsum are formed, because gypsum is less soluble than either of the two salts. Crystals can also be formed as a result of the chemical changes which a solution undergoes during the passage of an electric current. 6. Crystals may grow whilst suspended in a fluid mass which is more or less mobile, and which at the same time supplies the material for their growth; or they may be deposited on, and attached to, a solid base, when they can only grow on the side which is directed towards the freely open space. Snow and oc- casionally hailstones are instances of the former mode of formation. Sugar, crystallizing rapidly from a mass of syrup, is another. The imbedded crystals of leucite so often found in Vesuvian lava, and those of pyrites in slate, are possibly instances of the same kind. The attached crystals of calcite, quartz and orthoclase (KAlSi 3 O 8 ) found in crevices in limestone and granite are instances of the second kind. Imbedded crystals, which owe their origin to the first mentioned conditions, are generally completely developed on all sides, and form perfect individuals. Such completely developed crystals will be the subjects of our consideration. When a substance is only found in attached crystals, which are naturally incomplete, we determine their complete form by induction from examination of a large number of differently attached crystals. In the laboratory attached crystals are those most commonly obtained, for the growth is impeded by the sides or bottom of the dish. In many cases the incompleteness can be prevented by frequently turning the crystals so that they rest on the same side for only a short time. In this way very perfect octahedra of alum can be made by the evaporation of a concentrated solution. Another way of getting almost perfect crystals is by attaching a small crystal of the substance to a fine thread and suspending it in a solution which is depositing similar crystals. A third method is to add glue or other viscid body to the solution. Such a method, however, sometimes introduces restraints in the correct orientation of the particles which are passing into the solid state, as is manifested by the formation of complex individuals called twins, which are often distinguished by re-entrant angles (Chap. xvii.). Another method of obtaining completely developed crystals is by floating the solution on a liquid of greater density. It is HABIT OF CRYSTALS. 6 found that the more slowly the substance is deposited, and therefore as a rule the more slowly evaporation takes place, the more perfect are the crystals formed. It has not been found possible to form in the laboratory crystals of all substances, and those of the minerals common in Nature have in many cases not been obtained artificially, and very little is known of the conditions under which the latter have been, and are being, produced. 7. Crystals have, in many cases, the same form at every stage of their growth from the smallest to the largest. The growth here consists simply in the uniform deposition of the substance which passes from the fluid to the solid state. This gives rise to a structure in which the layers parallel to a particular face are completely similar and similarly orientated to each other, for the larger crystal has its faces and edges parallel to those of the nucleus. There seems in this case to be no essential difference in the growth of the crystal at different periods. In certain cases there is a tendency to grow more rapidly in particular directions or places, as is evidenced by the fact that some crystals are flat tablets, others long and columnar, &c. This tendency affects the general aspect of the crystal and is described by the word habit. The crystals of some substances have a fairly constant habit : as, for instance, sulphur. The habit in other substances is often very variable : as in quartz, barytes and calcite ; and this variableness of habit is one of the greatest difficulties a beginner has to encounter. It may depend, as is generally the case in quartz, on the exceptional development of particular faces at the expense of others \ the faces present being the same, or very nearly so, in all the crystals. Or, as is often the case in calcite, it may FIG. 3. FIG. 4. 6 TRANSFORMATIONS DURING GROWTH. depend on the different faces present on the crystals ; thus the crystals occur as stout six-faced prisms ending in two faces, Fig. 3, or in very flat rhombohedra, Fig. 4, or again in crystals bounded by twelve equal scalene triangles, Fig. 13. These crystals are discussed and drawn in Chap. xv. section (iii). The cause of the diversity of habit is unknown, but it is connected with the conditions under which the crystals have been formed. The habit is not subject to crystallographic law, and a full appreciation of it is only to be acquired by considerable familiarity with crystals. Another remarkable property of a crystal is its power of repairing itself. Thus, if a portion be irregularly broken from a crystal, and the specimen be then immersed in a crystallizing solution of the substance, the deficiency caused by the fracture is made up and a complete and perfect crystal re-formed. 8. During the growth of a crystal considerable transformations sometimes take place.' The first kind of change consists in the more rapid deposition on one part than on another. Thus, if regular octahedra of alum (Fig. 5) be allowed to grow with certain faces FIG. 5. FIG. 6. more exposed to the solution than others, the crystals become irregular in appearance, as shown in Fig. 6. The faces in these unequably developed crystals may be the same in number as in the equably developed crystals. Occasionally some of the faces may be completely obliterated. The relative position of the faces is the same as in the regular forms, i.e. the angles between the several faces remain constant. Other changes consist in the appearance of new faces, or in the disappearance of old ones. In this way transformations occur TRANSFORMATIONS DURING GROWTH. which result in a completely new form. Crystals of alum (Fig. 5) under certain circumstances develop new faces, replacing the edges and corners. The six corners may be cut off and replaced by squares as shown in Fig. 7. In some exceptional cases the change may be so extensive as to completely obliterate certain faces. When, however, new faces appear during the growth of a crystal, they always come (allowance being made for accidental irregularities) in such a way as to modify all similar corners and edges in the same manner. Further, in the disappearance of faces, like ones disappear together. The growth of crystals consists, therefore, partly in the deposition of layers of uniform thickness on each face so as to retain the parallelism of the earlier and later planes ; and partly in the development of new faces, or in the obliteration of old ones, in such a way that the symmetry is retained. CHAPTER II. THE LAW OF CONSTANCY OF ANGLE. 1. OBSERVATIONS made on the growth of crystals show that the faces when once formed can undergo considerable modifications in their development, but that their relative positions remain the same. Hence, many crystals, which seem strikingly dissimilar, are found on close inspection and measurement of the angles to be bounded by similar faces similarly orientated with respect to one another. Thus Figs. 8 and 9 represent two crystals of garnet. The only difference between the two figures arises from the fact that in the latter the vertical planes meet in edges whilst in the former they meet in points where two other faces inclined to the vertical also meet. FIG. FIG. 9. 2. The relative position of two faces is determined by the dihedral (i.e. between two faces) angle which they make with one another. This angle is that between the two lines in which a plane perpendicular to the edge of intersection meets the faces the angle ABC in Fig. 10. We may however take the angle to be NiON z , the angle between the two perpendiculars OJ\\, ON. 2 , drawn from a point within the crystal to meet the faces. For the angles NflN^ and ABC are supplementary, as can be seen from the fact that the lines ON U ON Z , AB and BC all lie in a plane perpendicular to the edge, which is supposed to be perpendicular to NICOLAUS STENO. FIG. 10. the paper. The figure N^ON^B can be divided into two triangles NflB, N. 2 OB. But the interior angles of a triangle make up two right angles. The angles of the figure NflN^B are therefore together equal to four right angles, and those at N 1 and N z are each 90. Hence NiONs+.ABC=\W. And AB being pro- duced to E, AC + EC=180. Therefore N^N^EBC. The lines ON lt ON 2 , drawn through an arbitrary point within the crystal are called the normals to the faces. Simi- larly the inclination of BC to a third plane CD in the zone is given by BCD or by N 2 ON 3 . The normal-angle between the faces AB and CD is then clearly N^ON^ = N^ON. 2 + N 2 0^ . The angles given in most modern works on Mineralogy and Crystallography are those between the normals, and normal-angles will be given in this book when the contrary is not stated. When precision of statement is needed to avoid ambiguity, the phrase normal-angle will be used in contradistinction to that of face-angle used to denote the Euclidean angle such as ABC. French mine- ralogists still employ the face-angle, and also Levy's modification of Haiiy's notation to represent the faces. 3. The first investigation into the values of the dihedral angles of crystals was made by Nicolaus Steno, whose work 1 was published in Florence in 1669. He cut a series of crystals of quartz of different habits one of which is given in Fig. 11, in two directions at right angles to each other, (i) perpendicular to the edge [mmj, and (ii) perpendicular to one or other of the edges [mr], [w*X]> [*W] ^ the figure. By placing the sections on paper and drawing lines carefully parallel to the edges he was able to measure the Euclidean angle. In the first set of sections he found that all the angles between the faces m on the same crystal were equal to 120, and that this was also the angle between similarly placed faces on other crystals. The normal-angle mm i is therefore 60. In the second series of sections he got figures which were not equiangular and of 1 An English translation of his work was published in London in 1671. It is hardly necessary to state that the figures and the values of /3 and 7, given in the text, are not taken from Steno's work. FIG. 11. 10 ROM DE L'ISLE. which Fig. 1 2 may be taken, to be a type. Two of the angles at the vertices had the same value /3=7626' in all cases; the four others had the same value y = 141 47'. He, there- fore, inferred that in all crystals of quartz corre- sponding pairs of faces are inclined to one another at constant angles. 4. To Rom6 de 1'Isle, whose ' Essai de Cristallo- graphie' was published in Paris in 1772, is due the credit of establishing, by an extensive series of measurements, the fundamental law of crystallo- graphy, viz. that the angles between corresponding IG- pairs of faces of similarly orientated crystals of one and the same substance are always equal. This experimental law we shall denote as that of the constancy of angles between corresponding pairs of He showed that the numerous suites of forms on such minerals as fluor (CaF 2 ) and calcite may be developed by the modification of edges and corners of a primitive form, which he assumed in the former case to be a cube, and in the latter a rhombohedron one of whose face-angles is 105 5'. When an edge or coign of the primitive form is modified by the introduction of one or more faces it is necessary to repeat the same process on each like edge or coign ; in fact to retain the symmetry characteristic of the primitive form adopted. It was seen, however, that the same suites of forms could be derived from different primitive forms. Thus, for instance, those of fluor may be derived with equal ease from either of the common forms, the cube or the regular octahedron. There is nothing to indicate the nature of the primitive form, except frequency of occurrence, or magnitude of development, or the simplicity of character of the form. 5. In the year 1784 the celebrated 'Essai d'une theorie sur la structure des crystaux' of the Abbe Haiiy was published, which fully established the correctness of Rome* de 1'Isle's law and placed the science on a firm basis, although his mode of interpreting the facts has long since been abandoned. Haiiy's attention was attracted to the cleavage of calcite by an accident which happened to one of his specimens, which led him to try a series of experiments on the cleavage of crystals of calcite of totally different shapes. He found that cleavage-rhombohedra could be obtained which had identically HAUY S THEORY. 11 the same angles, however different might be the shapes of the original crystals. This suggested to him the idea that the crystals were built up of small nuclei, or 'constituent molecules,' which had the form of regular rhombohedra having the same angles as the cleavage-rhombohedron. A cleavage-slice from a hexagonal prism of calcite, Fig. 3, can be again cleaved in directions transverse to the original cleavage, so that the fragments from the middle of the slice are rhombohedra, whilst the fragments at the edges are not rhom- bohedra, for they have one face of the original prism as part of their boundary. It is necessary, therefore, to assume either that the nuclei are very small and that the secondary face is of the nature of a slope produced by a series of extremely small uniform steps, or else that the secondary planes on crystals are due to a difference in their nuclei. Haiiy saw that the former assumption gave a simple solution of the derivation of secondary faces. Thus Fig. 13 shows a crystal of calcite of a shape frequent in Derbyshire, and Fig. 14 represents a model (drawn on a larger FIG. 13. FIG. 14. scale) by which such a crystal is by Haiiy derived from the cleavage- rhombohedron. In Fig. 14 a large rhombohedron, whose middle edges are the same as those of Fig. 13, is built up of layers of 12 HAUY'S THEORY. regularly disposed nuclei, each nucleus being similar and similarly placed to the large rhombohedron. A cap is then formed of three plates, one nucleus thick, and placed on the upper faces of the rhombohedron. Each plate is a rhombus similar to the face of the rhombohedron on which it rests, but it has one nucleus less in the edge than the face, and consequently two rows of nuclei in the face are left uncovered. Such a cap can easily be imitated by three equal rhombuses of thick card-board cut with angles of 101 55' and 78 5', and attached to one another along the edges so that the obtuse angles are at the apex. On this cap a similar cap of three plates is superposed in like manner so as to leave two rows of nuclei of the first uncovered. Each of the plates in this cap is one nucleus thick and has one nucleus less in the edge than the first cap. The same process is continued until the apex is reached. The plates in the last cap must clearly consist of four nuclei, as shown in Fig. 14. If now a series of exactly similar caps is placed in succession on the lower faces of the large rhombohedron, the model would present a similar aspect at the two ends. Suppose now the nuclei to be very small, as the ultimate molecules of bodies must be, then the discontinuity in the lines joining the corners of the caps, which is so conspicuous in the dotted lines of Fig. 14, becomes imperceptible and we have the crystal shown in Fig. 13, known as a scalenohedron of the rhombohedral system. Had the number of nuclei in the edges of the first cap placed on the large rhombohedron been the same as in its edges, the lines joining the projecting corners would not have converged, but would have been parallel. The same will clearly be true of any number of successive caps, for in this case they have the same dimensions. In this way a hexagonal prism {10T} terminated by the primitive rhom- bohedron can be easily obtained. 6. Crystals of galena (PbS) and rock-salt (NaCl) cleave along planes parallel to the faces of a cube. We can in these cases derive the secondary faces from a square plate formed in one of two ways, (i) The sides of the cubic nuclei may be parallel to the edges of the plate in chessboard fashion, or (ii) the sides of the nuclei may be diagonally placed to the edges of the plate. In the latter the edges of the plate will resemble a staircase in which the height of the step HAUY S THEORY. FIG. 15. is equal to the tread. If now to a plate of this kind similar plates are attached on both sides with a decrement of one nucleus on each side of the plate, and then to the outer surfaces of these plates similar plates with a like decrement of one nucleus are attached, and the process re- peated to the limit, we obtain a solid like that in Fig. 15, which, when the nuclei are very small, is the regular octahedron common on crystals of galena. Now the height of each layer is the edge of the cube, a, and the breadth of the decrement on each edge is clearly one-half the diagonal of the cubic face = a -r- J2. Hence, the in- clination of the face to the plane of the original plate, 0, is given by tan = height -j- breadth - ^2, /. = 54 44'. The Euclidean angle between the faces of the octahedron meeting in an edge is 2x54 44' =109 28'. The result of a decrement of one nucleus on each edge of a plate in the arrangement (i) is shown in the rhombic-dodecahedron, Fig. 16. In this the plates are shown un- divided into their constituent cubes, and the figure is supposed to start with a large cube, and not with a plate of single nuclei. The angle each face of the secondary form makes with that of the cube on which it is raised is 45, for the height of each layer is equal to the breadth of the decrement. Hence, the faces meeting at an edge of the large cube coalesce in one plane, and form a rhombus. Had we, however, taken a decrement on each whose breadth was double the height of the layer, each face would be inclined at 26 34' to the cubic face, for tan 26 34' = 1 -r- 2. We then get a tetragonal pyramid on each face of the cube, and the form is that fully described in Chap. xiv. Both these secondary forms are found on crystals of rock-salt. 7. In the manner illustrated by the preceding examples Haiiy gave an explanation of the constancy of the angles between cor- responding pairs of faces, and established the further fact that the angles, which the secondary faces make with those of the primitive FIG. 16. 14 HAUY'S THEORY. form are not accidental and arbitrary, but depend on the ratio of whole numbers involved in the decrements. This relation of whole numbers is now expressed by the phrase 'the law of rational indices' (see Chap. iv. Art. 9). It is not essential to Hatty's theory that the nuclei should be solid polyhedra exactly filling the space. It suffices to suppose that his nuclei are cells each of which contains a similarly placed molecule. 8. The difficulties in accepting Haiiy's theory are serious. (1) It does not follow that, because a body breaks up by cleavage into definite polyhedra, it has been built up by layers of these polyhedra. (2) It has not been possible to obtain cleavage polyhedra of all crystals. Some crystals cleave only in one, or two, directions, and some not at all. (3) It seems reasonable to expect that fairly compact, and therefore stable, bodies can be built up when the cleavage polyhedra are cubes as in galena, or rhombohedra as in calcite, or are four-sided prisms terminated by a pair of parallel faces as in barytes. For in all these cases the constituent polyhedra can be arranged so as completely to fill a definite portion of space. But in fluor the cleavage form is the regular octahedron, and the nuclei, when arranged in like orientation, will only touch one another at their edges and will leave a considerable portion of space unoccupied. Such a structure would probably be very un- stable and be easily deformed. Furthermore, the facts, (1) that we have a series of different forms all connected together, and (2) that the secondary faces depend on whole numbers, would still be explained if any other simple form of the suite were taken as primitive. CHAPTER III. SYMMETRY. 1. ATTENTION has been already called to the fact that the faces on crystals usually occur as pairs of parallel planes. When the two faces of each pair are coexistent and have the same physical character, i.e. the same lustre, striation or other marks, the crystal is said to be centro-symmetrical, or to be symmetrical with respect to a point within it, which is called the centre of symmetry. Figs. 1 and 2 are diagrams of centro-symrnetrical crystals. Professor Maskelyne expresses the relation by saying that the crystal is diplohedral, i.e. that every face-normal through the centre terminates on opposite sides in a pair of faces perpendicular to it. 2. There are crystals, however, like the tetrahedron, Fig. 17, in which the faces do not occur as pairs of ^___ ___^ parallel planes. From such figures it is easy to derive figures having parallel faces. Thus, by drawing planes as indicated by the small tri- angles, the octahedron, Fig. 18, can be obtained. Conversely, from the parallel-faced figure, such as Fig. 5, it is easy to derive a non- central figure like Fig. 17, by the omission of one of each pair of parallel planes. In more complex crystals there may be more than one way of selecting the faces to be omitted, so that we may get different non-central figures. Up to very recent times it has been usual to regard the parallel-faced crystal as the normal one and the non-central figure as caused by some deficiency in the symmetry. Other cases of a geometrical connection between different forms are also common, such that the form FIG. 17. PIG. 18. 16 CENTRO-SYMMETRY. having fewest faces manifests one-half, or one-quarter, the number observed on the most symmetrical cognate form ; the faces present being arranged according to definite rules and not being due to accident in growth. The most highly developed form was denoted as holohedral 1 , whilst the partial form was, and still is, called hemihedral* when it contains half the number of faces in the holohehral form. When the number of faces in the lower form is only one-fourth of that in the holohedral form, the partial form is said to be tetartohedral 3 . When it is not desired to express the exact numerical relation, the partial form is said to be merohedral* . These words do not accord with modern views on the symmetry of crystals, and are only retained for the sake of convenience. 3. From what has been said in Chapter I. as to the growth of crystals we see that, if the particles be themselves centro- symmetrical, and likewise the forces which cause them to cohere, a crystal growing freely in a liquid will grow equally well in opposite directions. If, however, there be a lack of symmetry, either in the particles or in the forces, the facility of growth may be on one side very different from that on the other side. It has been already pointed out that accidents of various kinds may modify the regularity of growth of a crystal. Hence the opposite and parallel faces of centro-symmetrical crystals need not be of the same dimensions. All that is needed is that the physical characters, as indicated by the lustre, stria tion, &c., and also the resistance to corrosion, should be of an exactly like kind. Further- more, it is clear that the deposition of matter on the opposite faces must under like conditions take place with equal facility. 4. Again the geometric law connecting crystal faces, known as that of rational indices (Chap. iv. Art. 9), shows that parallel faces are always possible, but does not require coexistence or identity of character. In non-central crystals parallel faces, which are not parallel to an axis of two-, four-, or six-fold symmetry (Arts. 7 and 8), have to be regarded as quite distinct faces, and often differ in lustre, striations and other physical properties. The minerals fahlerz and boracite give good instances of non-central crystals of shape like that given in Fig. 17. Crystals of quartz often appear to be 1 From SXos whole, and ZSpa. base. 2 From ^/xt- half, and HSpa base. 3 From T^ropros one-fourth, and HSpa. base. 4 From ptpos part, and ?5/>a base. PLANES OF SYMMETRY. 17 centro-symmetrical, for they are bounded by pairs of parallel faces, m, r and z, but the physical characters of all untwinned crystals of the substance and also the development of certain subordinate faces on some crystals, such as the faces s and x, in Fig. 1 1 , have conclu- sively proved them to have no centre of symmetry. 5. In simple crystals, such as are represented in the preceding diagrams, certain faces are observed to have the same physical characters, and, when the crystals are very regularly developed as in Figs. 1, 2, 5, 7 and 8 the similar faces have the same shape and magnitude. On quartz, Fig. 11, the faces m manifest this similarity in a most striking manner, for they are almost invariably all striated in directions perpendicular to their mutual intersections. Such like faces are found to be symmetrically arranged with respect to certain planes, or axes, or both, which are parallel to actual, or possible, faces or edges of the crystal. In complicated and irregularly developed crystals, such as Figs. 6 and 9, the similarity of shape of the faces is often lost, and it frequently happens that the physical characters are not sufficiently marked to enable us to discriminate those which belong to the same set of planes. The symmetry becomes obscured, and can only be established by measurement of the angles and careful criticism of the development of several crystals. Dodecahedra of garnet are frequent in which the symmetry is concealed by the unequal development of the faces. Fig. 9 shows one type of such irregular growth. The nature of the figure can, however, be generally perceived in such crystals by careful inspection of the physical characters of the faces and by comparison with a regularly grown dodecahedron like Fig. 8, and is at once established by measurement of the angles between adjacent faces which are either 60 or 90. 6. By symmetry with respect to a plane is meant the similarity of position of like coigns', edges and faces on opposite sides of the plane. A plane of symmetry divides a crystal into two like halves, which are to one another as an object is to its image seen in a mirror occupying the position of the plane. It thus bisects the angle between corresponding faces and edges. We shall often express the relation of two planes, or two lines, to a plane of 1 By coign is meant the solid angle or corner at which three or more faces meet. L. C. 2 18 PLANES OF SYMMETRY. FIG. 19. symmetry bisecting the angle between them by the statement that they are reciprocal reflexions in the plane. We shall also use the phrase to express that two crystals are related to one another as an object is to its image in a mirror in those classes of crystals in which such a relation exists. Thus the right and left hands are, when held out palm to palm, reciprocal reflexions of one another. Symmetry does not refer in crystals to the actual position, but to the relative angular position of corresponding faces and edges ; i.e. they are equally inclined to a plane of symmetry on opposite sides of it. In a regular figure, such as that of a crystal of barytes given in Fig. 19, corresponding faces, indicated by the same letters, will (if produced) meet in straight lines lying in the planes of symmetry. In the figure two planes of symmetry are perpen- dicular to the paper: the one bisecting the faces o, c, o' ; the other at right angles to the former and bisecting the faces u, d, I, c etc. The figure illustrates the repetition, in pairs, of faces inclined to the planes of symmetry, and also the fact that when the face is perpendicular to the plane of symmetry there is but one face, the two halves of which are reciprocal reflexions. It is extremely rare to find crystals so regularly developed as is indicated in diagrams, and the presence of a plane of symmetry, 2, is proved by the equality of angles between faces having similar physical characters and lying on opposite sides of 2. 7. An axis of symmetry is a straight line about which the faces, edges and coigns are regularly disposed, so that if the crystal be turned about it through a definite aliquot part of a complete revolution similar faces, edges and coigns are interchanged ; that is, any particular face or edge in the new position occupies exactly the place of a similar face or edge in the original position. We shall generally express the relation by saying that the like faces, edges and coigns disposed about an axis of symmetry are interchangeable or metastrophic 1 . to turn round, to change. AXES OF SYMMETRY. 19 8. If the least angle of rotation about an axis be 180, the faces, edges and coigns are associated in pairs, and it is clear that a second rotation brings the crystal back into its original position. Such an axis of symmetry will be called a dyad axis. The line through the centre perpendicular to the paper in Fig. 19 is a dyad axis ; also the lines joining the middle points of the opposite edges of the regular tetrahedron, Fig. 17. If the least angle of rotation giving interchangeability of like faces etc. be 120, then the faces, edges and coigns, occur in sets of three, i.e. in triplets or triads. The axis will be called a triad axis. The vertical line through the two apices of Fig. 11, and parallel to the faces m, is such an axis. The faces r, z, s and x, occur at each end in sets of three which are interchangeable on rotation about the vertical axis through 120. When the least angle which gives interchangeability is 90, the rotation can be effected four times before the crystal returns to its original position. The axis is in this case a tetrad axis. The vertical lines through the apices in Figs. 8 and 9 are tetrad axes. The only remaining axis of symmetry occurring in crystals is one of six-fold symmetry the hexad axis. The least angle of rotation is 60, and a crystal face or edge can occupy in succession six different positions. Such an axis is observed in well developed crystals of apatite and is shown in Fig. 20, where the hexad axis is parallel to the lines [ma] and perpendicular to the face c. An axis of pentad (five-fold) sym- metry, so common in flowers, is not possible in crystals ; that is, it would give a set of associated crystal-faces in- consistent with the laws which have been found to regulate the positions of faces. Several of these axes of symmetry of the same, or of different, degrees' may occur together in one and the same crystal; and, in fact, an axis of degree n is often associated with a set of n dyad axes all perpendicular to it. 1 The degree of an axis of symmetry indicates the number of like things disposed around it, or the number of times rotation giving interchangeability can be effected before the original position is regained. Thus a triad axis is of degree 3. 22 20 EXAMPLE OF SYMMETRY. 9. For the purpose of illustration the following table of angles on the crystal of barytes, Fig. 19, is given; we shall find the angles useful at a later stage. raX 22 10 \m 17 1 mk 28 34 kb 22 15 bk' 22 15 k'm' 28 34 m'\' 17 1 .X'o' 22 10 ~az 45 41' zy 18 18 (iv) yo oy> 26 1 26 1 y'z> 18 18 _z'a' 45 41 -aw ud a te el' I'd' d'u' ' u'a' 31 50' 19 19 16 55 21 56 21 56 16 55 19 19 31 50 w cr" r"z" _z"m" bz 55 18' 34 42 ' 34 42 z'"b' 55 18 25 41' 18 13 46 6 46 6 18 13 25 41 The angles in the columns connected by a straight bond are all measured in one zone and this method of giving the angles between a number of tautozonal faces will be frequently used throughout the book. For the sake of brevity 1 , the equal angles between the pairs of faces parallel to those shown have been omitted. The symmetry manifested in the drawing, and inferred from inspection of the crystal with respect to the central planes parallel to a and b, is now confirmed by the equality of the angles on opposite sides of these planes. Measurement of one zone, such as (i), is not enough to establish this symmetry, for in a centro-symmetrical crystal this zone will only show that there is at least one plane of symmetry parallel to a or to b, and one dyad axis perpendicular to the plane. But the equality of angles between corresponding pairs of faces in zones (i), (ii) and (iv) is only consistent with a plane of symmetry parallel to face a. Again the equality of angles in zones (i) and (v) entails a plane of symmetry parallel to b and at right angles to the former one. We shall see later on that (1) the lines of intersection of these two planes and (2) the perpendiculars 1 In actual measurement of a zone the student should always measure every angle and go completely round until he reaches the face from which he started. The second reading on this face should be recorded, for it will serve to control the accuracy of the angles. Any accidental displacement of the crystal during the observations will probably be indicated by a defect in ad- justment, and also by a considerable divergence from the first reading. FORM. 21 on them from the centre are axes of dyad symmetry. That the line through the centre perpendicular to the paper is an axis of dyad symmetry is clear from the figure, and is confirmed by the equality of angles on opposite sides of it in zones (ii) and (iii). As the crystal has parallel faces the plane of the paper occupies the position of a plane of symmetry. The student should bear in mind that the two halves into which a crystal is divided by a plane of symmetry cannot, as a rule, be brought into the same position. They are related to one another in a manner similar to that of the right and left hands, and may be said to be antistrophic 1 . Similarly the corresponding faces and edges may be said to be antistrophic. 10. Planes and axes of symmetry have been found to satisfy the following conditions: (1) They must be parallel to possible faces, or zone-axes, of the crystal. (2) A plane of symmetry must be perpendicular to a possible zone-axis ; and an axis of symmetry perpendicular to a possible face of the crystal. (3) When there are several planes of symmetry in a crystal the angles between any pair of them must be one of the four 90, 60, 45, or 30. These angles owing to their importance in determining the system, or type of symmetry, of a crystal have been called the crystallometric angles. (4) The angles of rotation possible about an axis of symmetry are 180, 120, 90, or 60. This may be stated otherwise : that axes of symmetry can only have the degrees, 2, 3, 4, or 6. The above and other relations between the elements of symmetry using this phrase to denote generally planes, axes and centre of symmetry form the subject of Chapter ix. The first relation has been found to be true in all the crystals which have been observed ; but in the case of a triad axis it is not capable of deduction from the purely geometric relations resulting from the rationality of indices. 11. The set of like faces, edges and coigns, which are connected together by the symmetry of the crystal and which have like physical characters, will be called homologous faces, edges and coigns. The simple figure, whether closed or not, which consists of one set of homologous faces will be called a form. Many of the figures so far given, such as Figs. 2, 8, 17, represent single forms, whilst 1 avTiffrptyfiv to turn to the other side. 22 FOKM. that of quartz, Fig. 11, represents five separate forms, whose faces are denoted by the letters m, r, z, s and x. The same letter will be used to denote the several faces of a form, and, as a rule, dashes or numerals will be used to distinguish the one from the other. The dashes of parallel faces will be the same but placed at the top and the bottom respectively. Occasionally some of the faces are left un- labelled (as has been done in Fig. 13) when no confusion is likely to arise from doing so, and this is more especially the case where very many like faces occur together. If the faces of forms on a crystal are approximately equidistant from a point within it, the faces belonging to the same form are similar in size and shape ; and each form is said to be equably developed. Any displacement, outwards or inwards, of one or more of the faces will clearly disturb this regularity of development. In the pairs of figures, Figs. 5 and 6 and Figs. 8 and 9, we see that the irregular figures are deduced from the regular ones by the displacement of some of the faces whilst they retain the constancy of their inclinations to the adjacent planes and therefore the parallelism of the edges. The new positions of these displaced faces are respectively parallel to those they would occupy in equably developed crystals. In theoretic discussions we consider the forms to be equably developed. The crystal-figures given in books, as also crystal-models, are, for the most part, such ideal representations of natural crystals. CHAPTER IV. AXIAL REPRESENTATION. 1. THE relative positions of faces on a crystal can be given by the axial method of geometry. For this purpose three planes parallel to three faces of the crystal which are inclined to one another at any finite angles, but are not in one and the same zone are taken as axial planes. The lines in which they intersect are the axes of reference, and are sometimes called the crystallo- yraphic axes or, shortly, the axes. The axes are supposed to meet at a point within the crystal, and this point is called the oi-igin. It is generally denoted by the letter 0, and the axes by the letters OX, OY and OZ. The projection on the plane of the paper, or perspective view, of such a set of axes is shown in Fig. 21. From what precedes, the axes are necessarily parallel to three edges, or zone-axes, which do not lie in one and the same plane. When they are shown in diagrams they will usually be given by interrupted lines consisting of strokes and dots. 2. A fourth face of the crystal, taken at any arbitrary distance from the origin and meeting all three axes at finite distances, cuts off from them respectively lengths, OA, OB and OC, which depend on the actual position of the face. But the ratios of these lengths for all planes parallel to the face are constant, and depend only on the angular relations of the fourth and the axial planes to one another. Furthermore the angles between the similarly placed faces on crystals of the same substance are constant (Chap. II. Art. 4). Hence the angles between the axes and the ratios OA : OB : OC are constant for all crystals of the same substance, 24- PARAMETERS. provided the faces selected for this purpose be similarly placed on the several crystals, or be (as we shall, for the sake of brevity, say) the same faces. In theoretical and general discussions the three lengths OA, OB and OC on the axes are denoted by a, b and c respectively, and are called the parameters of the crystal. In systematic descriptive treatises on minerals and chemical com- pounds, one parameter, usually b, is given as unity, and the other two are then the numbers which give the fixed ratios a : b and c : b. These ratios are determined by trigonometrical formulae, which will be given under the different systems, from the angular relations between the fourth (parametral) plane and the three axial planes to one another. The angles between these several planes are in many cases determined by direct measurement, or are calculated from angles measured between other faces by methods depending on the laws of crystallography. 3. In crystals which manifest no symmetry, or only centro- symmetry, planes parallel to any three faces of the crystal, which are not in a zone, serve equally well as axial planes. Such crystals are said to belong to the anorthic, or triclinic, system. In other classes of crystals the faces of the crystals are arranged in sets of homologous faces (forms) in a manner dependent on the elements of symmetry present. It is, therefore, convenient to select the axial planes in a manner which accords with the symmetry, and which enables us to represent the set of faces constituting a form in the simplest manner. 4. In order, however, to be able to represent planes on all sides of the crystal, and lying therefore in any one of the eight segments commonly called octants into which space is divided by the axial planes, we must adopt the further convention of analytical geometry, that lengths measured along OX are positive, and those along OX t are negative. Similarly lengths along OY and OZ are positive, whilst those along Y t and OZ t in the opposite directions are negative. The combinations of these various signs of the intercepts enable us to represent a plane lying in any one of the eight octants. Thus a plane in the upper left-hand octant to the front would in Fig. 21 have the intercepts on OX and OZ both positive, that on OY / negative. 5. If, as is frequently the case, several planes be so situated that they meet the axes on the same side of the origin, one of INDICES. 25 them, as ABC in Fig. 22 can be selected to give the three para- meters a, b and c. And any other, such as HKL in the same figure, can be represented as intercepting lengths, OH= a -=- h, OK- b + k and OL = c -e- 1, on the axes OX, (97 and 0^ respectively. The numbers h, , Z depend on the relative position of the new face HKL to the axes and the parametral plane ABC. They are called the indices of the face, and clearly fix its position when the positions of the axes and the ratios a : b, c 6. If the intercepts (a^-h, b -H k, c + FIG. 22. 6, are all known. on the axes are multiplied by any common factor r, i.e. are increased or diminished in any common ratio, the plane is simply displaced parallel to itself. Let HKL, Fig. 23, represent a face making intercepts, a -=- h, b -4- k, c + l, on the axes ; and let Then OH' : OH = OK' : OK; and by Euclid vi. Prop. 4 the line H'K' is parallel to the line UK. Similarly, OK' : OK = OL' : OL ; and the line K'L' is parallel to KL. Hence, the two planes H'K'L' and HKL have two lines of the one parallel to two lines of the other. They are there- fore parallel planes (Euclid xi. Prop. 15). FIG. 23. The face can be equally well represented by a -4- A, b -=- c + l; or by ran- A, rb -=- k, rc + l; since with its actual distance from the origin. Hence parallel faces on the crystal, which are, therefore, on opposite sides of the origin, can be deduced the one from the other by multiplying the intercepts, or the indices, by 1, i.e. by a simple change of signs of all the intercepts or indices. This is directly obvious if a model of the axes, similar to Fig. 24, be made in cane or wire, and the positions of the two planes be indicated by lines of Berlin wool. For we are not concerned 26 RATIONAL INDICES. 7. Suppose a face to be parallel to one of the axes, OZ (say), and therefore to meet this axis at an infinite distance. Such a plane can be represented as meeting the axes at ma, nb, oo c, or at a -=- h, b + k,c + Q, according as we take multiples, or submultiples, of the parameters to give the intercepts. The symbol oo represents an infinitely large number, which can also be expressed as 1 -H 0. 8. We have already pointed out that, in the case of planes meeting the axes in octants other than the first, the signs of the intercepts must correspond with the directions of the axes forming the edges of the octant. Thus a face passing through HKL of Fig. 24 makes on the axes the intercepts, a -f- h, - b H- k, c H- I. A face passing through HKL i has the intercepts a -=- h, - b H- k, -c-~l. The face through HKL the intercepts -a + h, b + k, c + l, and the face HKL i the intercepts, a -f- h, b -4- k, c -H I. In the two remain- ing octants we can have planes HKL given by - a + h, -b + k, c^-l; and HKL I given by - a -r h, - b -H k, - c H- I. The law of rational indices. 9. We are now in a position to enunciate this law, which is the same as that of decrements discovered by Haiiy. If a crystal be referred to three axes parallel to three edges of the crystal, which do not all lie in one plane; and if a face be taken which meets all the axes at finite distances a, b and c, respectively; then the position of all other faces possible on the crystal can be given by intercepts, a^-h, b -f- k, c + 1, where h, k and I, are commensurable (rational) numbers, positive or negative. Amongst these commensurable numbers zero has to be included. The three numbers are called the indices of the face. 10. A plane's position, given by the ratios 3a -=- 4 : b -f- 2 : 3c, can equally well be given bya-f-4:6-^6:c-=-l; the latter being obtained by dividing each term of the first set by 3. The latter representation is convenient for the following reason. Whewell proposed, and Miller brought into use, the convention that the parameters a, b, c should always occupy the same order, and that the ratios giving the intercepts should be so arranged that they consist of these parameters divided by integers which may be positive or negative, or one or two of them but not all three may be zero. The notation can then be simplified by omitting the parameters and writing only the integers which serve as divisors. Care must be taken in writing these latter that they are given in the FACE-SYMBOL. 27 order in which they refer to the axes of X, Y and Z, respectively. The sign must also be placed over the number when the intercept is negative. This position of the minus sign is not essential, but it has the advantage of condensation and avoids all risk of ambiguity. When the intercepts have been arranged in the manner required by the convention, so that for the intercepts a -f- /i, 6 -r k, c -f- 1, the numbers h, k and I are integers, the face can be represented by the symbol (hkl). A simple curved bracket is used to inclose the numbers if there is any likelihood of their being confused with other sets of integers. When letters are used to indicate numbers in general, they may be essentially negative although this is not shown. If, however, the same letters apply to planes meeting the axes on opposite sides of the origin then the sign must be expressed. Thus (hkl) is the symbol of the face parallel to (hkl). When particular numbers such as 1, 2, 3 90. The same holds for the other axes, and therefore equations (1) give the position of all possible normals. 19. Equation (4), which is all that is left of (1), when the normal is perpendicular to the axial plane YOZ, i.e. to the face (100), serves only to indicate that the normal to (100) makes an angle less than 90 with OX. This normal we shall denote by OA. The normal OA t to (100) must make an angle greater than 90 with OX: for OA and OA t are in the same straight line, being both perpendicular to the plane YOZ. The exact value of XOA is obtained from the general expression mentioned in Art. 16 when YOA and ZOA are made 90 and therefore cos AY = cos AZ = 0. The same reasoning applies to the normals OB (010) and OC (001). The normals OA, OB, OC do not coincide with the axes OX, OY and OZ, except when the axes are at right angles to one another, and the student should therefore be careful to avoid confusing the two sets of lines. CHAPTER V. ZONE-INDICES AND RELATIONS OF ZONES. 1. IN the preceding Chapters, we have remarked that crystal- faces are arranged in zones ; it is clear that the direction of an edge belonging to a zone, and therefore of the zone-axis, is known if the positions of the two faces meeting in the edge be given. It is also clear that a limitation is placed on the position of a plane when it has to pass through a known line, or, as it may be expressed crystallographically, has to lie in a known zone. This limitation gives, as we shall see in Art. 8, a simple relation between the indices of a face and the indices of a zone. 2. As a simple case of the intersection of two faces, let, in Fig. 28, the face ABC (111) intersect the face ELM (012) in the line EM. The two points B and M y common to both faces, are easily found, and the line is then known. But the line of intersection of two faces cannot always be so easily found. The general method of finding the direction of a zone-axis, OT, is to construct with edges along the axes a parallelepiped of which OT is the diagonal. This involves the determination of the lengths of the edges of the parallelepiped. We shall illustrate the method by finding the edges of the parallelepipeds OGMLB and OCANT, of which EM and the zone-axis OT are the diagonals. The first face meets the axes at A, B and C, where OA = a, OB = b, OC=c. The second meets OY at B, OZ at L, where OL = c + 2; it also meets the plane XOZ, or AOZ, in the line LM parallel to OA. But the line AC is also in the plane XOZ. Hence LUfsmd AC meet at M. The edges of the parallelepiped OGMLB, L. c. 3 34 ZONE-INDICES. of which BM is the diagonal, can be now found. The edge OG = ML, is obtained by Euclid vi. 4 from the similar triangles MLC, AOC. For ML : AO = LC : OC = 1 : 2; .'. OG = ML=a + 2. The edges of the parallelepiped are therefore a ~ 2, 6, c -=- 2. The same direction is given if the edges of the parallelepiped are all doubled. The line BM is then also doubled in length but not altered in direction. The direction is also given by any similar parallelepiped having its edges parallel to and in the same ratios as those of the original one, provided the diagonal joins corresponding corners. The required zone-axis OT, parallel to the edge BM, is the diagonal through of the parallelepiped having edges : OA = a, OB t = -20B = -2b,OC = c. The numbers multiplying a, b, and c, are called the indices of the zone-axis, or shortly the zone-indices, and OT is represented by the symbol [121]. 3. Let, in Fig. 29, the two faces (Aj&A), (hjcj,^ meet the axes at the points 7/j, K ly L l and 7/ 2 , K z , L 2 , respectively. It is required Fia. 29. to find the zone-axis through the origin parallel to the line of inter- section PQR. FIG. 30. The actual position of the face, on which its dimensions depend, is immaterial, provided the angles it makes with the axial and other planes remain constant. If then a face be shifted, remaining parallel to its original position, the line of intersection with an axial plane remains parallel to the original line in which the planes met. ZONE-INDICES. 35 Suppose the face (A^J,) to be shifted parallel to itself until it passes through the origin, and let its position be then given by the plane OK^ of Fig. 30. The face in its new position may l>e denoted as an origin-plane of the crystal. The lines 0/c n OXj are parallel to H^ and H^ of Fig. 29 ; for, by Euclid XL 16, parallel planes intersect a third plane in parallel lines. In Fig. 30 construct, with lengths OE, OF, 06 on the axes as edges, the parallelepiped OEM FT; and let OE = au = a (k^ - Ifa), OF=bv=b (l^-h^, OG = cw = c (hjc^ - kjiy). If the symbols of the two faces are known, the differences of the products of the indices given above are easily calculated, and the lengths OE, OF and OG are then known. Let now the lines OK I} 0^ meet the edges EM and ED in K n A.J respectively. Join K^, and produce it to meet MT. The line KjXj lies in the plane DEMT of the parallelepiped, and is the intersection of this plane and the origin-plane parallel to H 1 K 1 L 1 . If then KjXj can be shown to pass through T, the origin-plane OK^ passes through the diagonal OT. Now the triangle OE^ in Fig. 30 is similar to 0/7, j^ in Fig. 29, and OE Kl to II^OK^. Hence, by Euclid vi. 4, EX l 0Z, c A! A! EK, OK, b A, , A, E5=OH\' .*. = - 5 ^ ^ ......... (2). When the values of OH l , OK l , OL l are introduced, attention must be paid to the directions in which the lines are measured. From the figures, it is clear that EX l and OL^ are measured in opposite directions; as are also EK^ and OK^. Hence the intro- duction of the negative signs in equations (1) and (2). If we denote the directions of measurement by the order in which the letters are written, we can regard jEW, as negative, and ^E as the same length measured from KJ in the positive direction. We express this shortly by the equation K^E^E^. Hence Kl E = bu^ ......................... (2*). A 1 , Suppose the line K^, to meet the edge MT in a point j, the point S of Fig. 30. Then from the similar triangles ^EX^ x l MS l we have 32 ZONE-INDICES. Here the lengths on parallel lines are measured in the same direction and the two ratios have the same sign. Now - i -~ = 77j^ = j-f, and no change of sign is needed ... (4). Kj/V \JJ\.-^ 06] '*=w -<> But But (6). (7). . g M=--l w = -cw (8) For the edge MT was made equal to cw. Hence Sj^ coincides with T, and the plane OxjXj contains the diagonal OT\ To prove that the origin-plane 0*c 2 \2, parallel to the face (h.jtj^), contains the diagonal OT, we proceed in exactly the same way, and 1 The student will find it advantageous to get a tinman to join together, at right angles to one another, three pieces of tube, and to fit tightly into these tubes sticks of bamboo. These will represent the three axes, and the jointed tubes may be called axial brackets. Eight sets of them will be needed to form a parallelepiped. By fastening strings of differently coloured wool or thread to the sticks, models can be made to represent the diagrams in this and the following Chapters. Or a set of axes may be made by sticking rusty knitting- needles into corks. If the needles are smooth the threads slip. The axes in the text are supposed to make any arbitrary angles with one another, but it is difficult to get the sets made accurately at angles which are not right angles. ZONE-INDICES. 37 obtain exactly similar equations. These equations can be deduced from those given for the first face by changing the suffix 1 and replacing it by 2. Lengths of lines and ratios exactly similar to those referring to (hjch) are obtained. The sole difference is that h. 2 , & 2 > 4 take the place of h^ k l} ^ respectively. Employing letters with suffixes 2 to denote the corresponding points, and starred reference-numbers to denote corresponding equations, the final re- lations are : SJf_ \,E_ck, ~K 2 M-^E~'U 2 ' and K .,M = K 2 E + EM=bu J ~ + bv .................. (6*); K. 2 .: S 2 M= c ^(h 2 u + k 2 v) .................. (7*). But h. 2 u + k. 2 v .'. MS 2 = j-l. 2 w = cw = MT ................. (9*). Hence the origin-plane 0K 2 A 2 , parallel to the face (h.Jc 2 l z ), contains the diagonal OT. 4. The direction of the zone-axis of the two faces (A^Z,), (h. 2 k.J. 2 ), is therefore given by the following construction. With the origin as vertex, and lengths au, bv, cw along the axes as edges, construct a parallelepiped. Then the diagonal through the origin is the direction required. The numbers u, V, W are called the zone-indices, and are obtained by the following rule : *, VxV ..(10). Aj Each of the numbers u, V, W, is the difference of the products of the face-indices joined by the arms of the cross below it. It is only necessary to subtract in each case the product of the pair joined by the faint arm from that of the pair joined by the thick arm to secure that the signs are correct. The values of the zone-indices may be either positive or negative, and one or two, but not all three, may in particular cases be zero. Thus in the example given in Art. 2 the second index, referring to the axis of Y, was negative. 38 ZONE-SYMBOL. 5. Again, if in (10) the order of the two faces be altered the signs of all the zone-indices are changed ; thus u' v' w' A 2 & 2 ^ (10*), v' = IA - h.A = - (ifa - h&) = - r, w' = h& - h.k, - (AA - &A) = -w. The parallelepiped is clearly constructed in the opposite octant, but with equal edges. The diagonal is the same straight line produced through the origin. It follows that the direction of the line of intersection of the faces is independent of the order in which the faces are taken when the parallelepiped is constructed; as is also obvious from the geometry. 6. The numbers u, K, W, are necessarily integers, for the indices of the faces are integral. They are employed as multipliers of a, b and c respectively; and as the parameters are supposed to be known, it suffices to give u, V, W to enable us to construct the parallelepiped. They must be carefully distinguished from face-indices, which are divisors of a, b, and c. For this purpose it is usual to enclose them in crotclwts [ ]. Hence the above zone-axis is denoted by the symbol [uvw]. When the zone-indices have not been determined, the zone is often denoted by enclosing the letters denoting two or more of the faces in crotchets, and sometimes their symbols are introduced. Thus if P, be (A^i) and P 2 be (h), the zone-symbol may be given by [P.P,], or by [AM,_W], or by [uvw]. The zone- symbol of the example in Art. 2 is [121]. The distinction between face-indices and zone-indices must be carefully kept in mind. The points on the axes given by the former are points in the face, and the three lines joining them are three lines in the face. The points on the axes given by the zone- indices fix the edges of the parallelepiped, the diagonal of which from the origin is the direction of the edges of intersection of the tautozonal 1 faces. Thus (321) indicates the face meeting the axes respectively at distances a H- 3, 6-2, and c, from the origin. A parallel plane is obtained by taking points at six times the distance from the origin, i.e. with intercepts 2a, 36, and 6c. The lengths 1 From ratJT& "the same." WEISS S ZONE-LAW. 39 given by [321], are 3a, 26, c respectively, which form the edges of a parallelepiped, the diagonal of which from the origin is the zone-axis. And although, in consequence of the equations established in Chap. iv. Art. 15, we shall also use the face-indices to denote the face-normal, equations (1) of that article are very different from the relations of the diagonal to the edges of a parallelepiped. In the Chapter in which the relations of faces and zone-axes are treated analytically, it will be shown that, except in special cases and in the cubic system, a zone-axis cannot be perpendicular to a possible face. 7. If the zone-axis lies in one of the axial planes, XOY say, then clearly M T = cw = Q. Hence w = 0, and the symbol of such a zone-axis is [uvQ]. The axis is the diagonal OT of the parallelo- gram OEFT, Fig. 31, the edges of which are OE = au, OF = bv. If v becomes zero at the same time as w, then the length by of the side of the parallelogram mea- sured along OY becomes zero, and ET also becomes zero. The point T then coincides with E, and the zone-axis [(/OO] is the line OE, i.e. the axis OX. In this case it is immaterial what length is taken on OX, and the zone- axis may equally well be given by the symbol [100]. If, however, the second index to become zero is u, then OE and the parallel side TF become zero, and the zone-axis [QvQ] or [010] is the axis OY. Similarly a zone-axis lying in the plane XOZ is [wOlv], and the axis OZ is [OQw] or [001]. If the zone-axis lies in YOZ, its symbol is [Qvw]. Weiss's zone-law. 8. Suppose a third face (hkl) not parallel to either of the first pair to lie in the same zone with (/h&A), (M^a)- When it is shifted parallel to itself so as to pass through 0, it must contain the zone axis OT. It is required to find the relation between the symbols which corresponds to this geometrical fact. We may, as before, represent the line in which the transposed face meets the plane TME by T\K (using no suffixes). We shall clearly obtain ratios similar to those already given in (5), (5*), ' also KM = K E + EM=bu^ + bv ............... (6**), hu + kv) ......... (7**). But TM= - MT=-cW by construction, hence hu + kv + lw = ........................ (11). This important relation which connects the indices of any face in a zone with the zone-indices, as determined from any known pair of faces, we shall call Weiss' s zone-law. For the distinguished crystallographer, C. S. Weiss, was the first to call attention to the relations discussed in this Chapter. This he did in his trans- lation into German (1804-6) of Hatty's Mineraloyie. Weiss used the intercepts as the symbol of a face e.g. 5 : - : c so that his o 2i expression for the law was not in the form given in (11). His analysis of the relations is given in a memoir published in the Abh. d. Berlin Akad., 1820-1. The equation just given may be written out in full as follows : h (kj, - I,k 2 ) + k (1& - hj,) + I (hfa - kji. 2 ) = ...... (11*). This may also be rearranged, so as to make the indices of any one of the faces the factors multiplying expressions of the form of zone. indices. Thus (11*) is equivalent to AS (&! - flfcj) + k. 2 (Ott - hi,) + L_ (hk, - M,) - ...... (1 1**). It may also be given in the determinant form : h k I = To a student familiar with mathematics this latter form is far the most convenient one, for a glance is usually sufficient to show whether the determinant is zero, and whether therefore the three faces, the symbols of which have been introduced, are tauto- zonal. It also enables him to prove that the indices of any face (hkl) in the zone can be given by the following equations : h mhi + nh. 2 \ k = mk 1 + nk z I (13); I -. mli + ?i/ 2 j EQUIVALENT ZONE-SYMBOLS. 41 where m and n are whole numbers. The student can easily see that the values given by (13) satisfy the condition by introducing them into (11**). 9. It is clear from the analysis that any two non-parallel faces in the zone can be employed to determine the zone-symbol ; and like- wise that, when the zone-symbol is once known, any other face whatever, which lies in the zone, must have indices which satisfy the equations given in (11) or (12). Thus we can show that the following faces are tautozonal : (321), (112), (433), (2lT), (103). From any pair (say the first) we find 1 tf 12 = 3, V 12 = -5, W 12 - 1. The equation, which a face (hkl) in this zone must satisfy, is 3h-5k + l = Q (14). It is easily seen that the indices of the faces (433), (211), (103) will, when substituted for h, k, I in equation (14), reduce the left side to zero. The faces are therefore tautozonal. But attention should be directed to the fact that different pairs of faces will not give exactly the same values for u, V, W, but only values in the same ratio. Thus, if the third and fourth faces (433) and (211) are taken, we obtain (/34=- 6, ^34= 10, Ws4 = - 2. But f^ = ^ = ^ = _2... ...(15). "is ^a *M The variation in the values of u, V, W, means no more than that the size of the parallelepiped is altered, and that the length of OT given by one pair of faces is a multiple, or submultiple, of the length given by another pair; and likewise that it may be taken, as in the example just given, on the other side of the origin. We have already seen in Art. 5 that the signs of the indices, and the direction of OT as viewed from the origin, are changed by reversing the order in which the two faces are taken in the table. 10. Let [U 13 K 12 W K ], and [1/34 V^ W M ] be the zone-symbols of the same zone deduced from two different pairs of non-parallel faces in the zone, it is required to prove that = = (16). U 84 KM W u '" 1 When there are several faces in a zone, it is convenient to indicate the faces taken to give the zone-indices by attaching suffixes to the zone-indices. Thus f/u, &c., indicate that the indices of the first and second face are introduced in this order in the table (10) of Art. 4. 42 EQUIVALENT ZONE-SYMBOLS. Now the pairs of faces in a zone must meet in lines all parallel to the zone-axis. Hence the direction of the diagonal of the parallelepiped which gives the zone-axis cannot depend on the pair of faces selected. But the only variables are the lengths of the edges of the parallelepipeds, and these remain proportional to one another if the diagonals lie in the same straight line. The directions in which the edges are measured from the origin may, in some cases, be on opposite sides of it, but it will suffice to prove the proposition for the case in which they lie, as in Fig. 32, on the same side of it ; for by taking the second pair in the reverse order, we reverse the directions of the edges of the parallelepiped (Art. 5). Let OETF be the parallelepiped obtained by means of (AM), (hjc 2 l. 2 \ and OE'T'F' that obtained by another pair of faces in the same zone (AaVs), (AM)- Then OT and OT' are co- linear. Also OE=au n , OF=by l2 , MT=OG=cw n ; FIG. 32. But if a plane be drawn through OG and OTT', it will contain the edges MT, M'T' parallel to OG. The points M and M' must therefore lie on a straight line through 0. Hence we have the similar triangles OTM, OT'M', and by Euclid vi. 4, MT _ OM M'T'~OM'' Again, from the similar triangles OEM, OE'M' we have OE EM OM MT Hence, introducing the values of the edges, we have And .(16). The three ratios given in (16) are each equal to where h, k, I, are any three numbers whatever. If they be com- mensurable they may be the indices of a face. If this face do not lie in the zone [f/ 12 Ki 2 jf 12 ], the ratio has some definite commensurable value. It is clearly useless to take for A, k, I, the indices of a face in the zone, for the numerator and denominator then both become zero by Weiss's law, Art. 8. That the terms in (16) are equal to that given in (17) is proved as follows. FACE COMMON TO TWO ZONES. 43 Let each term iu (16) = r ; Adding together the left sides, as also the right sides, we have Af/i2 + ^Ki2+ ^12 = r (^34 + ^34 + ^34)- Hence r has the value given in (17). Face common to two zoties. 11. We can now show that a plane which contains two zone- axes is parallel to a possible face of the crystal, and that the intercepts this face makes on the axes are commensurable sub- multiples of the parameters. The face is clearly common to the two zones of which the indices are given. Let the two zones have the symbols [^KI^], [w 2 K 2 >V 2 ]; then, from equation (11), the face (hkl) lies in [w^M^] when hu l + kv, + lw, = (18), and in order that it should also lie in [W 2 K 2 W 2 ] we must have If (18) be multiplied throughout by W 2 and (19) throughout by Wi, and if the one equation be then subtracted from the other, we Similarly, by multiplying (18) and (19) by K 2 and V 1 respectively, and then subtracting, we have Hence, transforming the equations (20) and (21), we find l/U/ U/IS U/ // II LU II V V II *""\ /" Now tfj, K,, fec., are all commensurable numbers and, if derived by table (10) from face-indices, they are integers. Hence A, k, I, are also commensurable numbers and may be expressed by integers. The face (hkl), having commensurable indices, is therefore a possible face of the crystal. The relation between the indices /t, k, I of the face common to 44 FACE COMMON TO TWO ZONES. two zones and their zone-indices, given by equations (22), can be expressed by a table (23) similar in all respects to table (10). Thus h k I ! W l UT. XXX 2 W 2 W 2 The indices of a face common to two zones, viz. h= V^W^- M^, k=w 1 U 2 -U 1 W 2 , l-U^-V^, are obtained in a manner exactly similar to that by which zone-indices were by (10) obtained from the indices of two faces in the zone. The same precaution must be taken in both cases, and the product of the indices joined by a faint arm must be subtracted from the product of those joined by the thick arm. As an example let us find the face (hkl) common to [351] and to [102]. h k I By table (23) 5135 XXX 0210 .. h= -10, & = l-6= -5, 1=5, and rejecting the common measure 5, the face has the symbol (2ll). Had we however taken the zone [102] first, we should have found 7t=10, k=5 and 1= - 5, and the face to be (2ll). These two faces are parallel, and the first is necessarily common to every zone to which the second belongs. 12. The equation hu + kv + lw = 0, involving as it does the face- indices A, k, I, and the zone-indices u, V, W, in precisely the same manner, serves equally to connect: (1) all the faces lying in the zone [uvw], and (2) all the zones to which a particular face (hkl) is common. 13. By the aid of Fig. 33, similar in some respects to Fig. 30, and the expressions given in Art. 3 with reference to the lines in the latter figure, we can give a direct proof of the proposition that a plane parallel to two zone-axes is a possible face. Let C^be the zone-axis [uvw], and OST, the zone-axis [u^Wil and let TO K \T t be the origin-plane determined by them. We can find the ratios EK : HO, and EX : EO ; and can then, by a comparison with Fig. 29, find the intercepts on the axes of a parallel plane which will give the indices. From the similar triangles EK\, KMT we have E*_ = M*.__KM__ where u, V and w, are integers. Furthermore, every face common to two zones is given by j , T , j , where h, k and I are integers deduced from the zone-indices by the rule given in table (23). 15. Example. We may now apply the propositions in Chaps, iv. and v. to the solution, as far as the principles already established permit, of the crystal of barytes shown in Fig. 34. The lines of intersection of planes, parallel to a, b and c in the figure, drawn through a central point are taken as axes, so that face a is parallel to OY and OZ, face b to OZ and OX, and face c to OX and OY, These latter axes are shown in Fig. 37 which gives the faces m and c only. The indices of these faces are therefore at once known a is (100), a' (100) ; b (010), b' (010) ; c (001) and the parallel face below the paper will be c' (001). The face z is taken as parametral face, and its symbol is (111). The homologous faces are symmetrically placed with respect to the axial planes and axes. Hence z and z' meet in a line in the plane YOZ, and their intercepts on the axes Y and OZ must be the same. The axis OX is met by z at a distance a on the positive side, by z' at an equal distance on the negative side. Hence z' meets the axes at distances, - , b, and FIG. 34. EXAMPLE (BARYTES). 47 c, respectively, and its symbol is (111). Similarly, the symbol of z'" can be shown to be (ill), and that of z" to be (III). Again, if we note that the axis OZ is a dyad axis, and that therefore the pair of faces z and z", and also the pair z', z"', interchange places when the crystal is turned through 180 about this axis, we perceive the fact that the symbols of pairs of faces, interchangeable by rotation about a dyad axis per- pendicular to a pair of axes of reference, differ in having the signs of the indices referring to these axes both changed. This is clearly necessary, for the rotation about OZ interchanges, on each line perpendicular to it, equal lengths measured on opposite sides of the origin. We can now determine the symbols of the homologous faces of the forms denoted by the letters m, u and o. The faces m lie in the zone containing the faces a and b which may be represented by [ab] = [100, 010] = [001] the latter symbol being determined by table (10). Hence, by (11) of Art. 8, the last index of each face m is zero. This is obvious from the geometry, for the faces m, m', &c., are all parallel to OZ. But m and m" lie in the zone [c^] = [001, 111] = (by table (10)) [IlO]. Hence, if m be (likQ), we have from equation (11) - h + k = 0. This equation is satisfied by making h=l and fc=l; or by making h = l and k = l. The former value gives for m the symbol (110), for this face meets both the axes of X and Y on the positive sides of the origin. The second set of values gives (IlO) for the parallel face m", which meets the axes on the negative sides of the origin. Similarly, m' and m'" lie in [cz'] = [110], and their indices (hkO) satisfy the equation, h + k=0. Hence m' is (IlO), and m'" (IlO). The face u is determined in a similar manner from the two zones [ac] = [010], and [fcz] = [101]. From the first it follows that the middle index is zero, and the face can be represented by (/(O/). The second zone proves that, - h + 1=0, which is satisfied by making h = l = l, and the symbol (101). Or we may use table (23), thus 1001 XXX 01101. Hence u is (101). Similarly, from [ca] and [bz r ], u' may be found to be (101). From the zones [be] = [100], and [as] = [Oil], the face o is fixed and its symbol (Oil) determined. The symbol of o' is found, from [be] and [az"] = [011], to be (Oil). The symbols of the remaining faces will be determined in a later Chapter. CHAPTER VI. CRYSTAL-DRAWINGS. 1. CRYSTALS can be represented by models or by crystal-drawings, which give with sufficient accuracy the relative positions of the faces, those of the same form being usually equably developed. Models and drawings of crystals serve mainly to indicate their general habit, and aid us to distinguish the crystals of one substance from those of another : although, as already stated, the habit often varies greatly in crystals of the same substance; and, more especially, when they come from different localities, or are produced in the laboratory under different conditions. In models and drawings, irregularities in the size and shape of faces of a form are only admitted when they are needed to render manifest some peculiarity, either of general habit, or of a particular crystal. Drawings and models fail, however, to give that aid in the determination of symmetry which is afforded by the physical characters of the faces, for homologous faces on a crystal possess the same physical characters and show the same marks, such as striae, pittings, &c., which can only imperfectly be represented in drawings. 2. The crystal-drawings to be found in scientific works are not true perspective drawings, for the parallelism of the edges of intersecting tautozonal faces would be thereby obscured. A crystal is drawn as if all the rays proceeding from the coigns to the eye were parallel ; that is, the eye and crystal are supposed to be infinitely distant from one another. The crystal is, therefore, drawn in much the same way as it would appear, if viewed through a telescope adjusted for a very distant object. The crystal is held in front of the paper, supposed to be placed ver- tically, so that one of its zone-axes usually called the vertical axis lies in the vertical plane through the eye at right angles METHODS OF PROJECTION. 49 to the paper. The lines, or rays, which join the eye to the coigns are all parallel, and either (a) perpendicular to the paper, or (b) in- clined to it at a fixed angle not differing much from 90. In the former case the figures are orthogonal projections, in the latter they will be called clinographic drawings. The points, in which the rays meet the paper, are joined by straight lines in a manner corresponding with the edges of the crystal, and the net-work of lines forms a crystal-drawing. Drawings, which are orthogonal projections, may be divided into two classes : plans and elevations, described in section (i) ; and ortho- graphic drawings, treated of in section (ii). In the former an important zone-axis, which may, or may not, be itself perpendicular to an important face, is placed at right angles to the paper. Strictly speaking, in a plan the vertical axis should be perpendicular to the paper, and in an elevation it should be parallel to the paper. But we shall always speak of diagrams of both kinds as plans ; and shall specify the face which is parallel to the paper, or the zone-axis and faces which are perpendicular to the paper. In orthographic draw- ings the important edges and faces of the crystal are inclined at some arbitrary angles to the paper. In clinographic drawings, treated of in section (iii), the vertical axis is in, or parallel to, the paper, but the crystal has in other respects an arbitrarily selected position with respect to the paper. In this Chapter some knowledge of the relations which hold in the several systems is presupposed, so that the student should read the Articles referring to each system together with the Chapter on that system. As a matter of fact, an accurate drawing of a crystal is only made after the crystal has been completely determined. For working purposes a freehand sketch, which will enable the student to identify each face, is all that is needed. The student should, however, find no difficulty in following the description of the plans given in the next Articles and the example worked out in Art. 8. (i) Plans and Elevations. 3. Figures of this kind are largely used in Brooke and Miller's edition of Phillips's Mineralogy, 1852. The drawings are easily executed, and give a fair idea of the end of the crystal directed towards the observer. They, however, fail to render sensible the L. c. 4 50 ORTHOGONAL PROJECTION. inclination of the faces and edges to the line of sight, and they are deficient in the appearance of solidity. If a very simple crystal, such as a cube, or a tetragonal crystal of apophyllite bounded by three pairs of rectangular faces, be FIG. 35. projected on a plane parallel to one of the faces, it will appear as a square, Fig. 35 '. Similarly a hexagonal prism, like Fig. 3, will, if viewed endwise, be seen as the hexagon, Fig. 36. 4. A simple crystal of barytes, resembling a diamond-shaped lozenge, gives the plan, shown in Fig. 37. This crystal is bounded by four similar faces, perpendicular to the paper and denoted by the lines m. They are parallel to good cleav- ages. The rhombic face marked c is parallel to the paper, and is also parallel to a good cleavage. Two axes, OX and Y, are taken along the diagonals of the rhombus ; the third, 02, is perpendicular to the paper. 5. Fig. 38 shows a ditetragonal prism, terminated by planes parallel to the paper and at right angles to the prism-edges through the points F and K. Such a figure is readily made, if one of the face-angles F or K is known. If one be known the other is also, for F+K=2 f IQ. To make such a diagram, two lines, OX and OY, are drawn at right 1 Figs. 35, 36, 38-44, and some others used in later Chapters, are printed from Miller's original blocks which have been kindly placed at my disposal by Mr. H. Bauerman, F.G.S. Miller placed the axes X, Y, Z, in a different order to that adopted in this book, and his symbols, which have been left undisturbed, differ from those which are given in other diagrams. FIG. 38. TETRAGONAL PLANS. angles, and equal lengths, OK, are marked off on them starting from the origin. Lines KF are then drawn making with the axes angles equal to K -r- 2. Fig. 39 represents a ditetragonal pyramid having eight similar faces meeting at an apex on a tetrad axis, which is taken to be OZ and is perpendicular to the paper. To construct this plan, the angles between the similar edges, labelled L, have to be found, and then the ditetragon is drawn as in Fig. 38. The lines joining the corners to the centre are then drawn. Those labelled K are similar edges, and are interchangeable by rotation through 90 about OZ. The edges F are also similar to one another, i.e. they are interchangeable, and the FIG. 39. angles over them are all equal. The angles over the edges L are all equal to one another. But the angles of one of the above sets are never equal to those of either of the other sets. Two tetragonal pyramids, belonging to the same class of crystals, are shown in Figs. 40 and 41. In Fig. 40 the axes of X and Y are in the directions of the diagonals of the square formed by the edges L; in Fig. 41 they are parallel to the edges so marked. FIG. 40. FIG. 41. Each figure has two distinctly different sets of angles, those over the edges K and L in one, and those over F and L in the other. The angles L of the one pyramid differ from those of the other. In the preceding diagrams the distribution of faces at the back of the crystal is supposed to be exactly like that shown in front, and the paper is supposed to be parallel to a plane of symmetry. It is clear also that the three last figures will be the same whatever be the distance at which the axis OZ is met, for this axis is pro- jected in the central point. The figures fail, therefore, to show whether the pyramids are steeply inclined to the paper or not. 42 52 PLANS OF RHOMBOHEDRAL CRYSTALS. 6. Figs. 4244 show three simple rhombohedral crystals, in which the plane of the paper is perpendicular to the triad axis. The first represents a rhombohedron, and serves equally well FIG. 42. FIG. 43. FIG. 44. whether the rhombohedron is an acute one or an obtuse one, i.e. whether the apices are far apart or the reverse. If we suppose it, however, to represent a crystal of calcite, the symbols on the faces imply that the faces are steeply inclined to the paper. By placing on the faces the symbols on the upper faces of Fig. 4, the plan serves equally well for this obtuse rhombohedron. There is another objection to the plan. The hexagon bounding the figure represents edges which cross the paper and are not parallel to it. The parallel lines in Fig. 42 are, however, all inclined to the paper at the same angle ; and so indeed are all the edges of the figure. Fig. 43 represents a rhombohedron in a different azimuth. Its symbol may be {100}. From the fact that symbols are attached to the sides of the bounding hexagon, we are informed that prism-faces perpendicular to the paper are also present, opposite pairs of which are in a zone with two faces of the rhombohedron and with the faces below the paper parallel to them. All the edges of this figure are also equally inclined to the paper. Fig. 44 may be taken to represent the same rhombohedron but associated with a different hexagonal prism, the faces of which truncate 1 the edges of the prism in Fig. 43. The alternate sides of the hexagon, labelled 21 1, T2T, TI2, in which the faces of the prism meet each only one face of the rhombohedron, are edges parallel to the paper. Each of the remaining three sides represents two edges equally inclined to the paper. Such diagrams serve to show the development at one end of a crystal fairly well, and the method will be employed in Chap. xvi. to illustrate the dissimilar ends of a crystal of tourmaline. 1 A face situated in a zone with two faces and equally inclined to them is said to truncate their edge. EXAMPLE (BARYTES). 53 7. In the oblique system plans, or elevations, on the plane of symmetry are frequently employed by Miller. They are readily made when a few angles in the plane of symmetry and the symbols of the faces are known. Occasionally a plan of an oblique crystal is made on a plane perpendicular to a zone-axis lying in the plane of symmetry. An instance of this kind is given in the plans used, in Chapter XVIIL, in the discussion of twins of orthoclase. Similar plans are also occasionally employed for representing anorthic crystals ; as, e.g. the plan of oligoclase given in Chapter XL The paper is not in these cases parallel to a possible face of the crystal. 8. Example. Fig. 45 affords a good illustration of the use and method of construction of complicated plans. It gives the more important forms on barytes, which are not, however, often found associated together on the same crystal. The paper is parallel to the base c(001). The faces a, X, m, k, b are all perpendicular to the paper, and their projections are given by the lines so labelled. Two Hues are drawn, as in Fig. 37, at right angles for the axes of X and Y. Points B and B, on OY are taken arbi- trarily at any convenient distance, such that OB=OB t = b. From B and B t lines m m, m', &c. are drawn inclined to OY at angles of 39 11' = 78 22' -7- 2. (For this Fl - 45 ' and other angles see Chap. m. Art. 9.) We thus find points A and A f on OX, where OA = OA t = a. The corners at A, B, &c., are then truncated by lines a and b parallel to the axes, which represent faces parallel to them and to OZ. The corners between these lines and the intermediate m's are now modified by lines k and X, where k makes with OX the angle 22 15' and X makes with OYthe angle 22 10'. The length of these lines depends on their distance from the centre. They should be drawn so as fairly to represent the ratios of the breadths of the several faces to each other. The zones [yoz], [y'" o' z"] are then introduced by drawing lines parallel to OY and symmetrically situated on both sides of it. Their distances from one another should approximate to the impression of the width of the faces when the crystal is looked at endwise. The edges [ud], [dl], &c., are drawn in the same way parallel to OY. The edges [uz], [dr~\, \ry\, [co], &c., are drawn parallel to OX; for [zwz'"], [yrd..~], &c., are zones symmetrical with respect to the plane of symmetry parallel to the face b. The edges [7-2], \r'z'~\, c., are parallel to M, m', &c. ; for m, z, r, c are tautozonal. 54 EXAMPLE (BARYTES). So far the symbols of the faces have not been needed. But to get the edges [zd], [ro], and \lr] the indices of the faces are required. In Chap. v. Art. 15, the symbols of the faces were determined as far as they could be by the relations of zones, and in Chap. vn. those of the remaining faces are found by the method explained in that chapter. Hence, we may suppose all the indices to be known. The symbols are : s{lll}, e?(102), {104}, o{011}, y{122}, and r(112). The zone-axis [zd\ is [111, 102] = [211]. Now in the diagram, the axis OZ and every length upon it are projected in the point 0. Hence the parallelepiped with edges 2a, -6, -c, is projected in the parallelogram having sides 2a and b on OX and T. The diagonal through the origin of this parallelogram is the direction of the edge [zd]. The homologous edges are parallel to the diagonals of similar parallelograms. The re- maining edges can be obtained in a similar manner, and the figure completed. The student's attention is directed to the fact that the parameters a and b were obtained by simple construction from the measured angles. The length OB taken on OY was arbitrary, and the line m was then drawn making 39 11' with OY. The point where it met OX gave the point A at distance a on OX. Hence, from Fig. 46, OA + OB = a -=- b = tan ABO = tan A OM, where OM is the normal to the face m. But AOM=3S 11', and, if OB be the unit of length, we have OA = tan 39 11' = -8151. In a similar manner, if the angle bo, or co, be measured, the parameter c can be found. If OQ, Fig. 47, represents the normal to FIG. 46. B Flo. 47. the face o, and if BC is the line in which the face meets the plane containing OF and 02 ; then f\COQ= f\OBC=co. But OC+OB = c + b = tan OBC = tan COQ. By measurement, co = 52 43'. Therefore, when OB = 1, 00 = 1 -31 35. ORTHOGONAL PROJECTION. 55 (ii) Orthographic Drawings. 9. In drawings of this kind the rays are not parallel to a zone- axis, and the paper is not parallel to a possible face. We first project a cube, as explained below. The three projected edges, meeting at any coign, give the directions and lengths of three equal lines at right angles to one another, which can serve as rectangular axes with equal parameters, i.e. for cubic crystals. For equal lengths on all parallel lines are projected as equal lengths ; and unequal lengths on the same, or on parallel, lines in any given ratio to one another are projected as lengths having the same ratio. If the parameters are unequal, one of the edges is left unaltered, preferably the shortest in the projection ; the other two are then lengthened, or shortened, in the ratios of the parameters. When the axes are not rectangular, new lines of unit length must be found making the required angles with one another by one or other of the methods given in Arts. 15 20. We proceed to describe the principle by which Mohs obtained the projection of the edges of a cube. The cube seen with face parallel to the paper is projected- as a square, Fig. 35. If it is then turned about a vertical edge until the face to the right has a width one-third (or one-nth) of that to the left, the cube is seen as two rectangles, Fig. 48. If the angle of rotation about the vertical edge be p, then tan p --- tan CAA n = CA 11 * A A 11 (Fig. 49) = DD 11 ^ CD 11 = 1 H- 3; or generally, 1 +n. In the particular case when n = 3, p = 18 26'. The horizontal faces are still foreshortened into straight lines. To see the upper face, the crystal has to be turned about a hori- zontal axis, which may be supposed to go through the nearest bottom corner of the cube. During this rotation the vertical edge is main- tained in the vertical plane through the eye. Mohs and Haidinger adopted an angle of inclination to the horizon, such that the vertical distance between extreme points on the upper face, as seen on the paper, was one-eighth of the total breadth of the projected cube. 10. Figs. 49 51 are copied from Haidinger's memoir on Mohs' method of drawing crystals (Mem. Wernerian Nat. Hist. Soc. Edin., 1824). The first represents a horizontal plane through 56 MOHS CUBIC AXES. FIG. 49. DCA of Fig. 48 after the first rotation. ABDC is the base of the cube, the plane of projection passes through D n CA n , and the rays through the coigns A, B, D, are A n A, B U B, D n D. Through the corners of the base draw the square D ni A fI , having two sides in the rays AA 11 , DD"; and through A, , D, C, draw lines paral- lel to the sides of D UI A U . The triangles CAA a , ODD 11 , are equal in all respects. Hence, AA" = CD 11 = 3CA ", or, generally, nCA 11 . Let the cube be now tilted about D u A n so as to expose the top to the observer, and suppose the square D UI A U to be carried with the cube. During the rotation the coigns A, B, D, will remain in the vertical planes through A 11 A, B n B, D"D ; and, if the rotation were continued through 90, the square D m A n and the base ABDC would be brought into the plane of projection. If, however, the rotation is stopped when the coign B is projected in B 1 , where "*= D u A"+8 = n + 8, then the square base of the cube is projected in the parallelogram A I B I D I G of Fig. 49. To find the points A 1 and D 1 , suppose BA and BD produced to meet D ll A n , prolonged both ways, in M and N, respectively. We thus obtain two similar right-angled triangles BB U M and AA n M. On the other side of BB U we have a second pair of similar right- angled triangles BB"N and DD IJ N. Now during the rotation the straight lines BAM and BDN remain straight lines, and the pairs of similar triangles must remain similar. Therefore B I A I passes through M, and B*D* through N. Hence, the triangles A f A"M, B'B"M, are similar ; and (Euclid vi. 4), A 1 A 11 : B Z B U = A U M: B U M. Again, from the similar triangles AA U M and BB U M, A A 11 BB 11 = A U M B n M :. A 1 A 11 : B I B IJ ^AA" : BB 11 . But, B'B" = BB"+ 8 ; .'. A 1 A 11 = AA 11 + 8 = 3CA" -= 8. From the pairs of similar triangles BB n N, DD"N ; and D 1 D U N, we get similar proportions and find D'D 11 = DD 11 -5- 8 = CA u - 8. MOHS CUBIC AXES. 57 FIG. 50. 11. The length of the projection of the nearly vertical edge of the cube through C has now to be determined. When the cube and square D m A n have been rotated together through 90, the lines in Fig. 49 are brought into the plane of projection and one edge of the cube through C is in the line of sight. Let CC m and CH I} in Fig. 50, represent the line CC m of Fig. 49 and the edge of the cube when the latter is in the line of sight. If, now, the cube and Fig. 49 are ~ turned back to the final position of the cube, BB n and C m C are shortened to B*B n which is given by CL of Fig. 50. During this rotation the points C m and H 2 have traversed the circular arcs C ni C IV and Hfl. Draw the L perpendiculars C IV L and HG on the vertical c line HCC 111 . Then CL = B*B U = CC in -f- 8, and CH is the projected length of the cubic edge CG. If the angle of tilt, GCH=CC IV L, be denoted by o-, we have sin v = CL + CC IV = 1 H- 8 ; and o- = 7 1 1'. For any other angle of tilt, sin '8, each equal to A'H', may be measured off on the verticals from C, B and D' by means of a pair of com- passes. FIG. 51. 58 MOHS' CUBIC AXES. FIG. 51. . The lines CA', CD' and Cy, produced in both directions give the projections of a rectangular system of axes. Furthermore, the length CA' gives the unit of length on OX or on any parallel line, CD' gives the same unit of length on OY or on any parallel line, and Cy or, more correctly, CH, when theoretical values are involved gives the same unit of length on all vertical lines. The student will find it convenient to have a set of rectangular axes of equal length, shortly called cubic axes, projected in the manner just described. They should be drawn on a large scale on card-board or stout paper. He should notice that CA' is the negative direction of the axis of X and CD' the negative direction on OY. 12. The angles yCD' and yCA' can be easily calculated from the data of Figs. 49-51, and also the actual lengths of CA', CD', and CH in terms of any scale. D l D n 1 'D Tf C~^A A 1 A 11 _ 3 CA 11 ~8' Furthermore, CA' : CD' : CH= V73 : \/577 : N/630 = 8-5 : 24 : 25-1, nearly. It is easy to obtain, in the same manner, the projection of the axes for any other values of the rotation and tilt. Thus, for drawing cubic crystals, Professor von Lang (Lehrbuch der Krystal- lographie, 1866), uses axes in which D"C = CA n , and BB n = SB'B". Hence, if we assume Fig. 51 to represent a cube projected with these numerical relations, we have : sin 0-= 1-^-9, and o- = 623'-6. yCD' = 90 - 1 33'-5, yCA' = 90 - 23 58'. CA' : CD' : CH= N/97 : N/1296 : VI360 = 1 : 3-65 : 3-74, nearly. Having obtained a projection of cubic axes, all simple cubic forms can be drawn as will be shown in Chapter xv. ; and a little practice will then enable the student to draw combinations of several forms. For tan D'CD 11 tan 33'-3, and yCA' = 6926'-7. AXES OF TETRAGONAL AND PRISMATIC CRYSTALS. 59 ' C 2 FIG. 52. 13. To adapt the projected cubic axes to the requirements of tetragonal crystal, we only change the length of the vertical axis, as shown in Fig. 52. The parameter c on the principal axis is given by the equation, c - a tan E, where E is the angle 001 A 101. Thus, for cassiterite (SnO 2 ), ^=33 55'; .'. c = Cy tan 33 55', or, more correctly, = CH tan 33 55' ; for CH is the unit length measured on the vertical. If CH be taken to be 25-1 units of length, then c is 16-88 units of the same scale. The points A, A', C in Fig. 52 are at the parametral distances on the pro- jected axes of X, Y, and Z respectively. 14. In a prismatic crystal the length CA' of Fig. 51, measured along OX, has also to be changed and by a similar rule. In Art. 8 it was shown that, for barytes : a = b tan 39 1 1', and c = b tan 52 43'. But, in Fig. 51, CA' along OX is the same length as CD' measured along OY. Hence, we have to mark off on CA' a length = CM' tan 39 11' = CM' x -8151; and similarly, along Cy a length = CH tan 52 43'= CH x 1-3135. Similarly, for the axes of topaz, take a length CA' tan 27 50' along OX, CD' along OY, and a length OS" tan 43 31' along OZ. 15. To obtain the axes of an oblique crystal, we first replace CA' of Fig. 51 by a new line OX t , situated in the plane yCA' and inclined to Cy at an angle ft. This is done as follows. In Fig. 53, let Cy and CM, be two lines, each equal to CA of Fig. 49, and let yCM, = 90. With centre C describe the quadrant of a circle yQA t , and let the arc yQ = /\ yCQ = ft. ^ IG - 53. Then CQ, when projected in the plane yCA' of Fig. 51, is the direction of the axis OX t required: and CQ-CA = CA, is the unit of length. Draw QM and QN parallel to Cy and CM,, respectively. Then CM = CQ sin MQC '= CM, sin ft; and CN= CQ cos NCQ = Cy cos ft. Now, by the character of the projection, equal parallel lines are projected as equal parallel lines, and lengths in the same line are 60 AXES OF OBLIQUE CRYSTALS. projected as lengths having to one another the same ratio as the original lengths. Hence, OA,, OY, OZ, in Fig. 54, being the directions of CM', CD' and Cy of Fig. 51, construct in the plane ZOA t a parallelogram Omqn equal and similar to that of Fig. 53 ; where = CM' sin/3, of Fig. 51, --- -/- ~ On=Cy cos(3, . Join Oq, and produce it on both sides ' of the origin. Then Oq is the direction OX t , and is the unit of length measured in this direction. The lengths Oq and Oy have now to be changed in the ratios a -f- b and c -=- b ; or, as Oq is the shortest length, it may be more convenient to change CD' and Cy of Fig. 51 in the ratio of b H- a, and c -=- a, respectively. When such computations are made, the correct value of Cy, viz. CH of Fig. 50, is used, and not the approximate value AW. Example. To obtain the axes of orthoclase in which /3=6357', and a: b :c = -658 : 1 : -555, we have 0?=(L4'sin63 57' = 8-5sin6357' = 7-637 units. On = Cy cos 63 57' = 25 '1 cos 63 57' = 1 1-02 units. The parallelogram Omqn is now completed, and Oq drawn. Retaining Oq, the unit length on OX, as the unit parameter, a length OB t is taken on OY=24-i- -658 = 36-47 units of the scale. Similarly, OC on OZ is made equal to 25-1 x -555-f--658 =21-17 units of the scale. 16. A convenient set of oblique axes can be expeditiously obtained as follows. We start from the position in Art. 9, where the cube has been turned through the angle p but not tilted 1 . The positive cubic axes then form a cross, Fig. 60, with unequal horizontal arms, OD and OE ; where OD : OE =1:3, or, generally, 1 : n. The arm OE to the right is retained as the axis OY, whilst the cubic axis projected in OD has to be replaced by a new line in the vertical plane ZOX. The vertical arm OA" is retained as OZ, and its length is ODjlO, or OEjlO -=- 3, or generally, OE*Jl +?i 2 -3. Hence, in Fig. 55, OC and OB are drawn intersecting one another 1 This is the same position as that mentioned in Art. 23 before the eye is moved. The drawings are orthographic, for the rays are at 90 to the paper. The projection is only advantageous when /3 differs appreciably from 90, and when no face has to be shown which is nearly horizontal. AXES OF OBLIQUE CRYSTALS. 61 at right angles, and any arbitrary length OB is cut off on the horizontal arm to the right. This represents the unit of length on OY, just as OE does in Fig. 60. The continuation on the left, which may be denoted by OD, represents the cubic axis perpendi- cular to OB, and its length is OB -=- 3. With sides lying in OD and the vertical construct the parallelo- ^ gram OMXN, where j ON = OA" cos NOX = OB VlO cos 04-3. The diagonal OX of the parallelogram gives the unit length on the inclined axis 7F -Lr l c, OX of an oblique crystal, the lines OB and OC giving the positive directions of the axes of Y and Z. Fig. 55 gives the axes of orthoclase from which the figure, given in Chap, xu., has been drawn. Having obtained the directions of the axes, OA is now taken on OX equal to OX x -658, OB is the unit of length on OY, and OC = OA" x -555 = OB VlO x -555 - 3 = OB x -585. The parameters are known in terms of the arbitrary length OB. 17. In the anorthic system the angles between the axes, YOZ=a, ZOX=$, XOY=y, may have any values, and vary with the substance. We first find, in the plane A'CD' of Fig. 51, two lines inclined to one another at an angle >// = 100 A 010. This angle between the vertical faces is not to be confused with y, and is usually obtained by direct measurement. The obtuse angle, 180 \^, is usually placed to the right front. Suppose CD' of Fig. 51 to be unmoved; we can, by the process employed in Art. 15, find a line Og in the plane A'CD' inclined to CD' at the angle ^. The sides of the parallelogram along CD' and along the prolongation of CA' to the front are: Cd=CD'cos^, Cm = CA'sm^. The diagonal Cg lies in the plane D'CA', and its length is the unit length. Further, the plane yCD' is parallel to (100) and yCg to (010), and they contain the axes of Y and X, respectively. In the plane yCD' we form a parallelogram, the sides ON' along Cy, and OH' along CD', being given by ON' = Cycos a , OM' = CD'sina. The diagonal gives the direction of OY and the unit length on it. A similar parallelogram OMXN is formed in the plane yCg in which replaces a, and C replaces CD'. The diagonal is the axis OX and the unit length on it. When a and /3 are >90, the sides of the parallelograms are taken so that yCTand yCX &re >90. The unit lengths on OX and OZ are now multiplied by the numbers a-rb, and c-=-6, respectively. 62 AXES OF ANORTHIC CRYSTALS. 18. Suitable projections of the axes of anorthic crystals can be also obtained by a method similar to that given for oblique crystals in Art. 16, in which the cube has not been tilted. Starting with horizontal arms, OE to the right and OD=OE+3 to the left, we first change the length on the left so that it should represent unit length on a line making 180- ^, instead of 90, with the axis on the right. The direction of the line is still the horizontal arm to the left. Then = OEcos (^ p) -=- 3 sin p, where p = 1826'; and, as before, OA" on the vertical axis is 'equal to In the plane ZOE a parallelogram OM' TN' is made, where OM' = OE sin a, and ON' = OA " cos a. The diagonal OY is then the direction of the axis of Y, and the unit length on it. Similarly, in the plane Z0 the paral- lelogram OMXN is made, where OM= sin = OEcosty - p)sin^^-3sinp ; and ON=OA" cos /3. The diagonal OX is the axis of X and the unit length on it. The unit lengths on two of the axes are then easily changed to correspond with the parameters of the crystal to be drawn. Hence the lengths in the paper are all known in terms of the arbitrary length OE. Example. As an illustration let us take cyanite (Al 2 Si0 5 ) in which we know : ^ = 73 56', the angles: a = 905'-5, /3=1012'-25 ; and the parameters a: b :c = -899 : 1 : -709. Then 0= OE cos 55 30'-f-3 sin 18 26'= OE x -6 (very nearly) : OA" = OE x 1-1 (very nearly). The rectangular cross has the arms of 6 units, OE of 10 and OA" of 11 units of length. Again, since a=905'-5, the axis OY is left in OE, the error being imper- ceptible. OX is found from the parallelogram OMXN with sides : OM= sin 78 58' = 5-88, and ON=OA" cos 78 58' = 2-125. The final lengths on OX, OY, and OZ, are: OA = 6-27 x -899 = 5-63, OE = W, and OC= OA" x -709 = 7'8 units. The angle XOM can also be calculated for the above values ; for tan XOM = ON+ OM= 2-125-^5-88. Therefore XOM= 24 25'. 19. To draw rhombohedral and hexagonal crystals, a regular hexagon, 88J5'..., Fig. 57, is projected in the plane A' CD' of Fig. 51 which contains two lines of unit length at right angles to one another. Let 08, = 08' be the unit of length, and on the line at right angles to 08, take OM t = OM' = OS, tan 60 = OA>j3, where OA on OM t is the unit of length. Complete the rhombus M f 8'M'8 /} and AXES OF RHOMBOHEDRAL AND HEXAGONAL CRYSTALS. 63 bisect the sides at 8, 8 /f , 8, 8". Join 88 /t and 88", and produce them and the sides of the rhombus to meet at M, M it , fcc. Then clearly the small triangles 8M t 8 /f , 8^M8', &c., are all equal and equilateral ; and 8S /7 8 '88"8 f is a regular hexagon, each of the sides of which is the unit of length, for 88 7/ is parallel to, and one-half the length of, 88'. Also OM= OM = the face to the left is n times that to the right. So far the construction is the same as that of Mohs given in Art. 9. If the paper be turned through 90 until it co- incides with the horizontal plane, OA and OB are two of the axes, and their pro- jections on the horizontal line of the drawing are given by OD and OE of FIG. 58. Fig. 58. If n = 3, then tan p = OD-i-DA = OD -=- OE = 1 -f- 3 ; and p = 1826'. This value is generally convenient. Naumann now supposed the eye to be raised above the horizontal plane so as to look down upon the top of the cube. The vertical axis is unaltered in length and position, for it is in the paper. The extremities of the horizontal axes, originally projected in D and E, Fig. 58, are now depressed in the vertical planes. The amount of depression of each extremity is proportional to its distance from the paper, and varies with the height to which the eye is raised. Let the depression DA' of the extremity A on the axis OX be OD -=- s, and let o- be the angle the rays make with the perpendicular to the paper. The depression of A is seen from Fig. 59 (a), which gives the lines in the vertical plane through the eye and DA' of Fig. 58, to be D A' = DA tan cr. From Fig. 58, DA = OE=ODcotp; :. DA' = OE tan o-. By hypothesis, FIG. 59. and tan . L. c. 5 66 NAUMANN'S CLINOGRAPHIC AXES. The depressions DA' and EB' are, therefore, given in terms of the arbitrary length OE and the numbers n and s. We can now readily find the angles DOA' and EOB', and the lengths OA', OB'. For tan DOA' = DA' + OD=l+s, tan EOS' = EB' + OE - tan p tan a = 1 - n*s. Also, A'O* m OB* + A'D* = OD* ffO* = B'E* + OE 2 =OE*(l+ tan 2 p tan 2 a) The above expressions are perfectly general, and give the values for any rotation and elevation desired in terms of OE, or of OA the cubic edge in the paper. Naumann employed the values n = s = 3, in drawings of all systems, and these values are adopted in this book. Hence, p = BOE = 18 26'. Also, tan EOB' = tan p tan a = 1 -=- n 2 * - 1 - 27 ; /. EOB' = 2 7'. tanZ>04'=l+* = lH-3; :.DOA' = 18 26'. The angle ^' - n, generally. The line TS is drawn through T, parallel to NOE, to meet the vertical through E in B'. The line B'OY I intercepted between the two extreme verticals, gives the unit lengths OB' and OY t on the axis of Y. EB' DT DA' 1 For tan^'=^=^ = ^=^ = tanptanc, To find the length OA" on OZ, we take S on EB' where ES = DO. A length on OZ = OS, is the unit length OA" in the vertical, as is evident from either of the equal triangles BOE, AOD of Fig. 58. From a set of cubic axes thus projected those of tetragonal and other crystals can be obtained in the manner described in Arts. 1320. In Fig. 60, s has, for the sake of greater clearness, been taken - 3 -=- 2, though this value is rarely, if ever, used. If 1 -^ s = 0, DA' - 0, and the eye is not moved. The cubic axes are then projected in OD, OE, OA" of Fig. 60, and form the irregular rect- angular cross from which, in Arts. 16 and 18, the axes of oblique and anorthic crystals are obtained. The drawings are then par- ticular cases of orthographic drawings, for the rays are perpen- dicular to the paper. 24. The drawing of simple forms, or of simple combinations in systems in which the forms to be shown are not closed, offers little difficulty. But in drawing combinations, in which several forms occur, a good deal of judgment and practice in drawing is needed to know which forms should be drawn first. When a decision on this point is come to, the simple form should be completely drawn, a very hard pencil, or still better a stout needle mounted in a convenient handle, being used. When introducing the faces of the remaining forms, care must be taken to cut off proportional lengths on all homologous edges. This is readily done by means of proportional compasses. No general rule can be given as to whether the predominant form, i.e. that which has its faces most largely developed, or one of the subordinate forms, should be first 52 68 REDUCTION OF SCALE. drawn. Thus, if we desire to show the regular tetrahedron K [111} having its edges truncated by narrow faces {100} of the cube, it is best to draw the cube first ; the faces of the tetrahedron are then easily inserted by drawing lines through points on the cubic edges parallel to the diagonals of the cubic faces and modifying alternate coigns of the cube. To cut off equal lengths from the coigns on the cubic edges, the proportional compasses are fixed at the desired ratio say to cut off one-tenth the long legs being made to span each edge in turn, the short legs give the length from the coign on this edge. On the other hand, in a combination of *{!!!} with the rhombic dodecahedron {110} small, the tetrahedron should be first drawn. 25. Until some skill has been attained it is best to project the axes and to draw the crystal on a very large scale. The drawing can be reduced either by photography, or by the following method FIG. 61. illustrated by Fig. 61. All the coigns are joined to any point in the paper, which may be either at the origin, or at any point without the drawing. The proportional compasses are then set so as to give the required reduction in the length of any line. Thus, if the drawing were to be reduced by a-half, the compasses would be set to bisect each line. By spanning each line from a coign to the fixed point with the long legs and reversing the compasses we get a sei'ies of points, as shown in the diagram, which, when joined by lines parallel to those of the original drawing, give a diagram half the dimensions of the original one. In the diagram one of the smaller cubes is one-half, and the other two-fifths, of the large cube. Thus, Op : Q = Qp' : QB l =pp' : R&= I : 2. Similarly, Or : OR = Or 1 : OR 1 = rr l : RR 1 = 2:5. Clearly the process can be reversed and a small figure magnified in any required ratio. DRAWING EDGES AT THE BACK. 69 To avoid confusion the interrupted lines from O are not carried through the large cube, and the back edges of the interior cube have been omitted. 26. It is often desirable to show the faces at the back of a crystal. This is usually done by means of lines of dots or very short strokes. When the crystal is centro-symmetrical the edges at the back are very easily drawn as follows. All the front faces having been introduced, the coigns are pricked through on to tracing paper and the figure copied. The tracing paper is then inverted, that is to say, is turned half-way round in its plane, so that the edges bound- ing the figure again exactly fit. The coigns are now pricked through the tracing paper on to the original drawing and give the coigns at the back of the figure. It is clear that lines through the centre of the figure would join opposite coigns and be bisected at the centre, and that the process therefore gives the positions of parallel faces. CHAPTER VII. LINEAR AND STEREOGEAPHIC PROJECTIONS. (i) Linear Projection. 1. THE diagrams given in plans, orthographic and clinographic drawings, aid the student to realize crystal-forms, and to recognize the symmetry and habit of crystals of different substances ; but they do not enable him to grasp all the zonal relations of a crystal, or to determine the relations between the dihedral angles and the indices of the faces. For these purposes diagrams involving a more highly abstract representation of a crystal are generally used. 2. The first we shall describe is that known by German Crystallographers as linear projection. The plane of projection is parallel to some important face of the crystal, such as a plane of symmetry or an axial plane, or that perpendicular to the triad axis of a rhombohedral crystal. All faces and edges are shifted to parallel positions so as to pass through a fixed point, not lying in the plane of projection, which is called the centre of projection. The straight line in which the transposed face meets the plane of projection is called the trace of the face. Parallel faces will, when shifted, coincide in a plane parallel to each of them, and are therefore represented by a single trace. Two or more traces meet in the point in which the line of intersection of the corresponding planes meets the plane of projection. The line joining this point to the centre of projection is therefore a zone-axis, and the point is called a zonal point 1 . The points, in which different traces 1 It may be denoted by enclosing in crotchets the letters which indicate the traces meeting in it. LINEAR PROJECTION (BARYTES). 71 intersect the trace of any particular face P, are zonal points which fix the directions of zone-axes parallel to the face P; and all the zones to which this face is common must give zonal points situated in its trace. Again, the line joining any two zonal points is the trace of a possible face ; for the plane which contains the two zone- axes, joining the centre of projection to the zonal points, is a possible face. The projection is useful in several ways. It may be used : (i) for determining the directions of zone-axes, or edges of the crystal ; (ii) for testing whether any face is in a particular zone, for the trace must then pass through the zonal point ; (iii) for determining the symbol of a face parallel to two zone-axes. It can also be used for numerical calculations, but its suitability for such computations is so inferior to that of the stereographic projection that we shall not use it for the purpose. 3. Example. We shall illustrate the projection by giving the steps by which the linear projection, Fig. 63, is made. It represents the crystal of barytes of which Fig. 62 is a plan. A plane parallel to c (001) is taken for that of projection, and the centre is taken at a point C on the zone-axis, OZ, perpen- dicular to this face and at a distance c (the parameter) from the paper. The point C is not shown in Fig. 63, for all points on OZ are projected in 0, where OZ meets the paper. The faces a and a', when shifted parallel to themselves so as to pass through C, coincide in a plane through OZ which meets the paper in TOT,. This trace is labelled a (100). Similarly, the trace in which the plane through C, parallel to b and &', meets the paper is given by the axis XX, at right angles to TT,. It is, furthermore, clear that every face parallel to the zone-axis OZ must, when it is transposed so as to pass through C, meet the paper in a trace which passes through 0. Hence, from a knowledge of the angles given in Chap. in. Art. 9, we can draw the traces of the faces m, k, X, &c., by means of a protractor. The traces Jc and k' make angles of 22 15' with OX; whilst the pairs m, m' and X, X' are inclined to OT at angles of 39 11' and 22 10', respectively. Again, the traces of all tautozoual faces meet in the zonal point. If, 72 LINEAR PROJECTION (BARYTES). moreover, the zone-axis is parallel to the plane of projection, the zonal point is at an infinite distance, and all the traces are parallel. Hence, the traces FIG. 63. of z and z", which are in a zone with the faces m and c, are parallel to the trace of m, and make angles of 39 11' with OY. The trace z may be drawn through any point B on OY. The length OB is taken arbitrarily and gives the scale of the projection, and the distance c of the centre of pro- jection is determined by the length OB taken (Chap. vi. Art. 8). Again, from the triangle AOB, which the trace z forms with the axes, we have OA + OB= tan (OB A = 39 11'). Hence, OA is known in terms of OB, and is a fixed length. The face z" passes through B, and A, where OB,= -OB, and OA= -OA. The homologous traces / and z"' are now drawn through B and B t parallel to m', and clearly pass through A, and A, respectively. The symbols of the faces are attached to the traces. The faces a, z, y, o, y 1 , z', a' are all in a zone having CB for zone-axis, since B is the point in which the traces of a and z intersect. That a and a! are in the zone is obvious, for they are parallel to the plane of symmetry which bisects the angles yy 1 , z/, and is perpendicular to o. The face o is parallel to OX, and its trace must be the line through B parallel to OX. For a similar reason the trace of o' is the parallel line through B t . Their symbols are (Oil) and (Oil), respectively. LINEAR PROJECTION (BARYTES). 73 Again, the faces 2, u, z"' are in a zone with the faces b and &', which are parallel to the plane of symmetry bisecting the angle zz"'. The face u is also parallel to OT. Hence, its trace is the line u drawn through A parallel to OY, and its symbol is (101). The homologous face gives the parallel trace u' (TOl) passing through A,. The traces, o, m', u all meet in a point, and the faces are therefore tautozonal ; and similarly for the other sets which meet at the corners of the rectangle formed by the traces of o and u. Close inspection of Fig. 62 shows that the edge [ro] is parallel to the edges into which m! and m'" are foreshortened; and it is clear that all lines in these latter faces will be projected in the same lines. When we meet with such edges as [ro] in a plan, we know that the faces r and o must be in a zone with the faces m' and m'". Hence, the trace of r passes through the zonal point in which m' and o intersect. But r is also in the zone [wrc]. Hence, its trace is parallel to that of m. It must meet OX at a distance 20 A, for the triangle formed by the traces u, r, and OX is similar and equal to BOA. Similarly, the trace r meets OY sA 20B. The face r therefore intercepts on the axes lengths 20A, 20B, OC. A parallel face is given by OA, OB, OC+2. Hence, its symbol is (112). The traces of the homologous faces are now easily drawn. The edges of y, r, d, r"', y"' are all parallel to one another and to the lines 6. Hence, the faces are all tautozoual. They therefore all pass through the points in which the trace r meets OX. But d is parallel to OY. Its trace is, therefore, inserted; and the face has the symbol (102). But the trace d meets that of z at the point in which this latter meets that of o'. Hence, the faces 2, d, o', are tautozonal. The trace and symbol of y are now easily found, for y is common to the zones \zyo~\ and [dry]. Hence, its trace passes through B and through the point of intersection of r and d; and the intercepts of the face are 204, OB, OC. The symbol of y is therefore (122). The homologous traces can be now drawn. By drawing lines through A and B parallel, respectively, to X and k, the student can easily prove that the symbols of these faces are (210) and (130). The trace of X is seen to pass through the intersection of those of y and /, and that of k through those of r 1 and z". The symbols can therefore be obtained by Weiss's zone-law. The trace of I (104) has not been inserted, as it would unduly extend the diagram, for it meets the axis of X at a distance 40 A. The face c is parallel to the paper and cannot be shown as a trace. A linear projection is sometimes inconveniently large ; for a face, inclined at a small angle to the paper, meets it in a trace at a correspondingly great distance from the origin. In practice, this is a serious drawback. It should be noticed that a knowledge of the parameter c, or of the angle which the face z makes with the paper, has not been needed. The projection would be the same, except in scale, whatever be the length OC. 74 AUXILIARY LINEAR PROJECTIONS. 4. Linear projections are of great assistance in. making plans and other drawings of crystals. Thus the plan, Fig. 62, is readily made from the linear projection. In the preceding Article reference was made to the plan as standing for the crystal ; but in actual practice a linear projection would be drawn directly from the crystal after such measurements and determinations of zones as were needed had been made, and the plan would then be drawn from the linear projection. For the edges in the plan are the lines joining to the zonal points in which the corresponding traces meet. The edges are parallel to the traces when these are parallel to one another, for the zonal point is at an infinite distance. It is also clear that the edge of intersection of c (001) with any face is parallel to the trace of the latter face. Linear projections may be advantageously used in order to get the directions of the edges of a crystal, when an orthographic or clinographic drawing has to be made ; and the method is often less laborious than that given in Chap. v. Art. 4. All the faces are transposed parallel to themselves so as to pass through a point on one of the axes of reference at a distance from the origin equal to the parameter on this axis ; as, for instance, the point C on the axis OZ, where 00 = c. The traces are then drawn in the projected plane XO Y. If the axes of X and Y and the parameters on them have been correctly projected, the traces are as easily drawn as in the example worked out in Art. 3. The directions of the edges are given by the lines joining C to the intersections of the traces. The scale of the linear projection is, in this and similar cases, determined by the lengths of the projected axes. Or we may suppose all the faces to pass through A on OX, where OA = a, and the traces to be drawn in the plane YOZ. It may, occasionally, be convenient to use partial linear projections in more than one of the axial planes in drawing the same crystal. Such partial projections would be needed to give a few edges only. Care must, of course, be taken not to make mistakes as to the plane in which the traces lie, or as to the axis on which the centre of projection is taken. 5. Example. We shall illustrate the application of linear projection to the drawing of crystals, by describing the steps involved in drawing the simple crystal of anorthite, given in Fig. 65. The forms to be represented are: M {010}, P {001}, y {201}, I {110}, T {110}, p {111}, o {III}. The parameters and the following angles are known: Mh = 010 A 100 = 87 C', a = 93 13'-3, =115 5o''o. EXAMPLE (ANORTHITE). 75 Projecting the axes by the method of Chap. vi. Art. 18, we draw, in Fig. 64, MM' and CO, at right angles, and construct the parallelograms OMXN and OM'YN' with the sides given below. OE, or OM', being any arbitrary length i we take : cos (87 6' - 18 26') _ cos68 40' 3 sin lb c 26' 3 sin 18 26' ' OM = Of cos 25 55' -5= OE x -36 ; = OE x -46 ; OJ/' = cos 3 13'-3 = OE, very nearly ; O.N'=r y , each equal to a; and join Er, Er t . The points R and R t , in which these lines meet the FIG. 72. 62 84 PROJECTION OF ANY POLE. diameter OP, are the projections of points on the small circle. Bisect RR t at C, then C is the centre, and CR the radius, of the projection of the small circle on the sphere. 19. The preceding problem gives an easy method for finding the projection of any pole, when the angular distances from two known poles are given. For the pole must lie on small circles drawn with the given arc-radii about each of the known poles. The projections of these small circles are obtained by the preceding construction. The two points of intersection satisfy the data. To decide which point is that required is generally easy from knowledge of the crystal, and the position of the pole with respect to other poles. This construction is usually applied to make a stereogram of an anorthic crystal. The poles in the primitive are placed by means of a protractor, and then a few poles are projected within the primitive from a knowledge of their angular distance from any two poles in the primitive which are not the extremities of a diameter. The points P and p coincide when the pole is in the primitive. Consequently r and r t are marked off on the primitive at the required angles from P, and R, R,, and C are then found as before. 20. If, however, the angles from the unknown pole to those in the primitive exceed 45, the above construction is inconvenient, for the line Er t then meets the diameter at a very distant point. In such a case it is best to draw the tangents at r and r, to the primitive. The point of intersection is the centre C of the small circle. For, since p the centre on the sphere of the small circle lies on the primitive, the two circles cut one another at right angles, and the radius of one circle at the point of intersection is a tangent to the other. Example. Thus, in Fig. 73, the poles M, M t , T of anorthite are placed in the primitive, and the point n is required, where J/> = 47 24', and 7^=53 14' -5. Arcs M,a, M t a t on the primitive are mea- sured oft' equal to M t n, and the tangents at a, a, are drawn to meet at C. A circle with centre C, and with radius Ca (the length of the tangent), is a small circle, points on which are at 47 24' from M t . A similar construction is made on either side of T, the arcs Tfr, Tp a being 53 14'-5. A third point /3 on ZT can be found by joining Eft,, where T=90. The circle with radius 2)/3 and centre D cuts the circle aa, at the required point n within the primitive. EXAMPLE (BARYTES). 85 21. Example. We proceed to describe the construction of the stereogram, Fig. 74, of the crystal of barytes, already discussed in Art. 3 of this Chapter and in Chap. vi. Art. 8. The axis OZ, parallel to the faces a, m, b, is taken as the U0 diameter through the eye ; consequently the poles, a, \, m, &c., lie on the primitive. The pole a is placed at the lowest point, 6 at 90 to the right. The other poles X, t, k, &c., are put in by a protractor from the table of angles given in Chap. in. Art. 9. The zones \audlc], [hoc], [mrc], and \cr'z'] are then drawn as straight lines through the centre. The position of a pole, o, z, or d, on one of these lines is then determined by Prob. 1. Let us take o. Then angles = co (52 43') are measured on the primitive from the point a, , and the points so obtained are joined by straight lines to the pole a ; for the points a, a, are at 90 from the great circle [6c6,]. The points of intersection with [bcb,] are the poles o and o'. The circles in which the zones [azyo], [ao'aj, are projected can then be drawn. The centre of the circle through aoa, is easily found by drawing circles (larger than the primitive, so that lines of construction will not confuse the stereogram) about a, and o, as centres. The line joining their points of intersection meets the line be in the centre, x, required. The compasses are opened out to the extent xa and the circle described. The centre, x, is not shown; it is merely an auxiliary point in the construction. One of the needle points of the compasses is next placed at a and the centre on the line be of the circle ao'a, is found, and the circle itself then drawn. The intersections of these circles with the zones [me], [m,cj fix the positions of z, z', z", z'". The zone-circle [bzu] is next determined in a similar manner. Its centre is a trifle further from the point c than ', and the construction can be easily tested. The homologous circle [bz'u'] is drawn with the same radius. The poles ?t and u' are therefore fixed. By drawing the zone- circle [mtio'] and its homologues, the points r, r 1 , &o., are fixed. It is then possible to draw the zone-circles [byrd] and [by'r'd 1 ], which 86 EXAMPLE (BARYTES). by their intersections with the zones [am], [auc] fix the positions of y', Ac. and of d, d'. The student will find that the following zones exist, and that circles through the poles can be drawn; viz. \_kor"], \\zu'], [zly"], [o'ly], [\dr"], one of which has alone been drawn. The possibility of drawing a circle through three poles establishes the fact that the corresponding faces are tautozonal. 22. The indices of the faces on the crystal can now be all determined by the zonal relations of the circles and poles in the stereogram, and will be found to be those given in Chap. v. Art. 15 and in Chap. vi. Art. 8. The poles a (100), b (010), c (001) of the axial planes coincide with the axial points X, Y, Z since the planes are at right angles. The face of which z is the pole is the parametral plane (111). The homologous poles above the paper are a 7 (111), 2" (111), "' (111). The poles m, &c., in the primitive are those of faces parallel to OZ, and the last index is zero. The pole m is the intersection of the zone [001, 111] with_[aft] = [001]. Hence m is (110). The homologous poles are m' (IlO), m" (110), ^(110). Similarly, the pole o is the intersection of [&c] = [100] with [as] = [011]. Hence, o has the symbol (Oil), and o' (Oil). For similar reasons the poles u and u' are (101) and (101). Again, X, z, u' lie on the same circle and are therefore tautozonal. The symbol of the zone-circle is [111, 101] = [121]. Hence, since the face X is parallel to OZ, and the last index zero, its symbol is (210). The pole r is the intersection of the zone-circles [mc] = [l!o] and [m,0][lll]. Combining the two symbols by the rule, given in table (23) of Chap, v., we find for r the symbol (112) or (112). The pole r (112) lying in the first octant must be taken. The opposite pole (112) below the paper is necessarily in the same zone-circles. The indices of d and y can now be found, for both are in [6r] = [201]. The face d is also parallel to OY, or in the zone [oc] = [010]. Hence d is (102), and d' (102). Furthermore, y is also in the known zone-circle [az] = [011]. Hence y is (122). Again, when constructing the stereogram, k was placed by the protractor. It will be found to lie in the circle through the poles r and o', i.e. in [311]. The pole k has the symbol (130). Again, from the circle |>y"] = [431], I can be shown to be (104). Generally, the numerical values of a : b : c would be now determined. In this case, they have already been found in Chap. vi. Art. 8 from the ele- mentary geometry of the crystal. The student's attention may be called to the fact that the indices of a pole, situated in a zone between two known poles, are the sum of simple multiples of corresponding indices of the known poles (Chap. v. Art. 8, Equations (13)). Thus, the indices of I are the sum of those of y" added to double those of z. CHAPTER VIII. THE ANHARMONIC RATIO OF FOUR TAUTOZONAL FACES. LET there be four tautozonal faces l\(h l k l l l ), P 2 (h. 2 k 2 l 2 ), (hM), no two of which are parallel. Let H^BA and Jf&DC, Fig. 75, be the two planes P l and P 2 ; and let OX, Y, be two of the axes of reference, so that 011^ = a^-h l , OK^ = b-r-k 1} 0// 2 = aH-A 2 , OK. 2 -'b-^k 2 . We proceed to establish the relation, given in equation (9), between the angles made by the four faces with one another and the indices of the faces. Let the plane xOy of the figure be perpendicular to the zone- axis, OT, and meet the line of intersection of the faces 1\ and P., in e. Let the planes through OT and each of the axes meet the plane xOy in the lines Ox and Oy, respectively. In the planes TOX and TOY, draw the lines l^A, H^C, Kfl, A' 2 Z>, parallel to the zone-axis FIG. 75. OT ; and let them meet the lines Ox, Oy in the points A, C, B and D. Join AS and CD ; and draw US parallel to CD. From and A draw OM and A N both perpendicular to CD, and let them meet CD and BS in M and N, respectively. Let the angle between the normals to P l and P^ be denoted by < 12 , and let those between other pairs of normals be indicated in a similar manner by attaching subscripts to indicate the faces involved. 88 THE ANHARMONIC RATIO. Now /\AeC=-- A ABN, is equal to the angle between the normals to P l and P 2 , and is therefore < 12 . Hence, from the triangle ABN, we have From the similar triangles OMC, AN8, we have AN = AS _OS-OA OS OA OM OC OC ~ 00 00 ^ J' From the similar triangles OBS, ODC, -^ = and from the similar triangles OBK ODK^ ~ =^ =^. -* = ^ 2 (JD OK Z k^ k^ k^ . OS^k_ 2 " OC A^ ...(3). FIG. 75. Again, from the similar triangles OAH 1} OCH Z , we have OC ...(4), /. from equations (1), (2), (3) and (4), we have denoting by u at v u , w m the zone-indices of [P,PJ derived by taking (*!^4) and (M,i) in table (10) of Chap. v. Art. 4. In equation (5) the angle 12 and the indices \ , k,, h, k, are involved with OM (=p 2 ), the normal on P 2 , and the length AB intercepted in P, between the lines Ox and Oy. These two lengths are arbitrary and indeterminate, for the faces may be at any distance from the origin. They must, therefore, be eliminated THE ANHARMONIC RATIO. 89 2. A similar figure, and an exactly similar set of relations to those given in equations (1) (5), are obtained, if the face P 3 is combined with P 1 . By replacing P 3 by P 3 we do not change the axes, the lines Ox and Oy, or AS, for these do not depend on the position of P 2 . But 12 has to be replaced by < 13 , the angle between P 1 and P 3 , and p. 2 = OM by p 3 OM 3 , the normal on face P s . It is clear also that the indices h 3 , k 3 of P 3 replace A 2 , ^, those of P 2 . By comparing the present procedure with that followed in Chap. v. Art. 3, the student can easily write the five equations in which the face P 3 replaces P 2 . We need only give the final equation corresponding to (5). It is AB k 3 h 3 h^s-hhs W 13 ,. The zone-index, entering into this equation, is ^ 13 , that obtained by table (10) of Chap. v. Art. 4 from faces P l and P,. Dividing equation (5) by equation (6), we eliminate AB, the indeterminate length depending on the exact distance of P 1 from the origin, and have W^ ,_ '" 3. The length AB has been eliminated only by introducing a new indeterminate length p 3 . But the two indeterminate lengths in (7) depend on the distances of P 2 and P 3 from the origin and not on /V This suggests a ready method of eliminating both p 2 and p 3 . It is only necessary to take some fourth face P 4 (hjej,^ in the zone, which is not parallel to any one of the three already employed. If this be done, and faces P 2 , P 3 be combined with P 4 as they were with P lt it is clear that an equation exactly similar to (7) will be obtained, in which the only difference arises from the replace- ment of P l by P 4 . Representing the angles which P t makes with 7*2 and P 3 by ^ and $^3, and replacing h t , k by A 4 , k 4 , respectively, we have p^ sin p a sin (4 Dividing equation (7) by (8), we have 90 NECESSITY OF FOUR NON-PARALLEL FACES. This important relation 1 , connecting the angles between four faces in a zone with their indices, is the foundation of much of our future discussion. It will be extensively used in the solution of the problems which arise in the determination (1) of the symbol of a crystal-face, and (2) of the angles between faces of which the symbols are known ; and likewise in establishing various general relations. 4. The trigonometrical compound ratio forming the left side of the equation is known as the anharmonic ratio of four planes intersecting in, or parallel to, a common straight line, and clearly depends only on the angles between the planes. The numbers on the right side, being zone-indices, are whole numbers. Hence, we have established that the anharmonic ratio of any four tautozonal faces, no two of which are parallel, is a commensurable number. 5. If any one of the angles 12 , &c., becomes 180, the corre- sponding sine is zero, and the anharmonic ratio becomes zero, or infinite, according as the angle appears in the numerator or denominator. It is easy to see that the corresponding value of W on the right becomes zero at the same time, for the indices of parallel faces only differ in sign. Thus, if the first and third face are parallel, (hjc) is (hjcjj ; and w u = h s k t -k l h t = -h 1 k 1 + k l h 1 = Q. The equation then becomes indeterminate and meaningless. We can, also, show that the equation becomes indeterminate if the faces P 1 and P 4 , or the faces P 2 and P 3 , are parallel. Thus let us suppose the faces P 4 and P 1 to be parallel. Then < 42 = 180- 13 _ . sin 13 ' sin ^ sin < 13 ' sin (180 - ^ 3 ) ~ sin /> 13 sin <~ 2 ~ : and on the right side A-- - _ ~' 1 The proposition was established and first published by Miller in the Treatise on Cryst., 1839. It appears from manuscript notes, bearing the date 1831 but first published after his death, that Gauss also had discovered that the trigonometrical compound ratio was rational (Werke, 11. p. 308, 1863). He gives the proposition as a relation of four zone-circles, and his expression is equivalent to that given in (27) of Art. 19 of this Chapter. The elegant geometrical proof, given in Arts. 13, is due to G. Cesaro (Rivista di Min. e. Crist. Ital. v. 1889). ORDER IN WHICH THE FACES ARE TAKEN. 91 We have thus retained no angles on the left and no indices on the right, and the equation has reduced to the meaningless identity, 1 = 1. The same result can be easily shown to hold when faces P 2 and P 3 are parallel. 6. In distinguishing the four faces as P 1} P z , P 3 and P 4 , nothing has been said to limit the order in which they actually occur on the crystal. The faces may be taken in any order w/iatever, and the student can easily prove for himself that there are six different possible arrangements of the four faces. The only point which must be carefully attended to is this : that the numbers w must be calculated from the indices arranged in table (10) of Chap. v. Art. 4 in exactly the same order in which the faces occur in the angles on the left. The student will, however, find it convenient to select the extreme faces for P l and P 4 , and the pair occupying in- termediate positions for P 2 and P 3 . This is not the order most commonly adopted in mathematical text-books. The advantage of taking the faces in the order recommended arises from the fact that the anharmonic ratio is then positive; and difficulties as to the directions in which the angles are to be taken and as to the signs of the trigonometrical ratios are avoided. Errors may arise in other arrangements, when an angle has to be calculated by the aid of equation (9) from a knowledge of the symbols of the faces and certain angles. 7. If we take a stereographic projection of the crystal, we have for the zone a great circle with four poles in it, no two of which are at the extremities of a diameter. The angles 12 , l3) &c., are the arcs intercepted between the poles; and as we can transfer, by equations (1) of Chap. iv. Art. 15, the indices of the face to the normal and therefore to the pole, we may state the law of Art. 4 in a slightly different form as follows : the anharmonic ratio of any four poles in a zone-circle, no pair of which are at 180 to one another, is rational and is given by equation (9) of Art. 3. Hence, if the poles of tautozonal faces are projected on a stereo- graphic projection, we have, as will be shown in Art. 10, a ready way of determining the symbols of the poles, and therefore of the faces, when those of at least three faces in the zone are known and the angles between all the faces have been measured ; for we can, in turn, take each of the unknown poles with the three known ones to form an anharmonic ratio. 92 EQUIVALENT EXPRESSIONS. 8. Again, from Chap. v. Art. 4, we see that h 1 k. i -k 1 ht = W 12 is the zone-index referring to the axis OZ as deduced from the faces P l and P z ; and similarly, for the other numbers W 13 , &c. The two axes involved in the proof, given in Arts. 1 3, were OX and OY. Also the auxiliary lines Ox and Oy were the orthogonal projections of this pair of axes on the plane containing the normals to the faces. As far as the proof is concerned, any other pair of the axes and their projections on the plane of the normals might equally well have been taken. Thus, we might take OY and OZ, and the projections would accordingly be represented by Oy and Oz. We should have similar triangles, like those in the figure, connecting sin 12 with the intercepts on OF and OZ. The only change would clearly be to introduce the indices l lt 1 2 , &c., instead of h 1} A 2 , &c. And similarly, if the axes OX and OZ were taken, we should replace k by I throughout. Taking the first of the two pairs just given, we have = (changing signs throughout) ~ j-^ : 4 2 ~ *---' "1^3 ~ n*3 "^3 ^4^3 = g :^? (see Chap. v. Art. 4) ...... (9*) Similarly, when the axes X and OZ are taken together, sin<- (10). We shall also, for the sake of brevity, use the symbol A.R. P t to represent the anharmonic ratio of four tautozonal faces or of four poles in a zone-circle, and we shall generally suppose P l and P 4 to be the extreme faces or poles. 10. Since the indices of each face occur in the first degree both in the numerator and denominator of the right side of equation (9), or of the equivalent equation (10), it follows that an equation of the first degree in the indices of any one of the faces is given by it. If all the angles and the indices of three of the faces are known, we can, using the expression involving ^ 12 , &c., obtain an equation of the first degree connecting A,, k 3 , for the face ^3 (say) the indices of which are required. For, since the angles are known, we can 94 DETERMINATION OF FACE-SYMBOL. compute the left side of (9). Let the computed number be Then, from (9), h 4 k 3 - k 4 h s g^M, x n(say); where n is easily found, since the numbers on the right are all known. Hence, h s (nk^-k^ = k 3 (nh 1 -h t ) (11). But P 3 (h 3 k 3 l s ) lies in the known zone [u 12 V 12 W 12 ], .'. h 3 u 12 + k 3 v l2 + l 3 W 12 = Q (12). The simplest whole numbers satisfying equations (11) and (12) are the indices of P s . From equation (11) we have These equations are of the same form as (13) of Chap. v. Art. 8 ; and, when n has been computed, they give the symbol of P 3 . But, as already stated in Art. 9, the faces may be parallel to a zone-axis lying in XOY, when the zone-indices W 12 , W 13 , &c., are all zero. In this case equations (11) and (12) become identical, and one of the other equations, (9*) or (9**), must be taken. An equation of the same form as (11), but involving a different pair of the indices of P 3 , is then obtained which, combined with (12), gives the required indices. Example. Thus, in the crystal of anorthite, Fig. 76, M is (010), P (001), T' (110), and x (101) ; and the f&cep lies in the zones [Mx] =[101], and [TP2"] = [110]. Therefore, by Weiss's zone-law, the symbol of p is (111). By measurement of the zone [PpgT'], we have Pp=54 17', Pfif = 80 18', PT'=110 40'. Taking the faces in order to be those indicated in the formula by suffixes 1, 2, 3, and 4, and therefore taking g to be (h 3 k 3 l 3 ) or (hkl), since the subscript, 3, is no longer needed to indicate the particular face, we have 001 i 110 sjls. 001 110 FIG. 76. hkl I hkl sinPp sin Pg ' sin T'p "sin T'a ' INSUFFICIENCY OF TWO POLES. 95 On determining the zone-indices from [Pp] it will be seen that: W 12 = - 1, K 12 =-l, and W 12 = 0. We must, therefore, avoid the zone-indices w. Taking the u's, we find: u lt = -1, u 13 = -k, tf 42 = l and u^^l. Introducing the values of the angles and of the zone-indices into the equation, we have sin 54 IT sin 30 22' _ -1 l_ _ ^ sin 80 18' X sin 56 23' ~ ^k x I~ k ' L sin 54 17' = 9-90951 L sin 80 18' = 9-99375 L sin 30 22' = 9-70375 L sin 56 23' = 9-92052 1-61326 1-91427 1-61326 log 2= -30101 .-. k = 2l, which is satisfied by making A; = 2 and 1 = 1. But, since the face g is in the zone [PT'] = [110], h + k=0. Hence, the symbol of g is (221). The student should notice that the equation, h + k = 0, is the same as is obtained by making ^=0; and, since all the zone-indices w are zero, this equation might have been thus obtained and not from the direct application of Weiss's law. 11. When, however, two poles P l and P 3 (say) are alone known in the zone, equations (9) and (10) do not give us relations limited to poles. All they tell us is that, if either P 2 or P 4 is a pole, then they are both poles. But if P 2 is not a pole, P 4 is not one either. We can easily make any number of rational anharmonic ratios of which any two poles P l and P 3 are members but in which the other points are not poles. Thus, whatever be the angle P 1 P 3 , we get a rational anharmonic ratio if P 2 bisect the angle P 1 P 3 and P 4 be at 90 to P 2 . For, in this case, the left side of equation (9) becomes sin P l P 2 sin P 4 P 2 _ sin P 1 P 2 sin P 4 P 3 _ ~ ~~~ since sin P 4 P 2 = sin 90 = 1, and P 1 P 3 = 2P,P 2 . But P 4 P 3 - 90 - P 2 P 3 = 90 - P,P 2 ; and sin 2 P l P 2 = 2 sin P^ P 2 cos P l P 2 . sin PI P 2 cos P,P 2 Hence, m = - : =r = - =r-j? = \. 2 sm P a P 2 cos P X P 2 The value of the anharmonic ratio is therefore 1 -j- 2, a com- mensurable number. But it is absurd to suppose that we can have a pole bisecting the angle between any two poles tvhatever. For were this the case, we could go on continually bisecting the angle until at last the angles between adjacent faces made infinitely 96 EXAMPLE (BARYTES). small angles with one another, and this is only true of curved surfaces such as spheres. The assumption contravenes the funda- mental notion of crystalline structure and form. 12. Example. From the angles of the crystal of barytes given in Chap. in. Art. 9, the symbols of all the faces can now be easily determined. To the faces a, b, c and z, the symbols (100), (010), (001) and (111) have been assigned. The intersections of the first three are therefore the axes, and z gives the ratios of the parameters a : b : c. The symbols of m, o and u, were then found by the intersections of known zone-circles (Chap. vn. Art. 22) to be (110), (Oil) and (101), respectively. We, therefore, know the symbols of three faces in each of the measured zones. (i) In the zone [amb] we take a to be P lt m = P 2 , b = P 4 , and P 3 to be one of the unknown faces with symbol (MO). sin am sin bin sin aP 3 ' siniP 3 ,010 110 010 JikO The student will, by trial with the symbols of a and m, readily find that 13 , .P 4 P 3 = 43 , be known angles. Then, since the symbols of the faces are known, the right side of equation (10) can be computed ; call it m. Then sm0 13 sin { } ............ } sin 43 The terms on the right are known, and the value can be easily computed by means of logarithm-tables. It must be some number greater than zero, for by the arrangement of the anharmonic ratio we are at liberty to regard all the terms as positive. Now, any number greater than zero can be represented as the tangent of an angle, 6, less than 90. Furthermore, on computation, the expression on the right will be found to be greater or less than unity, according as the angle < 12 is greater or less than (fr^. If it is less than unity, the equation is in the most convenient form ; if it is greater than unity, an expression less than unity can be obtained by inverting the equation. We may, therefore, suppose < 12 < < 42 , and that the equation is in the required form. Now, and l Dividing the former by the latter, we have sin0 42 -sin0 12 1 - tan 6 sin^ + sin^l+tan^r 1 But, by well-known trigonometrical formulae (Todhunter's Trig. p. 55, 1859), sin 42 - sin 12 = 2 sin ^A 2 cos ^-^ 2 ; LZ^ 2 sin 2 sin 42 2 12 cos < * >42 * V Hence ' ITS T-TTT = tan(45-0), 12 tan (45 - 0), (15). EXAMPLE (ORTHOCLASE). 99 But < 42 + M are both known and < ]2 is needed, equation (10) is changed to sin 9 12 sin 9 40 = = m = = tan X (say). sin 9 13 sin 943 By the same transformations as were given in detail in the first case, we have tan 13 12 = tan 13 12 tan (45 - 0j). Here the known angle is < ]3 - < 12 = Hence, 9 J3 + 9 12 is determined. 15. Example. In the crystal of orthoclase, Fig. 77, the zone [cyx] is measured, and the indices c (001), c, (OOl), x (101), y (201), are known. It is required from the given angles, c,o; = 50 16'-5, c,/=80 18', to determine the 021 angle the face a (100) makes with c (001). The face a and its parallel a, (TOO) are not present on the crystal, but the angle ac is an important element of the crystal, viz. the angle between the axes OZ and OX t . Let us take a (100) to be P lt y to be P 2 , x = P 3 and c, = P 4 of the general equation (10). Hence, sin ay _ sin c t y sin ax ' sin c,x 100 1-21 FIG. 77. . sin ay = sin (c,y = 80 18') ' ' sin ax 2 sin (c,x = 50 16'-5) log 2= -30103 L sin 80 18'= 9-99375 L sin 50 16'-5= 9-88600 10-18703 10-18703 L tan 32 39'= 9-80672. .-. 0=32 39', and 45- = 12 21'. 72 100 DIFFERENT ARRANGEMENTS OF TWO KNOWN ANGLES. Also ax-ay-xy = 30 l'-5 is known. .. employing formula (16), tan \ (ax + ay) = tan 15 0' '75 -7- tan 12 21'. Ltanl50'-75= 9'42843 L tan 12 21' = 9-34034 .-. L tan J (ax + ay) = 10-08809 = L tan 50 46'-25. .-. aa; + ai/ = 10132'-5, ax-ay= 30 l'-5. .-. az=6547'. Hence, ac^ax + xc^UG 3'-5, and ac = 6356''5. It may happen that an arrangement of the four poles is taken which leads to disadvantageous angles. Thus, for instance, if the above poles are taken in the order {cayx}, so that the unknown pole a occupies a middle place, the angle of expression (15) is 44 34'. The angle 45 - 6 is then 26', and an error of a few seconds causes a considerable difference iu the tangent. It was for this reason that a was, in the example, taken as one of the extreme poles. Very small angles, or angles of nearly 90, should be as far as possible avoided ; for, in such cases, a slight error in the observation may become multiplied in the course of the computation. When disadvantageous angles cannot be avoided in a direct computation, the liability to error may be minimised by taking some intermediate possible pole with the known ones. The position of this pole being determined by the method of Art. 14, the pole can then be combined with two of those first given and the one required. An angle is thus obtained which gives, by addition or subtraction, the angle required. No general rule for such artifices can be given. 16. In computing the angle between a face with known symbol, and three other known faces in the zone, five different cases may be met with. The transformation of equation (10), necessary to compute the angle which the face makes with any one of the others, depends on the arrangement of the known angles. For the sake of brevity let the poles be indicated by the points 1, 2, 3, 4 in the zone-circles shown in Fig. 78. Suppose < 14 to be less than 180, and let the known arcs be those between the poles joined by interrupted arcs; then the different cases can be given as follows : (a) 13 , 034 being known, to find 12 ; () &3, 4>24 12 . Cases (a) and (b) are discussed in Art. 14, and case (b) occurred in the example of Art. 15. A. R. EXPRESSED IN TERMS OF COTANGENTS. 101 17. The expression for the anharmonic ratio can, in cases (a) and (6), be also transformed into an expression involving cotangents as follows : sin ls sin < 43 1 sin $43 sin < 12 _ sin ^ sin 13 - . "* sin ls sin 42 sin < 13 sin + sin $43 (sin < 13 cos 13 (sin ^ cos ^ + sin ^ cos $43) sin (43 sin fa sin ^ (cot ^ - cot ft 13 ) sin 13 sin ^ sin ^ ( c t $23 + c t 4*43) COt < 23 + COt 043 If then P 2 is the unknown pole, (m 1) cot < 23 = cot 0^ 4- m cot 13 (18). The angles and number on the right side are known ; hence 0^ can be found by the help of a table of natural cotangents. Similarly, for case (6), cot $43 is unknown and all the other numbers are known ; so that 0^ is found from the tables. 18. In cases (c), (d) and (e), the unknown angle cannot be obtained by either of the transformations given in Arts. 14 and 17. They can, however, be all dealt with by one and the same process. For, by taking in (c) the pole P l opposite to P lt we have the known arc P 2 A P l 180 12 , completely enclosing the arc $34. The case becomes therefore identical with (d). Similarly, (d) can be transformed to the arrangement (e) by taking the pole opposite to P 2 with the others. Now the A.R. is sin 12 sin 0^ 1 , . -~ x -r- p? = (say), sin 13 sin ^ m In (c) both the angles in the denominator, in (e) both the angles in the numerator, and in (d) all four angles, are unknown. Now 2 sin A sin B = cos (A-B)- cos (A + B) (Todh. Trig. p. 55). Applying this to the equation given above, we have 1 = cos (0 12 - ^43) - cos (0 12 + ^43) , 102 A. R. EXPRESSED IN TERMS OF COSINES. But, 0! 3 -042 = 012 + $23 -($23+ $43) ^^-'hiO 013 + 042 = $12 + $23 + ($23 + $) = 2 $23 + $12 + $43 ', $12 + $43 = $13 + $42 - 2 $23- Hence, in case (c), we have cos (2023 + 0i 2 + #43) = m cos ($12 + $43) + (1-m) cos (^-4.43)... (20). In case (e), we have m cos (0 13 + 42 - 20^) = (m-l) cos (0 13 - 42 ) + cos (0 13 + 42 ) . . .(21). The angles and numbers on the right side of each of these equations are known. The cosines can be extracted from tables of natural cosines. The number m is usually a simple one so that the computation is not laborious. Hence, the value of cos (2023 + 12 + 43 ), or of cos (0 13 + 42 - 20^), can be found from the tables, and the unknown angle 023 is determined. In case (d), the equation (19) is resolved into a form slightly different from that given in (20) or (21). For, 012 - 043 = $12 ~ ($41 - &2 - 023) = 2012 ~ 4>41 + 4>23 5 012 + 043 - 041 - $23 j 013 - 042 = 012 - $43 = 2012 ~ 041 + $23 j 013 + $42 =041 + 023- Hence, equation (19) becomes (m - 1) cos (20 12 - 41 + 023) = m cos (0 41 - m ) - cos (0 41 + 23 ) ... (22). The expression on the right, involving only the known value of the anharmonic ratio and known angles, can therefore be computed. Hence, the cosine on the left is found, and the unknown angle 12 can be obtained. Such cases as (c), (d) and (e), do not often occur in practice. For, in calculating angles from known indices, we generally have several angles in the zone either known or to be computed ; and the calcula- tions can be done in a series so that the poles used can in each case be brought under case (a) or (6), when the more convenient logarith- mic transformation of the equation can be adopted. Miller, in page 13 of his Tract on Crystallography, 1863, points out that we can frequently employ cases (c) or (d) when one of the known angles is a right angle. Thus, if 12 = 90 in case (c), equation (20) becomes 43 ) = (2m-l)sin0 43 (23). ANHARMONIC RATIO OF FOUR ZONE-CIRCLES. 103 If in case (d) 4l becomes 90, then equation (22) becomes (wi - 1) sin (20 W + < a ) = (+!) sin 14 sn 03, Pi-P If the reader computes the right side of this equation for the two sets of poles given for illustration, he will find the value to be -1. We shall retain the arrangement recommended in Chap, viii., which corresponds to the value 1-^-2. HARMONIC RATIO. 109 relation can also be expressed by stating that one pair of planes bisects the angle between the other pair internally and externally, the bisectors being, consequently, at right angles to one another. Thus we may suppose planes P 1 and P 3 to be at right angles, and P 3 to bisect the angle ^, so that 2s = si- The left side of equation (10) of Chap. vui. has in this case the value 1^-2, whatever may be the indices of the planes and the value of the angle &. For, making we have sin & _ L sin 13 sin ^ sin ^ 2 sin . Similarly, any other pole Q is repeated over (ANB} in a like pole Q', so that the zone-circle [QQ'] is at right angles to (ANB); and arc QN=NQ', N being the point of intersection of (ANB) and [QQ']. The two zone-circles [PP 1 ] and [QQ'], being both perpendicular to (A NB), must meet in a point T which is at 90 to (ANB) and must be a possible pole (Chap. VH. Art. 6). We have therefore established the first part of the proposition ; viz. that a plane of symmetry is parallel to a possible face of FlQ 82 the crystal. (&,) Again, in Fig. 82, T, P, P' are three tautozonal poles of the crystal, and M is a point in the zone such that MT =. 90, and PM = MP'. Hence, the four points form a harmonic ratio, and M is, therefore, a possible pole. In the same way the four points T, Q', N, Q form a harmonic ratio and three, T, Q', Q, are possible poles, no two of which are opposite. Therefore, N is a possible pole. Hence (ANB), containing two poles, is a possible zone-circle, and the normal to its plane through T is a possible zone-axis. (6 2 ) The same result can be obtained from a consideration of A DYAD AXIS IS A ZONE-AXIS. Ill the crystal-edges independently of the first condition. The edges are repeated over a plane of symmetry in pairs, which are reciprocal reflexions, such that lines drawn through a point in the plane 2 parallel to the edges lie in a plane at right angles to 2. Hence, the plane, parallel to two homologous edges which are symmetrical with respect to a plane 2, is parallel to a possible face and is perpendicular to 2. The same is necessarily true of every other pair of homologous edges. Hence, we must have a number of possible faces all perpen- dicular to the plane of symmetry ; and their lines of mutual inter- section must be all perpendicular to it. The plane of symmetry is therefore perpendicular to a possible zone-axis. 4. PROP. 2. To prove that an axis of symmetry is a possible zone-axis. Let the line OA in Fig. 83 be a dyad axis. Through any point in it draw a line OB parallel to any edge of the crystal which is neither parallel nor perpendicular to 0A, and suppose the plane containing OA and OB to be that of the paper. Then, by the definition, a semi-revolution about 0A must bring OB to the position OB t , which is parallel to an edge of the crystal similar to the first one. But OB t is necessarily in the same plane with 0A and OB, for a second rotation of 180 in the same direction will bring OB t back to OB ; and this is, clearly, not possible, if OB t stand out of the paper to the front or the back. Hence, OB and OB t being zone-axes, their plane is parallel to a possible face. Similarly, any other zone-axis OA, which is neither parallel nor perpen- dicular to OA and which is not in the plane OB&, is repeated in another zone-axis OA t \ and the plane AO&A t also contains OA. As the plane AO&.A t contains two zone-axes, it is parallel to a possible face. Hence, OA is parallel to two possible faces, and is, therefore, a possible zone-axis. COR. 1. In the case of a tetrad axis, we can turn the crystal about it through 90 four times in succession until the crystal returns to the original position. Hence, the edge LM t of Fig. 84 takes up three new positions, which are obtained by turning 112 AN AXIS OF SYMMETRY IS A ZONE-AXIS. FIG. 85. the crystal about the tetrad axis through 90, through 2x90= 180, and through 3 x 90 = 270. One of these positions, when the crystal has been turned through 180, is clearly LM' . But the face parallel to the edges LM t , LM', is also parallel to the tetrad axis OL. Similarly, the plane OLM is parallel to a possible face. Hence, a tetrad axis is a possible zone-axis. COR. 2. The same is also true in the case of a hexad axis ; for if the crystal is turned thrice in the same direction, each time through 60, a position is attained which is the same as that given by a single rotation of 180 about the same axis. Thus the edges VA t , VA, Fig. 85, are coplanar with the hexad axis 0V. Similarly, the planes OA V, OA it V contain two edges and are parallel to possible faces. Hence, a hexad axis is, also, a possible zone-axis. The same argument would apply to any axis of symmetry of even degree, but we shall see in Prop. 6 that no axis can have a degree higher than six. COR. 3. Suppose an axis of symmetry of odd degree, n, to exist in a crystal, n being > 3. Let the continuous lines VA, VA\ 3, it has been seen in Prop. 2, Cor. 3, that VM, VM 1 , &c., would be possible edges. Then VMM 1 , fec., would be possible faces belonging to a second pyramid. But, from Fig. 86, it is clear that the pair of opposite possible faces VAA\ VMM' intersect one another in a straight line parallel to AA 1 and MM 1 and perpen- dicular to 0V. The same is true of all the opposite pairs of possible faces. Hence, 0V is perpendicular to a number of possible edges, and is therefore perpendicular to a possible face. 6. It is not possible to establish, as a consequence of the law of rational indices, that a triad axis is, when alone, a possible zone- axis, or that it is perpendicular to a possible face. Let, in Fig. 88, VO be a triad axis, and VM a zone- axis of the crystal which is not parallel or perpendicular to 0V. Let MO be drawn perpendicular to the triad axis and meet it at 0. Rotation of the crystal about VO through 120 must bring VM to VM, and OM to OM, , ^ gg so that the triangle VOM t is not coplanar with VOM. A further rotation of 60 is required to bring OM t into the continuation Op of OM. But this is not a possible angle of rotation about a triad axis, nor is there an edge Vp. associated with VM. A second rotation of 120 in the same direction as the first brings the triangle into the position VOM ti so that OM it lies as far beyond Op. as OM t was short of it. A third rotation of 120 clearly brings the line back to VM. The proof of Prop. 2 depends, in the case of an axis of sym- metry of even degree, on the fact that two homologous zone-axes are coplanar with the axis; and in the case of an axis of odd degree, n > 3, the proof depends on the fact that two non-homo- logous edges, such as VA and VM, are coplanar with the axis. Here the homologous zone-axes form the edges of a trigonal pyramid, and no pair of them lies in a plane containing the triad axis. Nor can an auxiliary pyramid be formed by the edges of alternate faces. Again, the proof that a possible face is perpendicular to an axis PLANE AND CENTRE OF SYMMETRY CONJOINED. 115 of symmetry depends on the fact that pairs of possible faces intersect in lines perpendicular to the axis. The homologous faces, such as MVM t , M t VM lt in Fig. 88, intersect in VM t which is not at 90 to the triad axis, for OM t is at 90 to VO. Hence, we cannot prove that the triad axis is perpendicular to a possible face. But, although it is not possible to prove that a triad axis, when it is alone or associated only with a centre of symmetry, is a zone-axis perpendicular to a possible face, we can prove that it is so when it is associated with a plane, or with axes, of symmetry. It is also found to be an actual zone-axis, and perpendicular to faces, in all the known crystals in which triad axes exist. Hence, we shall assume that, where a triad axis exists, it is a possible zone-axis and perpendicular to a possible face. Hence, generally, an axis of symmetry is (1) parallel to a possible zone-axis, and (2) perpendicular to a possible face. 7. PROP. 4. If a crystal has any two of the three elements of symmetry a plane, a centre, and an axis of even degree perpen- dicular to the plane then it must also have the third element. (a) Let, in Fig. 89, ABCD be a plane of symmetry, 2 ; and let 0, a point in 2, be the centre of symmetry. Let OP be the normal on any face ABE, and OP l the normal on the corresponding homologous face ABF on the opposite side of the plane 2; and let 0A be the normal on the plane ABCD. ' The plane containing OP and OP 1 is perpendicular to 2, and must contain 0A. Now, since the crystal has a centre of sym- metry there must be faces at both ends of a normal continued through the centre. Hence, OP 2 and OP 3 , the continuations of OP and OP lt are perpendicular to faces FCD and ECD which are parallel to ABE and ABF respectively (Euclid xi. 14). The four faces form a four- sided prism, like the roof of a long building and its reflection in a horizontal plane passing through the eaves. It is clear that rotation through 180 about the normal 0A interchanges OP and OP 3 , OP 1 and OP Z , and the corresponding faces. For 0A is the external bisector of the angle POP lt and, consequently, bisects the angles POP S and PflP*. The same relations must hold for each set of four faces 82 116 CENTRE AND DYAD AXIS CONJOINED. symmetrical with respect to ABCD and the centre 0. They must be interchangeable in pairs about the normal 0A on ABCD. Hence rotation through 180 about the line 0A interchanges homologous faces. But 0A, being the normal on a plane of symmetry, is a zone-axis perpendicular to a possible face (Prop. 1). Hence, 0A is an axis of symmetry of at least twofold rotation. The possibility of rotations through 60, or 90, is not excluded. For three rotations in the same direction, each of 60, about <9A bring OP to the position OP 3 ; and similarly, two successive rotations, each of 90, are equivalent to a single rotation of 180 about the same axis. The axis may, therefore, be one of dyad, tetrad, or hexad symmetry ; i.e. it is an axis of symmetry of even degree. (6) When a crystal has a centre of symmetry 0, and an axis of symmetry of even degree 0A, it can, by the aid of the same diagram, be shown that the plane ABCD, perpendicular to OA, is a plane of symmetry. For rotations through 180 about 0A are always possible and interchange pairs of normals OP and OP Z , where all three lines are co-planar and OA bisects the angle POP 3 . Owing to the centre of symmetry each normal is diplohedral, and the faces at opposite ends of a normal are necessarily parallel. But parallel planes intersect a third in parallel lines (Euclid xi. 16). Hence the edges AB and CD are each parallel to the edge ^A, in which the faces ABE and DCE meet. The edges are therefore all perpendicular to the plane containing 0A, OP, OP 3 , and therefore to 0A. The plane, containing the edge AB and the line through in the plane POP 3 which is perpendicular to 0A, bisects the angle POP n and therefore the angle between the faces of which OP and OP l are the normals. A similar relation can be found for every other set of two pairs of parallel faces, which are also interchangeable, in pairs, by rotation through 180 about 0A. Hence, the plane ABCD is a plane of symmetry; for it bisects the angles between homologous pairs of faces, and is parallel to a possible face and perpendicular to a zone-axis. (c) If a plane of symmetry, ABCD, and an axis of symmetry of even degree, OA, perpendicular to ABCD coexist in a crystal, then parallel faces must be invariably present, i.e. the crystal is centro- symmetrical. The proof is obtained from the diagram, already used, by taking the relations of the lines and faces in a different order. OP and OP 1 are symmetrical with respect to ABCD which TRIAD AXIS AND CENTRE OF SYMMETRY. 117 bisects the angle between them and is perpendicular to their plane. Hence, the plane PflP contains 0A. Again, 0A bisects the angle POP 3 , and OP 3 lies in the plane containing OP and 0A. Hence, OP lt OP, OP 3 all lie in one plane, and POP 1 + POP 3 = 180. Therefore OP l and OP 3 are opposite directions on the same straight line. The faces ABF and DCE at the opposite ends P : and P 3 must therefore be parallel; and similarly, ABE and DCF are parallel faces. The same is true of every other set of faces which are homologous with respect to 2 and to OA. The crystal is therefore centro-symmetrical. Hence, the presence of any two of the three elements of symmetry a centre, a plane and an axis of even degree perpen- dicular to the plane of symmetry always involves the presence of the third element. 8. COR. The proof given above also establishes a very im- portant relation of the triad axis ; viz. that in a centra-symmetrical crystal having an axis of triad symmetry, the plane perpendicular to the axis cannot be a plane of symmetry. For the axis perpendicular to a plane of symmetry in a centro-symmetrical crystal must be an axis of symmetry of even degree. There is nothing to exclude the possibility of a triad axis and a plane of symmetry perpendicular to it occurring together in a crystal ; but in such a case the crystal cannot be centro-symmetrical, and parallel faces, if they occur, are to be regarded as belonging to different forms and will often be distinguished by a difference of character ; i.e. such parallel faces are not homologous. 9. We shall now consider the limitations to the number of planes of symmetry possible in a zone. We shall, thereby, see that the arrangements possible are such that the least angles between planes of symmetry are 90, 60, 45 and 30. PROP. 5. To prove that the only angles, less than 60, possible between planes of symmetry are 45 and 30. Let, in Fig. 90, P and Q be the poles of planes of symmetry such that no pair of planes of symmetry in their zone make with one another a less angle than PQ = < (say). Now, P must be repeated on the other side of Q in a similar pole P 1 where P^Q = QP ; and similarly, Q must be repeated in a similar pole Q.}. Suppose the planes of symmetry 118 ANGLES POSSIBLE BETWEEN PLANES OF SYMMETRY. to be perpendicular to the paper, so that the poles P, Q, P lt Ac., all lie in the primitive, where PQ = QP l = P^ = &c. = <. But, since P and Q are the poles of planes of symmetry, so must also the poles P lt $ 2 , &c., be poles of planes of symmetry. Let the successive poles be inserted in the zone until ft, the last pole before P, is reached, where P is the opposite pole to P. Now the next step gives either P, or a pole S beyond it. But P is also the pole of a plane of symmetry that first taken and RP, SP, are both less than RS = . This contravenes the limi- tation on the poles P and Q first taken; viz.- that no planes of symmetry in the zone make with one another a less angle than <. Hence, in proceeding beyond JK the next pole must be P, and RP . The angle < is therefore an exact submultiple of 180. Call the number n, then n< = 180. If, now, n is greater than 3, there must be at least three other poles between P and P, and we can apply the anharmonic ratio of four tautozonal poles to investigate the possible values of n and <. But, if (j> is 60 or 90, the anharmonic ratio is inapplicable, since pairs of poles are at 180 to one another. Supposing <60, we have = m > where m is some positive rational number. sin PP i sin QQ _ sin 2 2 = 45 and n - 4 ; and when 6 = 60, = 30 and n = 6. The value of the anharmonic ratio is, in the former case, m = 2 ; and in the latter, m = 3. 10. Proof. Series (3) (Todhunter's Trig., p. 226, 1859), and the properties of equations having integral coefficients, enable us to prove that the only angles <120, which are exact submultiples of 360 and have rational cosines, are 90 and 60. 2 cos nB = (2 cos 0) - ^ (2 cos 0)~* + n ( n ~^ (2 cos 0)-_,fec. . . . ( 3 ). m = ~ ANGLES POSSIBLE BETWEEN PLANES OF SYMMETRY. 119 The rth term is the coefficient of (2 cos #) n - 2r being integral. But, when nO = 360, cos nO = 1 ; and writing x for 2 cos 0, we have the equation a* - nx n ~ z + n( " U 2 ~ 3 ^ a"" 4 - &c. = 2 ............ (4). Further, the commensurable roots of an equation of the nth degree, which has integral coefficients and unity for that of the highest term, are integral and exactly divide the constant term. It follows that the possible values of x are limited to 2, + 1, and 0. If 2cos0 = 2, 6 is or 180. The former value is absurd, and the latter gives < = 90. This is a possible angle between planes of symmetry, but it is not admissible as a solution of equation (1), for it gives a meaningless anharmonic ratio. If 2cos0 = + l, 6 is 60 or 120, and is 30 or 60. The former establishes that the least angle possible between planes of symmetry is 30, and that we cannot have more than six such planes in a zone. The latter value (60) is a possible angle between planes of symmetry, but it is not admissible as a solution of (1) ; for, if four successive poles are taken at 60 to one another, the extreme pair are at 180, and the anharmonic ratio is meaningless. If 2 cos 6 = 0, = 90, = 45 and n = 4. This is the only other solution possible, and is that given when n is even and the last term of series (3) is 2. It follows that the greatest number of planes of symmetry in a zone is six in one case and four in the other 1 . Crystals may, also, have only two, or three, tautozonal planes of symmetry. 1 The proof of this important proposition given in the text is due to Professor Story- Maskelyne, who gave it in a course of lectures attended by the author in 1869. Professor von Lang (Kryst. p. 75, 1866) gives an expression, equivalent to (1), limiting the angles of isogonal zones, i.e. of zones in which equal angles recur. He points out that 30, 45, 60 and 90, satisfy the equation ; but he does not prove them to be the only angles which do so. In a classical memoir, published in Ada Soc. Sci. Fennicce, ix. p. 1, 1871, but read on 19 March, 1867, Professor Axel Gadolin finds that the angles of rotation about axes of symmetry are subject to equation (2) ; and he proves, by the method used in Art. 10, that 60 and 90 are the least possible angles. His method of arriving at equation (2) is, however, not altogether satisfactory ; for it is not applicable to the case of a tetrad axis. A different method has, therefore, been adopted in Art. 11. 120 DEGREES OF AXES OF SYMMETRY. Con. 1. No crystal can have five tautozonal planes of symmetry ; for the least angle between them is 36, and this angle does not satisfy equation (2). COR. 2. Planes of symmetry inclined to one another at angles of 45, and also at 30, cannot coexist in one and the same zone. It is clear that a zone-circle cannot be divided by poles at distances of 30 to one another and also at distances of 45 without some of the poles making angles of less than 30 with one another. 11. PROP. 6. To prove that tetrad and hexad axes are the only axes of symmetry of degree higher than three which are possible in crystals. Suppose A n to be an axis of symmetry of degree n (> 3) occur- ring alone in a crystal, and let it be placed vertically. By Props. 2 and 3, A n is a possible zone-axis perpendicular to a possible face. If then a face meets the axis at a point V, we have a pyramid of n similar faces all passing through the apex V and meeting the hori- zontal base in possible edges. We proceed to show that n possible faces can be found parallel to A n and inclined to one another in succession at an angle < = 180H- W . When n is even, n -f- 2 faces through opposite polar edges are possible, each of which contains the axis of symmetry. Thus, in FIG. 92. Fig. 91, the planes M t LM', MLO, both containing the tetrad axis OL, are possible faces inclined to one another at an angle of 90. Similarly, in Fig. 92, the planes A t VA', AVO, A JO, containing the hexad axis of symmetry OF, are possible faces inclined to one another at angles of 60. Generally, therefore, we have n-f-2 possible tautozonal faces inclined to one another in successive pairs at an angle 6 = 360 H- n. ISOGONAL ZONES. 121 If now the alternate faces of the pyramid are extended to meet one another, we obtain in Fig. 91 horizontal edges through L, parallel to OX and OY, which bisect the angles M t OM, MOM'. Through these edges and the axis OL possible faces can be drawn ; and we have four faces through the axis OL inclined to one another in succession at angles of 45. In the general case, when n is greater than 4, we obtain new polar edges, such as VM i of Fig. 92, through which and the axis w-=-2 possible faces can be drawn bisecting the angles between adjacent faces of the first set. Thus, for instance, the face OVM t bisects the angle between OVA and OVA t , We thus have, altogether, n tautozonal faces through A n inclined to one another in succession at an angle The same can be shown to be the case, when n is odd and > 3. Thus, in Fig. 93, the face through the axis and the possible edge VM bisects the angle between the pos- sible faces through VA 3 and VA* FIG. 93. parallel to the axis. Fig. 90 may now be taken to represent the poles of the n tauto- zonal faces through the axis A n , inclined to one another in succession at angles <, where n are restricted to 60 and 90 ; and the axis can only be a hexad or a tetrad axis. Similarly, 72 not being a possible value of 6, a pentad axis is inadmissible. The only axes of symmetry possible in crystals are therefore those of two-, three-, four-, and six-, fold symmetry. 12. Isogonal zones. Zones in which one of the crystallometric angles, 60, 45 or 30, recur are common in cubic crystals and in crystals having a principal axis. We can, however, show that, even when the faces are not planes of symmetry, no pair of faces can include an angle of 30, or 60, in a zone in which 45 recurs in succession ; and, vice versa, that 45 cannot occur in a zone in which either 30, or 60, recurs in succession. 122 ISOGONAL ZONES. Thus, in Fig. 94, let PQ=QP / = P i Q l =&c. = (j>, where is either 45 or 30. Take any possible pole R in the zone, and let f\PR = a. Also, let S be a point such that RS= a rational number. Similarly, from (6), we have where By expanding the left side of (11) and substituting for cot a from (10), we have n^ _ cot a cot ,-! _ m t cot 0, - v /3 V3~ cota + cot0 / ~ EULER'S THEOREM OF ROTATIONS. 123 If $, = 30, and ^ = ^J 3 . ... n,-3^ .................. (13). ^/3 771, + ti ^-{-O Hence, w, and n are rational, and a pole S is possible at 30 to any other pole R in a zone in which 30 recurs in succession. If, however, $, = 45, .*. cot$, = l ; and m-3 Hence, , is irrational, and therefore n. There cannot, therefore, be a pole S at 45 to a pole R in a zone in which 30 recurs in succession. By making $, negative in the above equations, we can obtain similar relations which hold for the case in which S is on the same side of R as P. Similarly, it can be proved that 45 and 60 do not occur together in a zone, in which one or other of the angles is repeated in succession. 13. PROP. 7. Euler's Theorem. Given two axes of rotation, OA and OB, intersecting at within a body ; it is required to prove that successive rotations about them of 2a and 2/3 respec- tively are equivalent to a single rotation through an angle 2y about a third axis OC, and to find the position of OG and the angle 2y. Let a sphere of any radius be described about the point of intersection as centre : and let A and B, Fig. 95, be the two points at which the axes emerge when the rotation about each of them appears to the observer to be in the same direction, either both with, or both against, the hands of a clock. Let AB be the arc in which a great circle through A and B meets the sphere. Through B draw the great circles BC, BA' and BC t , where A ABC = A GBA' = A A'BC = ft, one-half the angle of rotation about the axis OB. On A'B cut off an arc A'B-AB; and through A' draw the great circles A'C and A'G t on opposite sides of A'B, and each making with A'B an angle a, one-half the angle of rotation about A. Then OC is a line, rotation about which through the angle ACA'(= 2y) is equivalent to successive rotations, counter-clockwise, of 2/3 about OB and 2a about OA. The three spherical triangles ABC, CBA', A'BC, have their sides and angles equal, each to each. For, in ABC and A'BC, the two sides AB, BC are equal to the two sides A'B, BC ; and the included angle ABC to the angle A'BC. Hence, A'C ' = AC; and 124 EULER'S THEOREM OF ROTATIONS. A A'CB= A ACB, and A CAB = A CA'B. Again, in the triangles A'BC, A'BC t two angles of the one are equal to two angles of the other, and the adjacent side A'B is common. Hence, the other angles and sides are equal ; viz. A'C = A'C t and BC = BC r If the body and sphere (supposed to be rigidly connected together) are turned, counter-clockwise, about OB through 2(3, A is brought to A' and C to C r If, afterwards, the body and sphere are turned through 2a, again counter-clockwise, about OA' (the new position of the axis OA}, the point C 4 , where C had been left by the first rotation, is brought back to C. After both rotations the radius OC retains its original position ; and the body is in the same position as it would be in after a single rotation, counter-clockwise, about OC through the angle AC A' ; for A' is the final position of A. If, however, the rotations are taken clockwise in the same order, a new point C //} situated on the other side of AB, is the extremity of the equivalent axis of rotation ; where C tl is the vertex of a triangle equal and similar to CAB, having AB for base and the angles at A and B equal to a and ft respectively. If the body is first turned about OA and then about OB, the same two axes OC and OC lt are obtained. The former is the equi- valent axis, when the rotations are clockwise ; the latter, when they are counter-clockwise. Hence, A and B being the points at which the axes meet the sphere when the rotations about them appear to be in the same direction, the positions of the equivalent axes of rotation are given by C and (7 /; , the vertices of the two triangles which have AB for base and the angles at A and B equal respectively to a and (3, one- half the angles of rotation about the axes. The angle of rotation 2y is, in each case, double the external angle of the triangle at the vertex. The proposition has only been proved for pure rotations, but it holds true for axes of screw rotation. 14. It is now clear that, if there are two axes of symmetry in a crystal inclined to one another at an angle other than 180 the latter angle giving only two ends of one and the same axis there must be at least one other axis of symmetry, of which the position and the angle of rotation are determined by the above theorem. For suppose the axes of symmetry to be parallel to OA and OB of the preceding proposition ; and let rotations of 2a and 2(3 about OA and OB, respectively, interchange, in each case, homologous faces COMBINATIONS OF AXES OF SYMMETRY. 125 and edges. Then rotation about OB through 2/3 brings OA to OA', and interchanges homologous faces. The crystal has, therefore, the same aspect as at first. Rotation in the same direction about OA' (the new position of OA) through 2a also interchanges homologous faces, and leaves the aspect of the crystal the same as at first. But the final position of the crystal can be obtained by a rotation of 2y in the same direction about OC, which is, therefore, a third axis of symmetry. It is necessary to prove that the least angle of rotation about OC is, in each case, an exact submultiple of 360 ; and that OC satisfies the conditions for axes of symmetry given in Props. 2, 3 and 6. The same is true of the other equivalent axis OC t/ , which gives a fourth axis of symmetry when this line is not (as is often the case in crystals) in the continuation through the centre of the axis OC. The new axis, or axes when C and G tl are not extremities of a diameter may, when combined in a similar manner with each of the original axes, give the positions of new axes of symmetry. In order to obtain all the axes of symmetry, the process has to be repeated until every combination is shown to give an axis already established. It has thus been shown that the greatest number of axes of symmetry, which occur together in any crystal, is thirteen. 15. PROP. 8. If two planes of symmetry are present in a crystal, and are inclined to one another at the least angle, , between planes of symmetry in their zone, then their line of intersection is an axis of symmetry, the least angle of rotation about which is 2, and the degree of which is n = 360 -=- 2<. Let the two planes of symmetry meet the sphere in the great circles 2 and S, Fig. 96, and let their line of intersection be perpendicular to the primitive. Then over S the plane 2 must be repeated in a like plane of symmetry 2 t . Let the points in which the adjacent planes S, 2, &c., meet the primitive be A, B, A lt &c. Let P, P lt P s , P 3 , &c., be the projections of homologous poles sym- metrical with respect to 2, S, 2 1? &c. ; and let the great circles through C and each of these poles meet the primitive in Z), D lt Z) 2 , Z> 3 , &c. Since 2 and S are planes of symmetry, they are parallel to possible faces and perpendicular to possible zone-axes. Hence the plane of the primitive is parallel to a possible face, and the radius through C is a possible zone-axis the line of intersection of 2 and S. It, therefore, satisfies the conditions (Arts. 4 6) for an axis of symmetry. It is now necessary to show that rotation about it through a definite submultiple of 360 interchanges like faces. 126 AXIS PARALLEL TO SEVERAL PLANES OF SYMMETRY. Now the arc joining a pair of homologous poles, such as P l and P 2 , is perpendicular to, and bisected by, the plane of symmetry S with respect to which they are symmetrical. Hence, we have P 1 L=LP 2 , and CL is common to the two spherical triangles CLP^ CLP Z . The angle CLP l is also equal to CLP^ each being a right angle. The two triangles are therefore equal in all respects, and CP 1 = CP 2 , and f\P l CL = [\ P 2 CL. But these angles are the same as the arcs BD 1 , BD 2 measured on the primitive. Hence BD l = BD 2 . Again, for a similar reason,(7P= CP^ and the angle PCA = A PiCA, or the arc But, AB= Hence, FlG - 96 ' Also, D 1 D 3 =D 1 D z +D 2 D Hence, if the sphere is turned through 2$ about the diameter through C, the pairs of similar points A, A v ; D, Z) 2 ; B l , B ; &c., interchange places simultaneously, and the zone-circles CA, CA t ; CP, CP 2 ; CP^ CP 3 ; change places respectively. But CP=CP 1 = CP 2 = &c. Hence, the similar planes of symmetry and the homologous poles are interchanged at the same time. The line of intersection of the two planes S and 2 is, there- fore, an axis of symmetry, the least angle of rotation about which is 2$ ; and the degree is w = 360-f2<. COR. i. If the least angle between pairs of tautozonal planes of symmetry is 90, the line of intersec- tion is a dyad axis, and the distribu- tion of homologous faces is such as is shown in Fig. 19 and in Fig. 97. The two like planes 2 and Si of the genera] proof fall into the same plane, and the two planes S T and 2 are not like or interchangeable. In Fig. 97, one plane is XOZ, the other is YOZ. COB. ii. If the least angle between the tautozonal planes of symmetry is 60, the axis is one of triad symmetry. There are three planes of symmetry in the zone which are like and inter- changeable planes. The distri- bution of homologous faces with respect to the planes and axis of symmetry is shown in Fig. 98. FIG. 97. AXIS PARALLEL TO SEVERAL PLANES OF SYMMETRY. 127 khl FIG. 99. COB. iii. If the least angle between S and 2 is 45, there are four tautozonal planes of symmetry consisting of a pair of like S planes and a pair of like 2 planes. The like planes are inter- changeable with one another, but those of one set are not inter- changeable with those of the other. The line of intersection is a tetrad axis. The distribution of homo- logous faces is illustrated by the faces meeting at the apex L in Fig. 99. The planes of symmetry S and S' are LOM and LOM t , the planes 2 are LOH and LOK', and the axis OL is the tetrad axis. COB. iv. When the angle = 30, there are three like planes S at 60 to one another and three like planes 2 also at 60 to one another, so arranged that the planes of one set bisect the angles between pairs of the other set. The S planes are interchangeable with one another, as are likewise the 2 planes, but those of one triad are not interchangeable with those of the other. The line of intersection is an axis of hexad symmetry. The distri- bution of homologous poles and faces is given in Figs. 100 and 101. As will be explained in Chap, xvn, two sets of symbols, {hkl} and {pqr}, are needed to give all the homologous faces. In Fig. 100, the planes of symmetry 2 are those passing through the centre and the points X, Y and Z, respectively, on opposite FIG. 101. 128 AXES PERPENDICULAR TO W-FOLD AXIS. sides of which poles of {hkl} are symmetrically placed ; as also are the poles of {pqr}. The other triad of planes S bisect the angles between planes of the first triad, and in Fig. 101 pass through the polar edges V8, V8 t , V8 /t . They have a pole and face of the set {hkl} symmetrically placed to one of {pqr}. COR. v. Since 30 is the least angle possible between planes of symmetry, it follows that no crystal can have an axis of symmetry, in which planes of symmetry intersect, of higher degree than six. Moreover, since 36 is not a possible angle between planes of symmetry (Art. 10), a pentad axis cannot be the line of intersection of planes of symmetry. 16. PROP. 9. Given an axis of symmetry of degree n, not exceeding 6, and one plane of symmetry parallel to it, to show that there are (n - 1) other planes of symmetry all parallel to the axis. By rotation about the axis the given plane 2 is brought into (n 1 ) other positions which must be those of like planes of symmetry and the angle between which is 360-=- ft. =2$. But, when n is even, we thus get only -h2 different planes of symmetry ; for, if m = ?i-=-2, the result of m successive rotations of 2< is to turn the plane of symmetry through 180, when it again coincides with the original plane. But the last proposition has shown that the degree n of an axis of symmetry, which is the inter- section of planes of symmetry, is given by the equation, n = 360 -j- 2, where 20 is the least angle of rotation about the axis and is double that between adjacent planes of symmetry. Hence, since w< = 180, there must be n distinct planes of symmetry all intersecting in, or parallel to, the axis of symmetry. 17. PROP. 10. If a dyad axis 8 is perpendicular to an axis of symmetry A n of degree n, there must be (n- 1) other axes of even degree all perpendicular to A n . If the axis is a triad axis, rotation about it brings 8 into two new and different positions, so that there are three like dyad axes at right angles to A 3 . In the case of an axis A n of even degree, where n=2wi, each rotation about A n through 360 -i- n brings 8 into the position of a like dyad axis. But the mth rotation gives no new direction, for the whole rotation is m x 360-=- 2m = 180, arid the axis is in the prolongation across the origin of the original axis. Hence, rotation about A n , gives only m different dyad axes. But, by Euler's theorem, two rotations one about A n through 360 -=-n followed by one of 180 about 8 are equivalent to a single rotation about A, where A is the vertex of the triangle shown in Fig. 102. AXES PERPENDICULAR TO 71-FOLD AXIS. 129 The angle at A is clearly a right angle, and the axis A is at least a dyad axis perpendicular to A n . The same is true of a combination of A n with each of the 8 axes. There are, therefore, m like dyad axes A, each of which bisects the angle between pairs of 8 axes. The axes A, A, may, when n = 4, be tetrad axes. 18. PROP. 11. If two axes of symmetry of even degree parallel to a certain face are inclined to one another at the least angle, = 180 -z-n, between axes parallel to the face; then there are n axes of even degree parallel to the same face inclined to one another in succession at the angle 0, and an axis of symmetry of degree n perpendicular to the face to which the n axes are parallel. Let 8 and A be two dyad axes inclined to one another at the least angle, = 180 -rtt, between axes parallel to their plane. Describe a sphere, and take two diameters 88, A A, Fig. 102, parallel to the axes, and let them lie in the plane of the primitive and emerge at 8 and A. Rotation through 180 about the dyad axis A brings the axis 8 into a position in which it is parallel to the diameter 8,8,, so that we need only consider rotations about the diameters of the sphere. Simi- larly, a semi-revolution about 8,8, gives us the diameter A, A, parallel to an axis of symmetry. Repeating the process, a dia- meter R is reached such that at the next repetition the original diameter, but in a reversed direction, or a diameter T beyond it is attained. But if rotation about R gives T, then the angle between R and T is 0. But the original diameter lies between them and is also parallel to a dyad axis. The axes R and T are, therefore, inclined to the original axis at an angle less than 0. But this contravenes our first assumption, viz. that the pair inclined at the least angle was selected. Hence, the nth repetition must give the original diameter reversed in direction. The number of distinct dyad axes parallel to one plane is, therefore, = 180 -7-0 = 360-;- 20. Again by Euler's theorem, Art. 13, successive rotations about adjacent dyad axes 8 and A are together equivalent to a single rotation about the line perpendicular to their plane ; for the angles -4 n 8A, -4 n A8 are both 90. Furthermore, the angle of rotation about the axis emerging at A n is double the supplement of the angle 8 J B A. It is, therefore, 2 (180 - 0) = 360 - 20. But such a rotation about the diameter through A n brings the sphere and crystal into the same position as a rotation of 20 in the opposite direction. Hence, the least angle of rotation about the axis parallel to the diameter through A n is 20, and the degree n is given by 360 -r 20. L. c. 9 130 COMBINATIONS OF INCLINED LIKE AXES. The same proof applies when one of the original axes is a tetrad axis ; for two rotations, each of 90, in the same direction about it are equiva- lent to a single rotation of 180. For the purposes of the proposition we need only consider the latter rotation. Hence, the proposition is true when the adjacent pair consist of a dyad and tetrad axis. We shall see later on that it is needless to consider other arrangements ; for no crystal can have more than one hexad axis, or have two tetrad axes without a dyad axis midway between them. 19. PROP. 12. To find the possible arrangements of like axes of symmetry inclined to one another at finite angles. Suppose A l and J 2 to be like axes of degree n inclined to one another at the least angle ^ possible between such axes. About any point in A z as centre describe a sphere, and let A 2 meet it in 2 , Fig. 103. Through the centre draw a line parallel to A l meeting the sphere in a v Then the arc o 1 a 2 =^. Rotation about J 2 through ^a-^n brings A v to the position of a like axis A 3 and the radius Oo-, to a radius Oa 3 parallel to A 3 . Hence, a 3 a 2 =a 1 a 2 =^r, and the angle a l a^a 3 =Zrr-^-n. A similar rotation about A 3 brings A z into a position the direction of which is given by the radius to a t on" the sphere, where a 4 a 3 =a 3 o,j and the angle a i a 3 a. z =2ir-^-n. Proceeding in this manner the point a t is ultimately reached again and a closed polygon of p sides is obtained. The polygon encloses no point at which a similar axis to A t or A. 2 emergas and has no reentrant angles ; for its sides are the least possible between such axes, and on each rotation the interior angle, 2n-i-n, is less than 180, ex- cept when n = 2. In the latter case, the angle a 1 o 2 a 3 = 180, and the extremities of all the axes lie in a great circle. FIG. 103. Now the area of the spherical polygon is that of the p triangles formed by joining each of its angular points to a point within it, such as M the middle point But the area of a spherical triangle ABC is ~ (A + B+C- 180) ......................... (15). Hence, the area of the polygon ofp sides is jg^ (a 1 a 2 a 3 +a 2 a 3 a 4 +&c. + 360 -p . 180") = {2-p(-2)} ............... (16), where S is the surface of the sphere. COMBINATIONS OF INCLINFD LIKE AXES. 131 The area of the polygon is necessarily positive, and, consequently, the values of p are limited by the expression (17). 20. If the vertices a,, a 2 , &c., of the polygon are joined by great circles to the middle point, M, then a l M=a z M=*&c. = 6 (say) ; and we can readily find 6 and ^. Each of the triangles a^Ma^ given separately in Fig. 104, is an isosceles triangle, of which the angles at ctj and a 2 are 180" -7-71, one-half the angle of rotation about each of the axes, and the angle at M is 360 -rp. By Euler's theorem, OM is an axis of symmetry, an angle of rotation of 360 (1 - 2-T-/0 about which is equivalent to successive rotations of 2r-f- about A l and A. 2 . If, in Fig. 104, a great circle MD is drawn through M at right angles to a x a 2 , it bisects the angle a l Ma 2 and the arc aja 2 at the point D. Hence, by Napier's rules, =cot-cot- n p (18), (19). 21. We shall now consider the values of p given by expression (17) for the several possible values of n. i. Hexad axis. If n = 6, expression (17) becomes 12 - 4p > 0. Hence, the possible values of p are 1 and 2. The value p \ gives no repetition ; the polygon has only one side and is a circle. It is obviously inadmissible as a solution of the problem, for we supposed two axes A l and A t to occur. When p = 2, the polygon is a lune, i.e. the segment of a sphere bounded by two great circles. The sides are semicircles, and the apices are the opposite ends of one and the same diameter. Hence, no crystal can have more than one hexad axis. The value p=2, indicates that the distribution of faces about the two ends is similar, but the faces may, or may not, be interchangeable. By (16) the area of the lune is S+6. ii. Tetrad axes. When n=4, expression (17) becomes 8 - 2p > : and p = l, 2, or 3 are possible values. (a) The first gives, as before, no repetition. (6) The value 2 gives, as before, a lune but with area = S+ 4. We can, therefore, have a single tetrad axis with a like distribution of faces at the two ends, which may, or may not, be interchangeable. (c) The value /> = 3 gives for the polygon an equilateral triangle the area of which is S-r8. From equation (19) we see that ^=90. Hence, 92 132 COMBINATIONS OF INCLINED LIKE AXKS. we have three like and interchangeable tetrad axes mutually at right angles to one another. Also rotation about A z in the opposite direction to that first adopted clearly gives a similar quadrantal triangle. Hence, the new tetrad axis is the opposite extremity of that through a 3 . The middle point M of the spherical triangle is also the extremity of a triad axis ; and, by Euler's theorem, the point D of Fig. 104 is the extremity of a dyad axis. The whole arrangement of axes of symmetry, characteristic of two classes of the cubic system, is shown in the stereogram, Fig. 105. The three tetrad axes are indicated by letters T, the four triad axes by letters p, and the six dyad axes by 8. iii. Triad axes. When = 3, expression (17) becomes 6-p>0, and p may have the values 1, 2, 3, 4 or 5. (a) When p=l, there is, as before, no repetition. (6) When > = 2, the polygon is a lime with area S-r3. A single triad axis is therefore possible with a similar distribution of faces at both ends, which may, or may not, be interchangeable. (c) When p = 3 the polygon is an equilateral triangle, the area of which is S-r4. The sphere is divided into four equal triangles, the apices of which are at the coigns of the inscribed regular tetrahedron, Fig. 106. The triad axes, joining the coigns p in Fig. 106 (coincident with the points a ly &c. of Fig. 103) to the centre of the tetrahedron, will, when continued, pass through the middle points r of the opposite faces. Each of these continuations corresponds to an axis through the point M of Fig. 103. The radii in opposite directions along the same diameters are, therefore, to be regarded as dissimilar triad axes ; and the distribution of faces and edges at the opposite ends will be dissimilar. This character of an axis of symmetry is also described by saying that it is uniterminal. The angles between pairs of like triad axes Op' and Op,, and between adjacent dissimilar axes Or and Op', for instance, are obtained from equations (18) and (19), for ir-^n=ir -r-p=60. Hence cos (0=p'r) = l-=-3. Therefore p' Or =/>#?= 70 32'. But, since opposite ends of an axis lie in a diameter, p'0p // =180-p // 0r / = 109 28'. This is the same result as is obtained from (19) for cos ^/2 = cos 60 cosec 60 = 1 -7- ^3. Therefore FIG. 106. COMBINATIONS OF INCLINED LIKE AXES. 133 ^/2 = 5444', and 1 a i! =p'0p // = 109 28'. The arrangement of axes is characteristic of two merohedral classes of the cubic system. (d) When p=4, the polygon is a four-sided one having an area of S-r-6. Each of its sides is 70 32'; for from equation (19) cos^/2 = cos45cosec60 = >/2~-r l v/3. .-. ^-j- 2 = 35 16'. The figure inscribed in the sphere is the cube, the diagonals of which give the directions of the four triad axes p. We have here to distinguish between two possible cases according as A r and A t the axes first taken are (a) like and interchangeable, i.e. metastrophic, or () like but not interchangeable, i.e. antistrophic. FIG. 107. a. In this case the complete assemblage of axes is identical with that resulting from (ii c), and is shown in Fig. 107. The radii OT, &c., through the middle points M of the four-sided polygons are respectively perpen- dicular to the faces of the inscribed cube, and are three like and interchangeable tetrad axes, rotation about any one of which through 90 interchanges adjacent angles of the polygon, i.e. adjacent coigns of the cube. Furthermore, the radius to the point D of Fig. 104 is a dyad axis, and is parallel to the diagonal of one of the faces of the cube. For, by Eider's theorem, successive rotations of 120 about the triad axis A z and of 90 about the tetrad axis emerging at M are equivalent to a single rotation of 180 about the axis emerging at Z>, the vertex of the triangle a^MD. The dyad axes are the lines 08, 0V, &c., of Fig. 107. #. From the method of derivation adopted in Art. 19, the pairs of axes A l and A 3) A. 2 and A t , are necessarily interchangeable ; but the pairs A l and A<>, J 2 and A 3 , &c., though like axes, may be antistrophic. Again, Eider's theorem requires that M should be at least a dyad axis, for the angles aj J/a 2 , a 2 J/a 3 , &c., are each 90. This does not militate with its being a tetrad axis as in case (a), when A l and 4 2 are metastrophic. Hence, when 134 COMBINATIONS OF INCLINED LIKE AXES. A l and A 2 are antistrophic, the three lines through the middle points of opposite cubic faces parallel to the edges are only dyad axes ; and it can be proved that there are no other axes of symmetry. This arrangement of axes is characteristic of a subdivision of the cubic system of which crystals of pyrites give a good instance. (e) When p = 5, we obtain a five-sided polygon the area of which is S+ 12. The polyhedron inscribed in the sphere is the regular pentagonal dodecahedron, which is inadmissible amongst crystal-forms. For we have seen in Art. 20 that M, the middle point of the polygon, is the extremity of an axis of symmetry. Successive rotations about A l and A 2 are equivalent to a single rotation about the radius OM through 2 (180 -72), since, in this particular case, the angle ajJ/a 2 =72. The polyhedron always presents the same aspect after any number of rotations of this amount. But, two rotations give an angle of 2x360-4x 72 = 360 + 72. The polyhedron is, then, in exactly the same position as if it had been turned once in the same direction through 72 = 360 -4- 5, for complete revolutions of 360 cause no change in the position of the faces. Hence, the least angle of rotation about the axis through Mis 360 -4- 5, and the axis is a pentad axis. But this has been shown in Art. 11 to be inadmissible. No class of crystals can, therefore, show the arrangement of axes of this paragraph. iv. Dyad axes. When n=2, the expression (17) is independent of p, and the solution is indefinite. The axes all lie in one plane and the area of the polygon is a hemisphere. The possible arrangements of such dyad axes are governed by the relations given in Prop. 11. It is important to note that, in the case of tetrad axes, a crystal may have one or three, but no other number ; similarly, that a crystal may have one, or four, triad axes, but no other number; also that the angles between all the axes of symmetry associated together are fixed and definite. 22. So far the elements of symmetry have been discussed as if crystals were merely polyhedra the faces of which are subject to the law of rational indices. In recent years much attention has been paid to theories as to the internal molecular structure of crystals. It is interesting, therefore, to find that the uniformity of internal structure, which such theories presuppose, imposes the same restric- tions on the elements of symmetry as those established in the pre- ceding Articles. Considerations of this nature have also justified the acceptance of a triad axis as an actual zone-axis to which a possible face is perpendicular. We shall, however, only enter into the question to the extent needed to establish two propositions, viz. Props. 13 and 14. The assumption of the uniformity of internal structure of a crystal is based on the results of observation and experiment, which CRYSTALLINE STRUCTURE. 135 have established that a crystal has the same physical properties : (1) at all points on the same straight line, (2) for all parallel directions, and (3) for all homologous directions. Some of the physical characters have a higher symmetry than that manifested by the geometrical relations of the crystal-forms : as, for instance, the propagation of light. Thus, in a cubic crystal light is propagated with equal velocity in all directions. The cohesion is, however, different in directions which are not homologous ; as is indicated by the cleavages being limited to a few directions which are, in many cases, parallel only to homologous faces. Further, within the degree of precision attainable in estimating the perfection of a cleavage and the ease with which it is obtained, it is found that the cleavages in homologous directions are equally perfect and facile, but that, when cleavages occur parallel to faces of different forms, those in non-homologous directions generally show marked differences in their characters. The elasticity is another property which is only the same for directions which are parallel or homologous. The properties of cohesion and elasticity are, however, identical in opposite directions along the same line, though the facial develop- ment may be one which excludes symmetry with respect to a centre. Hence, such properties do not enable us to distinguish between related classes which differ, inasmuch as in the one class certain elements of symmetry occur alone, whilst in the other they occur in association with a centre of symmetry. The physical characteristics of crystals described in the preceding paragraph indicate that the internal structure is the same at all points within a crystal. Hence, the arrangement of the particles about any one of them must be the same as that about any other. This is only partially true of the particles at, or very near to, the surface, but the sphere of action between neighbouring particles is so small that only a very thin layer is affected. The surface- relations, known as surface-tensions, must exert a most important influence on the growth of a crystal, and must, more especially, be the determining cause in the development of the faces. M. P. Curie and Professor Liveing 1 have, independently, discussed the subject, and have endeavoured to estimate the relative values of the surface- 1 " Sur la formation des cristaux et BUT les constantes capillaires de leure differentes faces." BulL Soc. fran$. de Min. vm, p. 145, 1885. "On Solution and Crystallization." Tram. Camb. Phil. Soc. xiv, p. 370, p. 394, 1889; xv, p. 119, 1894. 136 POSSIBILITY OF HEX AD AXES. tensions of the faces of different forms in the same crystal and thus to account for the predominance of certain forms. It is, however, unnecessary for our purpose to enter into the questions which such considerations raise, and we need only consider the consequences involved in the regularity of internal structure. We shall suppose the size of the crystal to be indefinitely great in comparison with the distance between adjacent particles, or with the sphere of action of the particles on one another. PROP. 13. To prove that no crystalline structure, consisting of particles arranged in a regular manner at small but finite distances apart, can have an axis of symmetry of higher degree than six. The arrangement of the particles about any one of them being the same as that about any other, it follows that, if there is an axis of symmetry A whether it is an axis of pure, or of screw, rotation related to one set of particles, then every other similar set of particles must have a similar and parallel axis A similarly related to it. Further, the distance between the nearest parallel axes A cannot be made indefinitely small in comparison with the distance between adjacent particles. Hence, let us assume a pair of similar and parallel axes of symmetry, A 1 and A 3 , of degree n to meet a plane perpendicular to them at the points e^ and a a , Fig. 108, respectively; and let a^j be the least distance possible between any pair of such axes. Now, rotation about J 2 through the JQQ angle 27r4-n brings the axis A 1 to the position of a similar parallel axis A 3 which meets the plane at a 3 , where f\a 1 a. i a 3 = 2ir-r-n, and a 1 a 2 = a 3 a 2 . Draw the perpendicular a^d on a 1 3 . Then, from the right-angled triangle a^da^ we have a 1 c?=a 1 a 2 simr-=-. Hence, <* 1 a 3 = 2a 1 a 2 sin7r-=-w. But, by the selection of A 1 and A^ a^a z cannot be less than !. When a^a z is made equal to OjOg, sin TT-^W= l-=-2; and 180 + n = 30. Hence, 6 is the greatest value which can be assigned to n. 23. From the same considerations, we can show that the assumption of a pentad axis is inconsistent with the existence of a finite minimum distance between like axes of symmetry. PROP. 14. To prove that a pentad axis is inadmissible amongst the axes of symmetry possible in a crystalline structure. PENTAD AXIS INADMISSIBLE. 137 Let A t , A 2 be two parallel pentad axes having the least distance jor 2 between such axes. Let them meet the paper (placed at right angles to the axes) in a lt a 2 , Fig. 109. If, now, the crystal is rotated about A% through 360 -J- 5, the axis A l is transferred to A 3 , the position of a like pentad axis, and the point e^ to a,. Hence a* 2 a 3 =a 2 a l , and the angle 0^0203= 72. A similar rotation about A 3 brings the axis J, to the position of a like axis A 4 which meets the paper in 4 , where a 3 a 4 =a 3 a2, and the angle a 4 a 3 a 2 =72. It is clear that, if a 2 a x and a 3 4 are produced, they will meet, in V (say), for the two angles at a 2 and a 3 are each 72. The triangle Fa 2 a 3 is isosceles, and the portions a t a 2 and a s a 4 of the equal sides are equal. Hence aja 4 is parallel to a 2 a 3 ; and, by Euclid vi. 2, a t a t : 030%= Va^ : Foj. Hence, we have two pentad axes A l and J 4 , the distance between which is lass than the minimum distance 1 a 2 . Even if a 1 a 4 were selected as the initial pair, we can by the same process find pentad axes still nearer to one another, and this can be continued without limit. It is impossible, there- fore, to have a number of like parallel pentad axes separated from one another by a finite minimum distance. FIG. 109. CHAPTER X. THE SYSTEMS ; AND SOME OF THE PHYSICAL CHARACTERS ASSOCIATED WITH THEM. 1. THE principles laid down, and the relations established, in the preceding Chapters enable us to classify crystals ; and to show that only thirty-two classes are possible, which fall into seven larger groups called systems. The names adopted for the systems and classes by various authors differ considerably. We shall gene- rally adopt Miller's names for the systems and shall, likewise, give some of the synonyms employed by other crystallographers. The systems, and the classes included under each system, will be fully developed in the following Chapters. The systems may, however, be briefly defined as follows : 1. The anorthic (tridinic) system consists of two classes : I. Crystals with no symmetry; II. Crystals which have only a centre of symmetry. 2. The oblique (monoclinic, monosymmetric} system includes three classes:!. Crystals with a single dyad axis; II. Crystals with a single plane of symmetry ; III. Crystals in which a plane of symmetry, a dyad axis perpendicular to the plane of symmetry, and a centre of symmetry are associated together. 3. The prismatic (rhombic, orthorhombic, trimetric) system includes three classes of crystals, each of which has one set of three dissimilar zone-axes at right angles to one another, which are the most convenient lines to take for axes of reference. In class I the axes are dyad axes, and no other element of symmetry is present. In class II the three axes are, as before, dyad axes, but they are associated with a centre of symmetry and with three planes of symmetry, each perpendicular to one of the dyad axes. In class III one of the axes is a dyad axis, and is the line of inter- section of two planes of symmetry at right angles to one another ; DEFINITIONS OF SYSTEMS. 139 the two other axes of reference being the zone-axes normal to the planes of symmetry. 4. The tetragonal (quadratic, pyramidal, dimetric) system includes seven classes, and comprises all crystals having each a principal axis (p. 112) which is either a tetrad axis, or a dyad axis of special character. The special character of the dyad axis is due to the fact that like zone-axes at right-angles to it occur in pairs which are at right angles to one another ; but they are not inter- changeable, as is the case when the principal axis is a tetrad axis. 5. The cubic (octahedral, regular, isometric) system includes five classes. All cubic crystals have four triad axes, the directions of which are given by the diagonals of a cube : they have also three like and interchangeable rectangular axes, parallel to the edges of the cube, which are either dyad or tetrad axes. 6. The r/tombohedral system includes seven classes, the crystals of which are all distinguished by having each a single triad axis, which is a principal axis. 7. The hexagonal system includes five classes, the crystals of which have each a single hexad axis, which is also a principal axis. Miller doubted the correctness of regarding this last system as a separate one ; and, as will be shown in Chaps, xvi and xvn, the distribution of faces about the hexad axis is, in particular cases, the same as that of similar forms of the rhombohedral system. In other cases a single form of the hexagonal system can be represented as consisting of two correlative forms of the rhombohedral system, which can be interchanged by a rotation of 1 80 about the principal axis, and which are therefore connected together by a simple relation between the indices. On the other hand, foreign crystallographers have, until quite recent years, regarded rhombohedral crystals as forming important merohedral classes of the hexagonal system. The systems were first established by Weiss and Mohs from empirical observations of the development of the forms of crystals and approximate measurement of their angles. They seem to have arrived at the same main subdivisions independently, although Woiss's 1 classification was the first published. The subdivisions of the systems were only partly determined by them ; and the question, as to whether oblique and anorthic crystals constituted 1 De indagando formarum crystallinarum characters geometrico principal! dissertatio. Leipzig, 1809. Uber die natiirlichen Abtheilungen der Krystal- lisationssysteme. Abh. d. Berlin. Akad., 18141815, p. 289. 140 OPTICAL CHARACTERS OF CRYSTALS. independent systems, or were merely hemihedral and tetartohedral subdivisions of the prismatic system, remained a subject of contro- versy for a long period. Naumann (Lehrb. d. Kryst. II, p. 51, 1830) was the first to adopt oblique axes in. the representation of oblique and anorthic crystals, which he justified by strong reasons based on the observed differences in the forms commonly found on crystals of these systems and the prismatic. In the Chapters in which the systems are severally discussed, we shall see that it is not always possible to discriminate the class, or even the system, to which a crystal belongs by the geometry of the facial development. The physical, and more especially the optical, characters of the crystals afford useful tests which generally enable us to discriminate between crystals of different systems, and some- times between those of different classes in the same system ; and it is mainly on the optical characters of their crystals that we now rely in assigning to definite systems several minerals, such as, for instance, the humite group, harmotome, , D t , &c., when a : b : c and the angles between the axes, viz. TZ, ZX, and XY, are all known. It is necessary to find, from the known arcs YZ, ZX and XT, the angles YZX, ZXY, and XYZ (McL. and P. Spher. Trig, i, p. 47). Then 45= 180 - YZX, 5(7-180 - ZXY, and CA = 180 -XYZ. Hence, taking (3), for instance, we have sin BL = b sin AE sin CL ~ c sin CA ' All the numbers on the right side being known, the value of the term can be computed, and can, as in Chap, vui, Art. 14, be expressed by tan 0. If 6 is greater than 45, we invert the equation FORMULAE OF COMPUTATION. 163 before proceeding further. But we may suppose that BL is less than CL, and that 6 is, therefore, less than 45. . sin CL -unBL I- tan $ ' sinCL + sinBL l+tan0 Hence, as in the similarly formed expression of Chap, viu, Art. 14, we have tan | (<7Z -)= tan (45 -0) tan i (CZ + ^Z) (4). But CL + BL = CB is known. Hence the right side of (4) can be computed, and CL - BL found. Hence, CL and BL are both determined. If the same process is applied to (3*) and (3**), we can find the remaining angular elements. 20. Again, if the right sides of (3), (3*), (3**) are multiplied together, and also the left sides, we have &c a _sin.Z?Z sin CM sin AN sin CA sin AB sin BC cab~ u ~~ sin CL sin A M sin BN sin AB sin BC sin CA ' sin D sin E sin F t Hence, - =r- . = - =,- - 1 (5). sin D t sin E t sin F t We have therefore only five independent elements. 21. The chief problems of the crystallographer are : the deter- mination of the face-symbols and elements of the crystal when the angles are measured ; or, from a knowledge of the elements of the crystal and the face-indices, to determine the true values of the angles. The angles given in descriptive works are usually those calculated from the angles selected to give the elements. For the solution of these problems the anharmonic ratio of four tautozonal faces is the relation of most general applicability, and is one of great accuracy. In order to apply it to the solution of the first problem, the indices of at least three faces in the zone and all the angles must be known. Then the indices of every other face in the zone can be calculated by taking each in turn with the three known faces. For the converse problem, we must be satisfied as to the correct- ness of the symbols of the faces and of at least two angles which are not together equal to two right angles. If the given angles are adjacent, we use the transformation of the A. R. given in Chap, viu, Art. 14. If the two given angles are not adjacent, we must employ 112 164 ANORTHIC SYSTEM. the less convenient transformation given in Chap, vm, Art. 18. In crystals, such as that of anorthite, Fig. 120, having numerous faces, the solution can be carried out in a systematic way so as to determine, by the transformation of Chap, vm, Art. 14, most of the angles from a few known ones. The remaining angles, which have to be computed, can then, generally, be found by the solution of spherical triangles. 22. If, however, a face does not lie in a conspicuous zone, and if the angles, which it makes with two or three known faces, can alone be measured, the determination of the symbol involves some- what laborious computation, in which oblique-angled spherical triangles enter. We shall only give the case, in which the angles between it and two of the axial planes are known. The position of a pole P (hkl) is fixed, if its arc-distances from two of the axial poles are known ; for, by the construction given in Chap, vii, Arts. 19 and 20, we can then place it on the pro- jection. Let us suppose that the arcs BP and CP are known. Produce, in Fig. 123, the zone-circles [BP] and [CP] to meet the opposite axial zones [CA], [AB] in M^ (hQl) and JlTj (MO) respectively. The symbols of M 1 and ^ are obtained by Weiss's zone-law, for each of them is the inter- section of two zone-circles. Suppose the great circle XP to be drawn and to meet [BC] in Aj. The great circle XP is not a zone-circle ; and Aj is not a pole, but is a point at 90 FlG - 123 - from X useful in the calculation. Hence, Similarly, the great circles YP, ZP are drawn to meet [CA] and [AB] in p. lt vj, respectively; of which the former is alone shown in the figure. Then cos YP = sin Pp.^ and cos ZP = sin P Vl . But, since X is the pole of the great circle [BO], the angles at Xj are right angles. Hence, by Napier's rules, cos XP = sin PA a = sin BP sin PEC = sin CP sin PCS. \ Similarly, we have right angles at /^ and v 1 and cos YP = sin P^ = sin CP sin PC A = sin AP sin PAC ( ' ' ' ^ cos ZP = sin P Vl = sinAP sin PAB = sin BP sin PBA J FORMULAE OF COMPUTATION. 165 But the equations of the normal P (hkl) are XP b cos YP c cos ZP substituting from (6) for cos XP and cos YP, we have a sin CP sin PCB b sin CP sin PGA (7); .(8) or cancelling the common factor, sin CP, a sin PCB ^b sin PC A By similar substitutions from (6), we have bsinPAO c sin PAB c sin PBA a sin PBC The rule of these three equations is clear and simple. If the elements of the crystal are given, either in terms of a, b, c and the axial angles, or as angular elements D, D t , (10). sin BN 100 110 si 010 110 A & sin B^ sin 100 hkO BL 010 A&O atfZ sin BL l sin CL t ~ I ' sin CM sin AM I sin CM-L sin AM^ h' from A. R. {BLLj/0}, and from A. R. 24. We can prove directly that each of the expressions (10) is equal to the corresponding expression in (9). For, in the triangle SNjC, we have sin BN^ _ sin NB _ sin PCB sin EG sin EN-fl sin BN^G ' and in the triangle AN \C, we have sin ANi sin N-f/A sin PC A sinCA = sin AN = sin BN ' since ^^C = 180 - BN. Hence, dividing the first by the second, we have sin EG sin PCB Therefore sin AN^ sin CA sin PC A ' h _ sin ^^ sin BN^ _ sin AN sin 5(7 sin PCB k ~ sin BN sin 4 JVi ~ sin BN sinCA sin PC'^ ' which is the expression given in (9). The other expressions can be proved to be identical in an exactly similar manner. 25. The converse problem of finding AP, BP and CP, when the symbol (hkl) of P is known, is solved by determining the arcs BL,, CM lt AN lt Ac., and the angles PBC, PJBA, &c. FORMULAE OF COMPUTATION. 167 From equations (9) we have sin PCB h b h sin BN sin CA -, = -y , or = -=- -r-f. =r^. = tan v. sin PC A k a k sin AN sin BC By hypothesis we know either a, b, c, or the equivalent angles AN, BN, &c. Hence, the right side can be computed, and the auxiliary angle 9 is determined. Then, by the transformations employed in adapting a ratio of sines to logarithmic computation (Chap, vin, Art. 14), we have tan | (PC A - PCB) = tan (45 - 9) tan $ (PC A + PCB). But PC A + PCB = EGA is known, or can be calculated from the elements. Hence, tan \ (PGA - PCB) = tan (45 - 6) tan EGA. ^ Similarly, tan \ (PAB -PAC) = tan (45 - 0) tan | CAB, [...(11). tan \ (PEG - PBA) = tan (45 - $) tan | ABC. J k c k sin CL sin AB Where tan d> = T T = T -. =^ 7 =- i , I b I sin BL sin CA I a _ I sin AM sin BC he Tt, sin CM sin AB ' The angles 9, , and i/f must be arranged so that they are, each of them, less than 45. The angles PEC, PEA, &c., can then be all computed. The arcs AN,, BN, and the two other similar pairs can be computed by a like process. For, from equations (10), we have sin BN, h sin BN . = = tan v, sin AN, k sin AN :. tan (AN, - BN,) = tan (45 - 9,) ten^AB; similarly, tan \ (EL, - CL,) = tan (45 U - ft) tan J BC ; ' '" ^ 2) ' tan \ (CM, -AM,) = tan (45 - ft) tan \ CA. Care must be taken that 9,, ft and ft, obtained from the similar expressions to that given in full for 9,, are each less than 45. Since AN, BL, CM, &c., are all known, the angles 9,, ft, ft, can be readily computed for any values of h, k, I, which may be taken. Hence, the arcs -4-^, BN,, BL,, &.c., can be all computed. 168 ANORTHIC SYSTEM. 26. The values of AP, BP and CP, are now found as follows. From the triangle ACN lt we have sin CNi sin CAB sin ^~ sin PCM ' and we obtain similar relations from each of the six triangles into which ABC is divided by AL 1} BM l and CW lt Hence, . sin . _ r sn -. j^-r^ = sin C7Z, - sin P.4-6 sin . . , T sin CAB . ,_ . = sin ANj. -. ^-r = sin BN, -.- sin PC A l (13). Hence, AL it BM lt and CN^ can be computed. Again, from the triangles .4PC7 and P^A, we have sin C7P _ sin PAG sin CA ~ sin 4P~C7 and ...(14), ,. (15). But .'. sin APC = si Hence, dividing (14) by (15), sin CP _ sin CA sin PAC sin P^ ~ sin AN^ sin P^15 sin^P sin AB sin PBA Similarly, sin PLu sin .//! sin^P sin BC sin PCB sin PJfj .(16). M-i sin P(7-4 The values on the right side can be computed and made equal to tan , tan , tan *, respectively. Hence, tan I- (PNi - CP) = tan (45 - 0) tan C^ j ] tan (PA - ^P) = tan (45 -*) tan \ AL,; 1 ...... (17). Hence, AP, BP, CP, PL lt &c., can be computed. EXAMPLE (OLIGOCLASE). 169 27. Example. To illustrate the application of some of the preceding for- mulae, we take the crystal of oligoclase described by vom Bath (Pogg. Ann. cxxxvm, p. 464, 1869). The forms observed were: P{001}, M {010}, /{130}, Z{110}, fe{100}, TjllO}, z_{_130}, y{201}, r{403}, *{I01}, e{021}, n{021}, p {111}, g {221}, o {III}, u {221}. The zonal relations of these forms are shown in the plan, Fig. 124, and in the stereogram, Fig. 125, already employed in the discussion of the crystals of anorthite. Apparently, the most trustworthy angles measured were : MP = 86 32', M'T = 61 40', Pr=6848', Pw=8457' and M'u=S8 13'. It is required to determine from these five angles the parametral ratios and the angles between the axes. The equations of the normal u (221) are : a cos Xu_b cos Yu _ c cos Zu -2 -2 : ~~1 ' or taking X' and Y', the axial points opposite to X and Y : a cos X'u b cos Y'u c cos Zu 2 But from equations (6) of Art. 22, cos X'u = sin M 'u sin PM'u = sin Pu sin M'Pu, cos Y'u= sin Pu sin h'Pu =sin h'u sin Ph'u, cos Zu = sin h'u sin M'h'u = smM 'usinh'M ' r'pu, ) Ph'u, ( i'M'u.} ..(18). ..(19). All the sides of the two triangles M'Pu, M'PT being known, we can compute all the angles by the well-known formula (McL. and P. Spher. Trig, i, p. 47). The hloo U-y. these and die known angles APT and ATP, we find, from the APQ, ^=6 y, P= 63= 54-75', APQ = 6= 52-25', and of the crystal are OZ=93 c 4', 0=ZOJT= 116= 22-6", 7= : * : e=-321 : 1 : -5524. EXAMPLE (OLIGOCLASE). 171 We also know the sides of the triangle formed by the axial poles, viz. AIT =88 23-3', ATP=8632', and P/i=6334-75'. It is therefore easy, by the formulas (3) and (4) to calculate the angular elements D, D,, E, E,; and, hence, the angles Pe, Pn, Px, Py, &c. The element F=hl=hQ + Ql, can be found from the right-angled triangle IPQ, of which the side PQ and the angle IPQ are known. For A IPQ =180- TPu-TPQ = 25 24-25'. The element F' is then known. Instead of using the auxiliary point Q and the right-angled triangles to which it gives rise, the reader will find it simpler to deduce hP and hT, by Napier's Analogies (McL. and P. Spher. Trig, i, p. 118), from the triangle PhT, in which we know PT=6848', hPT=32lfrl' and hTP = 73 34-4'. The results only differ by an insignificant fraction of 1' from those given above. CHAPTER XII. THE OBLIQUE SYSTEM. 1. THIS system includes three classes : I. That in which a dyad axis occurs alone ; II. That in which a plane of symmetry occurs alone ; III. That in which a dyad axis, a centre of symmetry, and a plane of symmetry perpendicular to the dyad axis are associated together. In Chap, ix, Prop. 4, it was established that a centro- symmetrical crystal, having a dyad axis or a plane of symmetry, must have the third element which also occurs in class III. From Chap, ix, Props. 1, 2 and 3, it follows that, in crystals of all three classes, there is one plane which is parallel to a possible face and perpendicular to a possible zone-axis. In classes II and III this plane is a plane of symmetry, the normal to which is a possible zone-axis. In classes I and III the zone-axis is a dyad axis, and is, therefore, perpendicular to a possible face. 2. Hence, in the possible face just mentioned there are a number of possible edges which are all perpendicular to the zone-axis, but are not necessarily, and indeed only very exceptionally, at 90 to one another. The prominent zone-axis, which is either a dyad axis or the normal to a plane of symmetry, will in all cases be taken as the axis of Y. As the axes of X and Z, it is most convenient to take lines parallel to a pair of edges at right angles to OY. Any pair will do, but for the sake of simplicity in the symbols, the most conspicuous zone-axes are usually taken ; and the positive directions are taken to include the obtuse angle. The choice being arbitrary, the angle between these axes is an element of the crystal which must be specified. The acute angle XOZ is denoted by /8 in HEMIMORPHIC CLASS. 173 Dana's Mineralogy and in Hintze's Mineralogie, whilst the same letter is used in Groth's Zeitschr. f. Kryst. u. Min. to denote the obtuse angle XOZ. We shall take ft to be the acute angle XOZ. Any face of the crystal meeting the axes at finite distances from the origin may be taken to determine the parameters a : b : c. Hence, the elements of the crystal are : XOY = 70^=90, XpZ=$-, and a : b : c. But, in general, only those elements which vary with the substance are given; the constant elements being known from the system. Thus, in oblique crystals the angle ft and the parameters are those which have to be specially given in the description. 3. The parameters a : b : c may have any values, and are not, as a rule, in any simple ratios to one another. Their values may, by the aid of the equations of the normal (Chap, iv, Art. 15), be determined from any face meeting all three axes at finite distances from the origin. It may, occasionally, happen that different crystallographers adopt different edges for X and Z, and also a different parametral face. A difference in the stated value of ft, or in the indices ascribed to faces inclined to one another at the same angles, will show the reader whether the same axes and para- meters are used or not. I. Hemimorphic class ; a [hid], 4. Crystals of this class possess a single dyad axis and no other element of symmetry. The faces occur in pairs which change places on rotation through 180 about the dyad axis, except in the single case in which the face is perpendicular to the axis. We know that the dyad axis is perpendicular to a possible face (Chap, ix, Prop. 3), the position of which remains the same when the crystal is rotated about the axis. This face constitutes the form called a pedion. The dyad axis being a possible zone-axis, faces parallel to it are possible. The pair of such interchangeable faces are parallel and constitute a pinakoid. Such forms appear to show a centre of symmetry, which will not be the case with other forms. 5. We shall throughout, in each class and system, use the term special forms to indicate forms, the faces of which are parallel to planes or axes of symmetry, or are perpendicular to them, or 174 OBLIQUE SYSTEM. FIG. 126. are related to the elements of symmetry in such a way as to give rise to a peculiarity of figure distinguishing the forms from those of the general case. We shall speak of the general form, when we wish to emphasize the fact that the faces have no exceptional relation to the elements of symmetry, but occupy any general position. The general form may be said to have the characteristic configuration belonging to the class. 6. The general form of this class consists of two inclined faces which intersect in an edge per- pendicular to the dyad axis. Such a form is represented by the two faces of Fig. 126, in which, how- ever, the dyad axis is placed ver- tically. By transposing the figure, we may suppose the faces to be brought into a position represented by Fig. 127, in which the dyad axis now taken to be OY is perpendicular to the paper, and meets the edge EF (represented by a continuous line) at a distance b -4- k from the origin : the axes of X and Z lie in the paper, and meet the faces at the points II, H t and L, L t , respectively the traces of the faces being given by the discon- tinuous lines. Hence, if OH = OH t a-^h, and OL = - OL t = c + l, the two faces have the symbols (hkl), (hkl), respectively. For it is clear that a rotation through 180 about the perpendicular to the paper, Y, interchanges equal lengths on XX t , measured on opposite sides of ; and, likewise, equal positive and negative lengths on ZZ t . The form a {hkl} consists, therefore, of the faces hkl, hkl. 7. The special forms are : (1) pedions, (2) pinakoids. 1. The pedion is either (010) or (010), according as it meets the axis of Y on the positive, or negative, side of the origin. Both are possible, for they are parallel to two zone-axes OX and OZ ; but the presence of one does not involve that of the other. Owing to the geometrical relation between them, they are called complementary pedions. FIG. 127. PYRO-ELECTRIC CHARACTERS. 175 2. The pinakoid {hOl} consists of the two faces hQl and hOl; these faces being parallel to one another and to the dyad axis. Particular cases of the pinakoid, such as {100} and {001}, differ in no essential respect from {hQl} : they only indicate the pairs which have been selected to give the axes of Z and X. 8. The crystals have a different facial development at opposite ends of the dyad axis, which is one of uniterminal symmetry. Such axes are also called polar or hemimorphic. We shall use the latter word to denote this class of the system (Groth's sphenoidal class). The crystals belonging to this class, which have been examined, have been found to manifest, during change of temperature, opposite electrifications at the two ends of the dyad axis, which is then called a pyro-electric axis ; and this character may be expected to distinguish all crystals of the class. It is found that, at one end, positive electrification is manifested with rise of temperature and negative electrification with fall of temperature. This end is known as the analogous pole. The opposite end is called the antilogous pole; and is that, at which positive electrification is manifested whilst the temperature is falling, and negative electrification as the temperature rises. By carefully dusting the crystal with a mixture of red lead and flowers of sulphur, forced through a very fine gauze, the different electrification is rendered very manifest. In passing through the gauze, the red lead becomes positively electrified, the sulphur negatively electrified. Hence, as the mix- ture falls on the electrified crystal, the red lead is attracted to the one pole, and the sulphur to the opposite pole. If, as is generally the case, the electrification is examined whilst the crystal is cooling, the red lead is attracted to the neighbourhood of the analogous pole whilst the pale yellow sulphur is attracted to the antilogous pole. Solutions of crystals of organic substances of this class, also, rotate the plane of polarization of a beam of light ; and it is found that solutions of opposite rotation can be obtained from crystals formed of the same chemical constituents combined in the same proportions. Thus, crystals of tartaric acid (C 4 H 6 O 6 ), Fig. 128, an be obtained, showing the forms: c{001}, r{101}, a {100}, x {10T}, m = a{110}, q = a. {Oil}, and m t = a {ll0}. In these crystals the analogous pole is at the end of the horizontal dyad axis to the left at which the pair of faces m t occur alone, and the antilogous pole at the end where both the forms m and q occur on the right. A solution of such crystals rotates the 176 OBLIQUE SYSTEM. FIG. 128. plane of polarization to the right, i.e. with the hands of a watch to an observer receiving the light. The crystals may be described as those of dextro-tartaric acid. Crystals of tartaric acid having the same general appearance but in which the development at the ends of the dyad axis is reversed, can be obtained. Their solution rotates the plane of polarization to the left, i.e. against the hands of a watch to an observer re- ceiving the light; and. for equal paths traversed in solutions of equal strength, the amount of rotation produced is the same for crystals of both kinds. The forms c, r, a and x, are developed as before, but on the right the faces w, = a{110} occur alone; and this end of the dyad axis is the analogous pole. The antilogous pole is on the left, where the faces w ( ~ a { 1 1 0} and q t = a {01 1 } occur together. Were a crystal of this latter type placed with the faces c, r, a, x all parallel to those of the one shown in Fig. 128, it would be seen to be its reciprocal reflexion. Hence, the crystals of dextro- and of Isevo-tartaric acid are enantiomorphous. These two bodies and pairs of similarly correlated organic substances, which give dextro- and laevo-gyral solutions, are called stereo-isomers. The elements, and optical characters, of the crystals are the same whether their solutions rotate the plane of polarisation to the right or to the left. The elements are : /3 = 7943'; a : b : c= 1'2747 : 1 : 1-0266. The plane of the optic axes contains the dyad axis, which coincides with the obtuse bisectrix. The acute bisectrix lies in the acute angle ZOX t , and makes with OZ an angle of 71 18' for red light, and of 72 10' for blue light. 9. Fig. 129 represents a crystal of cane-sugar (C 12 H 22 O n ) having the forms: a {100}, c{001}, r{10T}, i = a{110}, m, = a{110}, J,-a{Oll}, o = a {111}. ^ = 76 31'; a : b : c= 1'259 : 1 : -878. The corrosion-figures on m and m t are different ; and the antilogous pole is situated at the end of the dyad axis at which the faces o, q t , and m,, are developed. o. A. || XOZ ; Bx a A OZ= + 67 '75. The angle of the optic axes varies considerably , - L 100 m 110 ^ r 101 Fio. 129. GONIOID CLASS. 177 FIG. 130. with change of temperature ; and, likewise, the positions of the bisec- trices. Des Cloizeaux found that with red light Bx a was displaced through an apparent angle in air of 1 35' between 17 and 121 C. The solution rotates the plane of polarization to the right. Fig. 130 represents a crystal of quercite (C 6 H 12 5 ) with the forms : c{001}, r{10l}, m = a{110}, m, = a{lTO}, q = a{0ll}. = 69 50' ; a : b : c=-7935 : 1 : -7533. O.A. || XOZ\ Bx a A OZ= -11 49' (lithium flame), = - 11 22' (thallium flame). Des Cloizeaux gives 2E = 55 30' (red light), 58 25' (blue light), at 19 C. With rise of tempera- ture the angle diminishes rapidly ; and at 121 C. 2^=37 12' (red light). The acute bisectrix under- goes an apparent displacement of 1 26' between the same temperatures. II. Gonioid class ; K {hkl}. 10. The crystals of this class have a single plane of symmetry 2 and no other element of symmetry. The plane 2 must be parallel to possible faces, which form a pinakoid. It is also perpendicular to a possible zone-axis which is taken to be the axis OY. Hence, we get a series of possible faces all perpendicular to 2, which are necessarily divided symmetrically by it, and constitute special forms which are pedions. Two of these pedions are selected to give the axial planes XOY, ZOY ; and have therefore the symbols *{001}, *{100}, and the angle between them /? is the angle X t OZ. Any other pedion will be K {hQl}, and consists of the single face (hOl). The special form {010} is a pinakoid consisting of the faces (010), (OlO) parallel to 2. 11. Any other face (hkl) is "repeated in a like face over 2 in such a manner that, if both are equally distant from the origin a point in 2 the line of intersection lies in 2. The plane of symmetry also bisects the angle between them. The faces, therefore, meet the axis of Y at equal distances on opposite sides of the origin. Hence the form K {hkl} consists of the faces (hkl), (hkl) ; and is a hemihedral form with inclined faces. The class will therefore be called the yonioid class, from yoma, an angle : it is also known as the hemihedral, clinohedral, or domatic, class. .. c. 12 178 OBLIQUE SYSTEM. 12. Crystals of potassium tetrathionate (K 2 S 4 O 6 ), Fig. 131, belong to this class. The elements are: /3=7544'; a :b :c='93 : 1 : 1'26. The forms are : a = K {100}, a. = K {TOO} L c = * {001} FIG. 131. The optic axes lie in the plane 2, and one optic axis is inclined at a small angle to the normal (100). The faces of the complementary forms m and TO' should show differences in the corrosion-figures and also in the electrification during change of temperature. Hydrogen trisodic hypophosphate (HNa 3 P 2 6 .9H 2 O) has, also, been placed in this class (see Art. 31). The crystals of clinohedrite, (ZnOH) (CaOH ) Si0 3 , have been shown by Messrs. Penfield and Foote ( Am. Jour, of Sri. [iv] v, p. 289, 1898) to belong to this class ; and to have the elements : /3=76 4' ; a : b : c= '6826 : 1 : '3226. The crystals show numerous forms ; and have a perfect cleavage parallel to (010). O.A.JL010, Bx a AOZ=-62. The crystals are pyro-electric ; and as they cool, the portions in the neighbourhood of *c{lll}, K {101} are positively electrified, whilst those in the neighbourhood of K {131}, K {121} and K {TOT} are negatively electrified. III. Plinthoid class; \hkl\. 13. In crystals of this class the three elements of symmetry, which occurred singly in the two preceding classes and in the piiiakoidal class of the a K anorthic system, are associated together; viz. a centre of symmetry, a plane of symmetry 2, and a dyad axis 8 perpen- dicular to 2. The class will be called the plinthoid (prismatic, hololwdral) class, from irXivOos, a brick. A ~ F 14. The general form consists of four FIG. 132. tautozonal faces, as shown by the faces of Fig. 132, which are perpendicular to POP, and P^P^. The four faces are divided into pairs in three ways: (1) pairs of parallel faces, (2) pairs which are metastrophic (p. 18), i.e. inter- changeable on rotation through 180 about 3, (3) pairs which are antistrophic (p. 21), and are reciprocal reflexions with respect to 2. The edges of intersection are all parallel to 2, and per- pendicular to 8. PLINTHOID CLASS. 179 15. The special forms are pinakoids; (1) the faces of one pinakoid being parallel to 2 ; (2) those of the others being per- pendicular to 2 and parallel to the dyad axis. Of these latter pinakoids there may be several on one and the same crystal. 16. The axes are related to one another as in the two pre- ceding classes; viz. OY coincident with the dyad axis, and per- pendicular to the axes OX and OZ which lie in 2 at an angle (3 to one another. This relation is not affected by the fact that 8 and 2 are now both elements of symmetry. The symbols of the pairs of faces interchangeable by rotation about the axis OY are connected by the same rule as was established in Art. 6, for the pairs of metastrophic faces of class I. Hence, if (hkl) is one face, (hkl) is the symbol of the associated face. Since the crystal is centro-symmetrical, this pair is associated with a pair of faces, (JJd) and (hkl), each parallel to one of the first pair. This second pair obeys the rule for the change of signs of a pair of faces meta- strophic about OY. The four faces can be divided into two pairs (hkl), (hkl) and (hkl), (hkl) which obey the rule connecting pairs of faces symmetrical with respect to the plane XOZ. Hence, no other faces are associated together; and the form {hkl} consists of the faces : hkl, Ttkl, hkl, hkl. It constitutes what is called an open form, for it does not by itself enclose a finite portion of space. No completely developed crystal of the oblique system can, therefore, consist of a single form. 17. If the face through HL of Fig. 127 is parallel to OF, it is perpendicular to the paper and the two faces through HL, on opposite sides of the plane of symmetry, become coincident. Hence, we have only the two parallel faces (hQf) and (hQ~l) through HL and HL t . They constitute the pinakoid {hOl}. The particular case, obtained by drawing planes through H and H t parallel to OZ, gives the pinakoid {100} with faces (100) and (TOO). Similarly, by drawing faces through L and L t parallel to OX we obtain the pinakoid {001} with faces (001) and (OOT). These particular cases only differ from the pinakoid {hOl}, inasmuch as they are selected to give the directions of Z and X, respectively. Usually the most conspicuous faces in the zone [010] are taken for the purpose; but this selection is often modified by other characters of particular sub- stances. Thus, in amphibole, and also in orthoclase, the face (100) is absent or inconspicuous. But in each of these substances a very 122 180 OBLIQUE SYSTEM. conspicuous form occurs to which crystallographers assign the symbol {110}; and the faces of the forms thus placed in a vertical position, and called prisms, are inclined to one another at angles of 55 49' and 61 13', respectively. The remaining pinakoid {010} has its faces parallel to 2, and includes the faces (010), (OlO). 18. The general relations of the axes and parameters are the same for all three classes of the system, and in each of them the angle ft and the parametral ratios a : b : c vary with the substance. The form a {hkl\ of class I is geometrically connected with the form {hkl} of class III, inasmuch as the latter may be regarded as made up of the pair of faces in the former and the pair of faces parallel to them. Thus, a {hkl} consists of the pair of faces (hkl) and (hkl) which lie on one side of the plane XOZ. Taking with them the parallel faces (hkl) (hkl) a four-faced figure is obtained which is geometrically the same as {hkl} of class III. But, when the crystal belongs to class I, the second pair of faces constitute a form a {hkl}, the faces of which have a different physical character from those of a {hkl} ; and the two forms do not necessarily occur together. The two forms a {hkl}, a {hkl}, are known as complementary forms ; but, in using this term, the student must understand that it implies no more than the geometrical relation given above. Similarly, the form K {hkl} of class II consists of the pair of faces (hkl), (hkl) symmetrically situated with respect to the plane XOZ, which is now a plane of symmetry. By taking the parallel faces, we get a complementary form K \hkl\ which consists of (hkl), (hkl). These two complementary forms make up a four-faced figure geometrically identical with {hkl} of class III. Similar geometrical relations are found to connect forms of different classes in each of the other systems. The form of the class of greatest symmetry was said to be holohedral ; those of the other classes being stated to be hemihedral when the number of faces was one-half of that in the holohedral class, and tetartohedral when the number was one-fourth. As this mode of expression is still used in descriptive works, we shall in all cases indicate the classes which were regarded as holohedral, and give the corresponding name for the classes having merohedral forms. 19. The reader should notice that the special form {hOl} has the same pair of faces, and the same development, whether it PLINTHOID CLASS. 181 belongs to a crystal of class I or class III ; and similarly, that {010} has the same development, whether the crystal belongs to class II or class III. Hence it may not always be possible to distinguish from the development of the crystal-forms the class to which a crystal really belongs, and physical considerations, such as those of pyro-electricity, the corrosion of faces by solvents, and, dividing throughout by sin SP, we have a sin CS _ b cot SP cam AS Similarly, for the pole (111) we have, since [SI] meets CA in sin CN 6cot El c sin AN Hence a : b : c = . : D> : ...(4). sin (7^ cotltf sin 4^ For this reason Miller used the three angles CN, SI and AN, as the angular elements of the crystal. It is clear they also give /?. ForAN+CN=AC = . 184 OBLIQUE SYSTEM. 24. There is no advantage in giving a series of formulae of rare application. The above formulae (2) and (3) enable us to connect the angles with the parameters and face-indices in all cases. It is, however, frequently more advantageous to select a face, such as m, to be (110). When ft is known, a knowledge of the angles which the homologous faces m make with one another suffices to fix the ratio a : b, but leaves c indefinite. To determine c we can assume some other pole to have known indices consistent with m being (110). The most convenient pole to take is one lying in one or other of the zones [BC], [GA], or [Cm]. The formulae, applicable in these cases, are easily deduced from the general equations (2) and (3), or they can be determined directly by elementary geometry. Thus, m being (110), we can take the section of the crystal in the plane AOB, when we get the relation shown in Fig. 136. Here AOB is a right angle, OB is the axis of Y, and OA is a FIG. 136. normal and not an axis. Now the angle BAO = 90 OBm = BOm, an angle supposed to be known. Hence, OA + OB = cot BA = cot Bm. But, if a section, Fig. 137, is made in the plane XOZ, we have the lines showing the axes and the line A A', parallel to OZ, in which the face m meets the plane of section. Now f\AA'0 = ($; and OA = OA' sin AA'O = a sin 8, since OA' = a. k ^""icf \ft -Jr"^ ' . a O4' OA I _ cotBm f " b~ OB ~ OBsinB~ sin ft " FIG. 137. This same expression may be also obtained from equations (2) by making h=k=l and 1=0; and remembering that the pole S becomes coincident with A when P coincides with m. Equations (2) then reduce to a sin CA =b cot Bm; / . = co ^ Bm (5} b sin /? ' Similar figures to Figs. 136 and 137, showing the lines in which the face Z(011) meets the planes COB and XOZ, give the lengths on the axes OY and OZ ; and establish an equation between the parameters b and c similar to (5). It is c cot BL FORMULA OF COMPUTATION. 18o This can also be obtained from equations (2) by making P coincide with L, and S with C. If P of equations (2) is made to coincide with ^V(lOl), we have a sin CN=csinAN (5**). This last equation is easily established by Fig. 138, which represents a section of the crystal by the plane XOZ. The face (101) meets the axes of X and Z in the points A', C", where OA' -a and OC' = c. The lines OA and OC are the normals (100) and (001), and are at right angles to OC' and OA' respectively. We shall, when- ever it is necessary, distinguish between the poles A, (7, and the points on the axes at dis- tances a and c from the origin by denoting the FIG. 138. latter points by A', C', if measured in the posi- tive directions ; and by A /f (7,, if measured in the negative directions. Hence a _ sin A'C'O _ sin (90 - C" J Q) _ sin AON c ~ sin C'A'O ~ sin (90 - A'CO) ~ sfoCON ' since the angles at N are 90. 25. To find the position of P, when the symbol (hkl) and the elements of the crystal are known, we proceed as follows. Eliminating a, b, c from (2) and (3) we have sin CS _ cot BP _ sin AS fi , h sin CN ~ k cot Bl ~ I sin AN '" sin CS h sin CN ., .*. -^ = -. j-=. (known numbers) = tan (say) (7). sin AS I sm AN v Hence, CS and AS can be computed by the process of trans- formation given in Chap, vin, Art 14. Assuming CS, c as axial planes and n as (111), the symbols of the faces, with the exception of a few of those in the zone [ac], can be found by Weiss's zone-law. We assume the symbols of all the forms to have been determined; they are: a {100}, b {010}, c {001}, e {101}, i {102} , r {101} , I {201} , /{301}, u {210}, z {110}, * {012}, o {Oil}, d {111}, n {111}, q {22l}, y {211}. The face c(001) is parallel to a perfect cleavage, a (100) to an imperfect cleavage. By the aid of the stereogram, Fig. 145, the student can follow the analysis ; the poles A (100), e, C (001), t, r, l,f being in the primitive. L. c. 13 194 OBLIQUE SYSTEM. r cd 52 19-3' cz 75 45 Ed 49 52' o 77 2-6 cq 90 18 n 110 57-6 L*cn'" 104 49 y' 134 53-5 run'" 70 29 bn 35 14-5 EM 35 30 z 54 58 6 90 ae 29 54' *ac 64 37 ci 34 21 cr 63 41-5 cl 89 25-7 c/ 98 37 L/a,16 46 i. Zone [AC]. The angle Cr is first found from the right-angled spherical triangle Cm, for rn = 90 - Bn, and Cn=180-Cn'" = 75ir. Hence, cos Cr=cos(Cn=75ll')^-cos(rn=54 45-5'). Lcos 75 11' =9-40778 L cos (54 45-5') = 9-76120 L cos (63 41-5') = &&4S58. .: Cr = 63 41-5', and Ar = 180 - A C - Cr = 51 41-5'. The angles between the other faces in the zone can now be found by the A. B. of four faces. Let T be (hOl). .: A. B. {ATrC} gives ilOO m loo 101 FIG. 145. sin Ar ' sin Cr ..(a). -. tan | (CT - AT) For /(301), _Zsin(4r = 5141-5') sinCT ~fcsin(Cr = 6341-5')~ : tan (CT+ AT) tan (45 - 6) = tan 57 41-5' tan (45 - 0). sin 51 41-5' I = 16 16' and 45 - 6 = 28 44'. For Z(201), 3 sin 63 41-5" .-. tan i(C/- Af) = tan 57 41-5' tan 28 44'. C/-I/=8110-8', and C/=9836'9', 2/=1646-l'. ^ = 23 38-3', 45 -6 = 21 21-7'. .-. tan (Cl - Al) =tan 57 41-5' tan 21 21-7'. .-. C7-Zz=6328-4', and Cl = 89 25-7'. If T is t (102), equation (o) must be inverted, sin Of sin 63 41-5' .-. tan %(Ai - Ci) = tan 57 41-5' tan 15 16'. .-. Ai-Ci = 46 41-4', and Ci = 34 20-8'. The angle Ae is found in a similar manner from A. B. { AeCr] . EXAMPLE (EPIDOTE). 195 ii. The parameters can now be obtained from equations (2) by taking P to ben (111). .-. a sin Cr = c cot 35 14-5' iii. Zones [Bz] and [Bo]. By formulae (5) and (5*) the angles Bz and Bo can now be found ; and then by the A. B. the angles Bu and Bk. Hence .: z = 352', and Bo = 31 32'. Also, .: Bu = 54 30', and Bk = 50 49-5'. iv. Zone [Aon]. The angles Ao and An can now be found by the right- angled spherical triangles AoC and Anr ; and then Ad and Ay' can be obtained from the A. R. of four poles. The formula are : cos Ao=cos(AC =64 37') sin (Bo = 31 32'), .-. 4o = 772-6'. cos An = cos (lr= 51 41-5') sin (Bn = 35 14-5'), .-. An = 69 2-4'. From A.E. {Adon}, we have sin Ad sin (Ao = 11 2-6') = ' = t /. by computation, nd-Ad = U 13-8', and Ad = 4Q 52'. Similarly, Zj/' is found from sin Ay' _ sin (An = 69 2-4') sin oy' ~ 2 sin (no =33 55') ' v. Zone[Czq]. The angle Cz is found from the right-angled triangle CAz; cos Cz = cos (^C = 64 37') sin (#2 = 35 2'). .-. (7* = 75 45-25'. If now the A. K. {Cdzn'"} is taken to find Cd, an auxiliary angle of less than 4' enters into the computation. It is, therefore, best to compute Cd from the right-angled triangles Cde and CzA, in the way adopted in a similar calculation for gypsum ; thus cos eCd = ta.neC cot Cd, cos ACz tan A G cot Cz. .-. tan Cd = cot (A C = 64 37') tan (Cz = 75 45-25') tan (eC = 34 43'). .-. Cd = 52 19-3'. The angle Cq is then found to be 90 18' from the A. B. {Cdzq}. 132 196 OBLIQUE SYSTEM. 31. Crystals of hydrogen trisodic hypophosphate, HNa 3 P 2 6 . 9H 2 0, are described by M. H. Dufet (Bull. Soc. franc,, de Min. ix, p. 205, 1886) as having only a plane of symmetry; and belong, therefore, to class II. He determines the elements of the crystal from the following three angles : NX =101 A 101 = 87 52', Cm =001 A 110 =83 25', (70 = 001 A 111 = 65 21'. To obtain the elements, we proceed as follows : A freehand stereogram of the above poles, of /(111), and of the poles of the complementary forms is first made. i. From A. R. { Clmo, } , we find Cl = 55 24-5'. ii. From the right-angled triangle CNl, cos NCI = tan CN cot Cl ; ,, Cox, cos ( Cox = NCl) = tan Cx cot Co. = i ^(co=ll?2r) =tan (* =33 38 - 4/ >- t&nCN sinCNcosCx But tan Cx sin Cx cos CN ' Gin Hf f*r\a /7A7" _ ain /7 \T f*r\a Hr 1 _ -f.an fi = tan(45-0). sin Cx cos CN+ sin CN cos Cx 1 + tan 6 " .: sin (Cx - GZV) = tan 11 21-6' sin (Cx + CN) = tan 11 21-6' sin 87 52'. Hence, by computation, Cx- CN=11 35'; and Cx = 49 43-5', (7^=38 8-5'. iii. From the several right-angled triangles, we have : cos (Cox = NCl) =tan (Cx=43 43-5') cot (Co = 65 21'), cos NCl=i&n CA cot (Cm =83 25'), sin Bl = cos Nl = cos Cl+ cos CN, sin Bo = cos ox = cos Co -=-cos Cx, sin Am = sin Cm sin NCI. Hence, M7Z=5712-5', ^C=/3=7758-2', Z=46 12-5', Bo = 4010-6', ^m=5637-6'. The parameters can now be found by equations (2) ; and are a: b :c = l-552 :1: 1-497. CHAPTER XIII. THE PRISMATIC SYSTEM. 1. THE prismatic system includes all crystals having one set of three dissimilar zone-axes which are, at all temperatures, perpendi- cular to one another, and are the directions of the principal axes of the wave-surface for light of all colours. The system includes the three following classes : 1. The sphenoidal (bisphenoidal (Groth), hemi/iedral) class, the crystals of which have three dissimilar dyad axes at right angles to one another, but have no other element of symmetry. II. The bipyramidal (holohedral) class, in which the three dyad axes of class I are associated with a centre of symmetry and with three dissimilar planes of symmetry, each perpendicular to one of the dyad axes. III. The acleistous 1 pyramidal (pyramidal, hemimorphic) class, the crystals of which have only one dyad axis associated with two dissimilar planes of symmetry intersecting at right angles in the axis : the normals to these planes are zone-axes, but are not axes of symmetry. 2. The three rectangular zone-axes of the several classes can be selected as the axes of X Fand Z\ and no other set of zone-axes is equally advantageous. The angles, 90, between the three axes of reference are now fixed by the symmetry, and will remain constant as long as the crystal is subjected only to strains, such as that due to the expansion caused by change of temperature, which affect the substance in a manner consistent with the retention of the 1 From dCK\ijv, a wedge. 6. The face of any sphenoid can be selected to give the parameters a : b : c ; and we may for the present suppose them to be known. Let ^, in Fig. 146, be (/*&), then OAT = a - A, OY=b+k, OZ=c + l. Rotation about OX interchanges p with p t , and at the same time 200 PRISMATIC SYSTEM, CLASS I. equal positive and negative lengths on each of the axes OY and OZ perpendicular to OX. Hence, 6> 7,=- OY=b+k, OZ^-OZ^c+l. The point X is common to both faces. Hence p t , passing through XY t Z , has the symbol (hkl). Similarly, by rotation about ZZ t , the face p changes places with the face which passes through LM and the parallel line XY^,. The intercepts are: OX f = a^Tt, OY t = b + k, OZ=c + l; and the symbol of the face LMQ is (hkl). Again, by rotation about YY', the face p is brought into a position in which it meets the axes at OX t ~ a, -f- h, OY =b -f- k, OZ f = c+-l. The symbol of the fourth face MNQ is (hkl). These four faces complete the sphenoid a {hkl} ; which, therefore, consists of : hkl, hkl, hkl, hkl (a). In a stereogram of the sphenoid a {hkl}, two poles above the paper occupy opposite quadrants, and two poles below the paper, represented by circlets, occupy the remaining pair of opposite quad- rants. Such a stereogram is shown in Fig. 152, Art. 13. 7. Similarly, the sphenoid, Fig. 147, which consists of the faces parallel to those in the first, and which is consequently called the com- plementary form, has the symbol a {hkl}, or a {hkl}. It consists of the faces : hki, hki, hki, hkl (b). This latter sphenoid is clearly pos- sible when a {hkl} exists, for the symbols of its faces conform to the law of rational indices, and to the rule connecting the faces interchangeable by rotation through 180 about the axes of refer- ence. Since each of the faces of the one sphenoid is parallel to a correspond- ing face of the second, the angles between the pairs of faces on the one are equal to those between the pairs of parallel faces of the other. The sphenoids cannot, however, be similarly orientated, or, as we may express it, superposed ; for they are reciprocal reflexions of one another in planes parallel to any pair of the axes. The complementary sphenoids are therefore enantiomorphous. 8. The rule, established above, connecting the symbols of pairs of faces interchangeable by rotation about a dyad axis perpen- dicular to two axes of reference is perfectly general. For any line PRISMS AND DOMES. 201 perpendicular to a dyad axis remains parallel to itself after rotation through 180 ; and, if it meets the axis, equal lengths on opposite sides of the axis are interchanged. Hence, a pair of homologous faces makes equal intercepts on the pair of axes of reference meeting them on opposite sides of the origin. The corresponding indices differ therefore only in sign. 9. The special forms (Chap, xn, Art. 5) are of two kinds. 1. When a face is parallel to one of the dyad axes, it is clearly brought into a parallel position by rotation through 180 about that axis ; but into new positions inclined to its original position when it is turned about each of the axes which it meets. The two faces obtained by rotations about these latter axes will necessarily be parallel to each other and to the first axis. We, thus, have a set of four tautozonal faces intersecting in edges parallel to one of the dyad axes ; and such that the angles between pairs of the faces, or their normals, are bisected by each of the two other axes. If the faces are parallel to ZZ^ the vertical axis, the form {hkQ} is called a prism. If parallel to the makro-axis YY tt the form {hOl\ is called a makrodome ; and if to the brachy-axis XX t , the form {Okl} is a brachydome. Figs. 148 150 show three such forms, which, not enclosing a finite portion of space, cannot occur alone. They are in the figures cut short by pinakoids, the faces of which are perpendicular to the zone-axes of the four-faced forms. The prism, Fig. 148, with vertical faces may be represented as a {MO}, or {MO}; for, as we shall see later on, there is in this case no advantage in keeping the Greek prefix before the brackets. It includes the faces : hkQ, hkO, hkO, hkO (c). The alternate faces are parallel ; whilst the first two, and the last two, interchange places by rotation about YY/, the middle pair, and also the extreme faces, interchange places by rotation about XX t . The faces of the particular z case {110} we shall in this and the tetra- gonal system always denote by the letter m. Similarly, the makrodome {hQl}, Fig. 149, includes the faces : Z, The student can easily arrange them j- ia 149 202 PRISMATIC SYSTEM, CLASS I. in pairs interchangeable about each of the axes, as was done in the case of the prism. The brachydome {0^}, Fig. 150, includes the faces: Okl, Qkl, Okl, Okl (e). Although, in the above special forms, one index alone changes sign in the deduction of the symbol of a face from that of an adjacent face, we have not contravened the rule given for the general form a (hkl\. For, if one of these latter . ' ' , FIG. 150. indices becomes zero, there can be no distinction between its positive and negative value. 2. Lastly, we have the three special forms, in each of which the faces are perpendicular respectively to one dyad axis, and are therefore parallel to the two other axes. In such a case the form is a pinakoid consisting of a pair of parallel faces. Thus we have the makro-pinakoid {100} including (100) and (TOO), both of which are perpendicular to XX , and parallel to 77, and ZZ f Similarly, the brachy-pinakoid {010} includes (010) and (OlO). The pinakoid {001} has the horizontal faces (001) and (OOl), and is often called the basal pinakoid, and one or other of its faces is called the base. One, two, or all three, pinakoids may be present on any one crystal of the class. The faces of the different pinakoids will as a rule have different physical characters, such as striae, .fee., so that it is often easy to distinguish one from another. 10. Crystals of any particular substance may present com- binations, i.e. have several forms of the same or of different kinds associated together ; and must be combinations if any special forms are present. We may, for instance, have the three pinakoids alone, forming a figure distinguishable from a cube only by the facts that the three sets of parallel faces have different physical characters and that the crystals are optically biaxal. We may also have crystals, like those shown in Figs. 148 150, and also combinations of the prism with one or more domes. Crystals on which special forms alone occur cannot be distinguished geometrically from similar crystals belonging to the next class. 11. Crystals of the following substances belong to this class : Magnesium sulphate (epsomite), MgS0 4 . 7H 2 0. a :6:c=-990: 1 : -571. EXAMPLES OF SPHENOIDAL CRYSTALS. 203 The crystals are usually prisms {110}, terminated by a (111/ and oc- casionally also by {101}, {Oil}, o. A. || (001) ; Bx. || OF (Chap, x, Art. 8). Zinc sulphate (goslarite), ZnS0 4 . 7H 2 0, isomorphous with the preceding salt. a : b : c = -980 : 1 : -563. o. A. || (001) ; Bx. || OF. Sodium potassium dextro-tartrate (seignette salt), NaKC 4 H 4 6 . 4IL/). a : b : c = -8317 : 1 : '4296. The crystals are usually combinations of several prisms : {110}, {210}, {120}, with the pinakoids {100}, {010}, {001}. The horizontal edges made by these forms are sometimes modified by small faces: o = a {111}, = a{211}, q {021} and r {101}. O.A. || (010); Bx, || OX. The isoinoq)hous sodium ammonium salt forms similar crystals. O.A. || (100) ; Bx. || OZ. Hydrogen potassium, dextro-tartrate, HKC 4 H 4 6 , and the isomorphous ammonium salt, form crystals in which special forms and sphenoids occur together. In the first salt, a : b : c = 7148 : 1 : -7314; in the second, a : b : c = '6931 : 1 : -7100. In both O.A. || (001); Bx. || OT. 12. Potassium-antimonyl dextro-tartrate (tartar-emetic), K(SbO)C 4 H 4 O 6 . a : b : c= -9556 : 1 : 1'1054. The forms, Fig. 151, are: o = a{lll}, w = a{lll}, m {110}, c{001}. O.A. || (001) ; Bx. || OF. Similar crystals of the laevo-tartrate have been obtained, in which the parameters are the same, but in which the faces =a{lll} give the predominant form ; and the crystals are enantiomorphous to those of the dextro-tartrate. The drawing is made as follows. On the projected cubic axes (Chap, vi, Art. 11), lengths OA = -9556, and OC= 1-1054 are cut off. The sphenoid a {111} is then drawn with edges passing through A, B, C, tanF; and = 61 0-6'. Zone [mm']. From (6**) tan^=2tan.F, .-. ^=46 35-3' Zone \m'ri\. From the right-angled triangle Bnm", we can now determine (a) m"n, and ()3) the angle nm"B = A mm'o. Then, from the right-angled triangles mom', Ixm', we can determine (7) m'o, and (S) m'x. The formulae are : (a) cos m" = cos Urn" cos L'n = sin 27 51-5'cos46 21'; .-. m"n=71 11'. (P) eotnm"B = awBm"+ta.nBn=coa27 51-5' cot 46 21'; .-. nm"B = 49 51 '. (7) tanm'o = tanmm'-f-cosnm'm=tan55 43'-=-cos49 51-3' ; .-. m'o 66 16-4'. (8) tan m'x = tan m'l-i- cos nm'm= tan 74 26-8'-=- cos 49 51-3'; .'. wi'z = 7949-6' The pole d", Fig. 171, lies in [&"] = [301] and in [m'n] = [lll]. By Chap, v, Table (23), its symbol is found to be (143). The angle nd" can be now determined by the A.R. of four poles. (e) From the A.E. {d"nom'}, we have 143 sin d"n sin d"o Oil 112 110 Oil Ho J112 EXAMPLE (TOPAZ). 223 sin d"n 2 sin m'n -4 sin 71 11' _ ' = sin d"o ~ 5 sin m'o ~ sin 66 16-4' L sin 71 11' =9-97615 log -4 =1-60206 9-57821 L sin 66 16-4'= 9-96165 L tan (6 = 22 28-U 1 ) = 9-61656. .-. tan i (d"o - d"n) = tan (45 - 6) tan (d"o + d"n). But d"o - d"n = ore = 42 32-6', .-. tan(d"o + d"n)=tan21 16' -3 -4- tan 22 31-86'. L tan 21 16-3' =9-59030 L tan 22 31-86' = 9-61789 L tan 43 11 =9-97241. .-. d"o + d"n=8622', d"o-d"n = 4232-6'. .. d"o = 6427'3', d"m=2154-7', and ro"d" = 49 16-3'. Zone [com]. From the triangle mom', we have cos om cos om' -f- cos mm' ; .: om = 44 24-7'. Hence, by equation (7), tanc*=2tanco4-3; .-. cs = 34 14'. Zone [Bdxs]. From the right-angled triangle Bsm, we have cos Bs = cos sm cos Bm = sines sin Am: .-. Us = 74 45-5'. But the point bisecting ss' is a possible pole (103) at 90 from B: call it t. Then by an equation, similar to (7), which is derived from the A.B. {Bzst}, where z is any pole (hkl) in the zone, we have tan Bz = ^- tan Bs. Hence, for x (123), tan Bx = % tan 74 45-5'; and for d (143) , tan Bx = tan 74 45-5'. .-. fix = 61 24-7', Bd = 4232-2'. The angle xl is now easily determined, for cos xl = cos Bx -T- sin A I ; .: xl = 48 49'. Zone [cm/]. In this zone tan cy = 2 tan (en = 43 39'). .'. cy = 62 20-3'. Many of the above angles are brought together in the following tables : Br 32 14-3' Bl " 1" *4 *, ! &612M Bm 62 8-5 I ". "~ " I B, 74 45-5 mjft' 55 43 'm'o 66 16-4' m'x 79 49-6 m'n 108 49 m'd" 130 43-7 CHAPTER XIV. THE TETRAGONAL SYSTEM. 1. THE crystals possible in this system fall into seven classes, and have each a principal axis which is either a tetrad axis, or a dyad axis perpendicular to which pairs of like edges occur at right angles to one another. The classes will be discussed in the following order : I. The scalenohedral (sphenoidal-hemihedral) class, the crystals of which have three dyad axes at right angles to one another associated with two like planes of symmetry, S and S', intersecting in one of the axes and bisecting the angles between the two others. The axis AA y , Fig. 172, in which the planes S and S' meet, is the principal axis; and the two axes 08, 08' are like dyad axes, though not interchangeable. II. The diplohedral ditetragonal (holohedral, ditetragonal-bipyramidaT) class, the crystals of which have a tetrad axis T '; four dyad axes, 28 and 2 A, perpendicular to T 7 ; a centre of symmetry C ; and five planes of sym- metry, each perpendicular to one of the axes of symmetry. The planes may be denoted by II, S or 2 according as they are perpendicular to T, A or 8. The above elements of symmetry may be shortly given as follows : T, 28, 2A, C, H, 22, 2S. III. The acleistous tetragonal (pyramidal, hemimorphic-hemi- hedral) class, the crystals of which have only a tetrad axis T. SCALENOHEDRAL CLASS. 225 IV. The diplohedral tetragonal (bipyramidal, pyramidal-hemi- hedral) class, in the crystals of which T is associated with a centre of symmetry C and a plane of symmetry II perpendicular to T. V. The trapezohedral class, in which four dyad axes 28 and 2A are associated with the tetrad axis T. VI. The acleistous ditetragonal (ditetragonal-pyramidal) class, the crystals of which have four planes of symmetry, 2S and 22, intersecting in the tetrad axis T. VII. The sphenoidal (splienoidal-tetartohedral) class, in which every form is a sphenoid, save when the faces are parallel or perpendicular to the principal axis. One pair of opposite edges of each sphenoid are at right angles to one another and to the principal axis which is a dyad axis; and the crystals have no other element of symmetry. I. Scalenohedral class; K{hkl}. 2. It is clear from the definition given in Art. 1 that the planes S and S' are like planes of symmetry at 90 to one another, for they change places when the crystals are turned through 180 about either of the dyad axes 08, 08'. Again, the axes 08, 08' are like axes at 90 to one another, for they are reciprocal reflexions in each of the planes S and S' which bisect the angles between them ; but, in this class, the axes 08, 08' are not interchangeable. They differ also from the dyad axis AOA,, for two planes of symmetry intersect in the latter ; and, as we shall see later on, similar edges occur in pairs which are perpendicular to it and are at right angles to one another. This axis AOA / is a principal axis, and in translucent crystals coincides with the direction of the optic axis. 3. It is obviously convenient to select the three dyad axes as axes of reference, for they remain at right angles to one another at all temperatures. The principal axis is always taken as OZ. Again, since the axes of X and Y are reciprocal reflexions in each of the planes of symmetry S and S', equal lengths on them must always correspond to one another. If the representation of forms possible on a crystal is to accord with the symmetry, equal lengths on OX and Y must, therefore, be taken for parameters. It follows that L. c. 15 226 . TETRAGONAL SYSTEM, CLASS I. only one element varies with the substance in crystals of this class. This element may be taken to be the ratio of the parameter c measured on the principal axis to that measured along either OX or OY, which we shall call a. We shall take 08 to be OX, and OB' to be OY; and when we desire to denote that lengths equal to a, or to an- A, are measured on OY we shall denote them by a t , or a ( H- h. We shall see in the course of the Chapter that, as a consequence of the principal axis, the crystals of each class of the system can be referred to three rectangular axes of which OZ is the principal axis; and that equal parameters can be taken on the axes of X and Y. 4. The pinakoid, {001}. The simplest form on a crystal of this class is a pinakoid, the faces of which are perpendicular to the principal axis and to the planes S and S'. Such faces are possible (Chap, ix, Prop. 3) : they are parallel to the axes of X and Y, and are interchangeable by a semi-revolution about either of them. The form is {001}, and comprises the faces : 001, GOT (a). The possible crystals shown in Figs. 174, 176, are each terminated by the pinakoid. 5. The tetragonal prism, {110}. By Chap, ix, Prop. 1, a plane of symmetry is always parallel to a possible face. Let us suppose, in Figs. 173 and 1 74, a face m i to be drawn through A on OX parallel to S ; and let it meet OY f at A f . Then since S bisects the angle XOY, and AA t is parallel to the trace of S on the plane XOY, the angles OAA t , OA t A are equal, each of them being 45. Hence OA t = OA. These lengths, being taken as the parameters on OX and Y, may be denoted by a. But the face m t through AA t parallel to S is also parallel to OZ ; and its symbol is therefore (iTO). Similarly, there must be a parallel face m on the opposite side of S which has the symbol (TlO). These two faces also interchange places by rotation through 180 about OZ. FIG - 173 ' Again, rotation about OX through 180 interchanges equal positive and negative lengths on OY, and also the planes S and S'. The face m i is brought into the position given by m, and ni' into that given by m". The two new faces have the symbols (110), (TTO). TETRAGONAL PRISMS. 227 Again, since m is parallel to S' and m, to S, it follows that the angles mm,, Dim', &c., are all equal to 90. The form Fig. 174, is, therefore, a rectangular prism having the faces : 110, TlO, TTO, 1TO ....................... This prism differs from the corresponding prism of the prismatic system inasmuch as the angles are permanently 90. In a prismatic crystal the axes retain their direction whilst the temperature varies, but the coefficients of expansion along the axes are different ; hence the angles of the prism and domes vary slightly with the temperature. In crystals of the class now under consideration the relations connect- ing the planes and axes of symmetry do not vary with the substance, or with the temperature as long as it is not raised to a point at which the crystalline structure is destroyed. The parameters on the axes of X and Y therefore remain equal when the temperature is changed ; but the ratio of a : c changes with the temperature. The same holds for crystals of all classes of this system. 6. The tetragonal prism, {100}. A face parallel to the principal axis and to one of the other dyad axes, OY (say), is clearly possible, and is necessarily perpendicular to the remaining axis OX. The face may be drawn through A on OX, when its symbol is (100); and its trace on the plane XOY is given by MAM t in Fig. 175. The planes of symmetry meet the plane XOY in the traces marked S and S', and the face (100) in vertical lines through M and M t . But through these vertical lines homologous faces pass which are inclined to S and S' at the same angles as (100). Faces of this form therefore pass through the traces MA'M', MA t M", where A A' MO = A OMA, and f\AM t O= I\OM,A. But Hence f\AMA' = A AM t A t = 90; and the new faces are parallel to one another and to OX. Their symbols are therefore (010), (OTO). Similarly, each of these faces is repeated over the planes S' and S 152 TETRAGONAL SYSTEM, CLASS I. in the same face (100) which passes through the trace M' AM", where the angles between the faces meeting at M' and M" are 90. We have therefore a second rectangular prism {100}, Fig. 176, which has the four 100, 010, 100, 010 (c). The faces of this prism truncate the edges of the prism {110}; and vice versa, the faces of {110} truncate the edges of {100}. FIG. 176. 7. The ditetragonal prism, {MO}. If a face (MO) parallel to OZ occurs on a crystal, it meets the plane XOY in a line HK, Fig. 177, where OH = a + h, OK = a, -r k. Let HK meet the trace of S in M. Through M a new face can be drawn parallel to OZ and inclined to 8 at the same angle as (MO) makes with 8. Let the new face meet the plane XOY in the trace H'K'. Now in the two triangles HOM, K'OM, we have f\HOM = f\MOK', each being 45; and f\HMO= /\OMK', the equal angles on opposite sides of a plane of symmetry : also the side OM is common to both triangles. Hence the remaining angles and sides are equal. Therefore A OHM = A OK'M ; and Again, in the triangles HOK, H'OK', we have f\OHK= f\OK'H', and f\HOK= 90 common ; also the side OH= the side OK'. There- fore the remaining sides and angles are equal; and OH' = OK= a -=- k. The face through the trace H'K' has therefore the symbol (MO). A semi-revolution about OX interchanges positive and negative lengths on OY, and brings the traces HK, H'K' into the positions given by HK it and H'K^ where OK ll = -OK=a l + k, and OK =-OK' = a^h. The vertical faces through the traces HK it and H'K t are (MO), (&AO), respectively. DITETRAGONAL PRISMS. 229 Again, by a semi-revolution about Y, the above four faces are brought into the positions of faces which meet OY at the same points as before, but in which the signs of the intercepts on OX are changed. The four faces taken in succession from K'M'H /t are : MO, MO, MO, MO. The form {MO}, Fig. 178, is therefore a ditetragonal prism of eight faces which have the symbols : MO, MO, MO, MO, MO, MO, MO, MO ......... (d). The alternate angles F over the edges passing through the points M, M', &c., are all equal, and so are the angles over the ^ edges through H, A", H t , K r But the angles F are never equal to those of the K, other set. Further, alternate faces are at right angles to one another. This important relation is easily proved from Fig. 177. Let H ti K' be produced to meet HK in p IG> N. Then the external angle K'NH= t\NKK' + [\NK'K. But A NK'K = A OK' H ti = A OK'H' = A OHK. .% f\K'NH= f\OKff+ f\OHK= 180- ^1 z ^>. hkp \hko kho U \ l -\ The reader should observe that two faces symmetrical with respect to S or S' have the indices h and k in reverse order. This is obvious in the case of the adjacent faces which meet in lines through the points M. But it is also true of faces like those through HK and K t H ti which, if produced, will meet in the plane S'. These two faces' have the symbols (MO), (MO), and are reciprocal reflexions with respect to AS". Again, if OG is the normal to (MO), it is parallel to H ti K'N, since K'NK=W. :. A XOG = A OH K' = A OKH. #-afir='= ......... (1). Hence, the inclination of a face (MO) to one of the vertical axial planes, and therefore also to the planes of symmetry, is determined as soon as the indices are known, and is independent of the parameter c on the vertical axis, which is the only element varying with the substance. The angles of ditetragonal prisms can therefore be calculated once for all, and will be the same for all 230 TETRAGONAL SYSTEM, CLASS substances crystallizing in the tetragonal system. It is easy to construct a table of angles for such prisms, similar to the following, which gives the angles for a few cases of common occurrence. {MO} 100 A MO MO A 110 F=hkQ/\khQ. {310} 18 26' 26 34' 53 8' {210} 26 34 18 26 36 52 {320} 33 41 11 19 22 38. 8. When the faces of the forms are inclined to the vertical axis and to the horizontal plane, the forms are closed figures, which differ from the preceding inasmuch as each single form completely encloses a finite portion of space. The tetragonal bipyramid, {hQl}. Let one of the faces be parallel to Y and meet OZ at L, where OL = c -=- 1. Such a face can be supposed to be drawn through MAM t of Fig. 175, where OA is now replaced by OH a-^h. A semi-revolution about Y brings the face into a parallel position, where it passes through M' M" and L t , and meets the axes of X and Z at distances a H- A, c -=- 1. The two faces have therefore the symbols (hQl), (hQl). If, again, the crystal is turned through 180 about OX, the trace MAM I remains the same, but the points M and M t , and the points L and L t are inter- changed, we therefore have the face MM t L t of Fig. 179, the symbol of which is (hQl) : the parallel face is (hQl). The above four faces are now repeated over the edges LM, ML t , &c., in which they meet the planes S and S' in four new faces which are parallel to OX and meet OY at H', H t , where OH'^a^h. The symbols of the new faces are: Qhl, Qhl, Qhl, Qhl. The form {hQl}, Fig. 179 is a tetragonal bipyramid, the horizontal edges of which are at 90 to one another ; it consists of the faces : Ml, Qhl, hQl, Qhl] hQl, Qhl, hQl, Qhl)" Each face is an isosceles triangle, and the angles F over polar edges (p. 1 1 2), such as LM, are all equal : so are also the angles over the horizontal edges MM t , MM', &c. ; but the angles over these latter edges are in no case equal to the angles F. The face e (101) of the pyramid {101} occupies a position similar THE CRYSTAL-ELEMENT. 231 to that of r(101) of a prismatic crystal. The equation connecting the parameters on OZ and OX must be the same as that given for c : a and the face r in (4) of Chap, xm ; for a section of the pyramid by the plane XOZ gives a right-angled triangle LOH, in which the angle LHO is equal to the angle between OZ and the normal to e. Hence, tan ZOe = tan LHO = OL + OH. But for e(101), OL = c, and OH=OA = a. Hence, denoting the pole of (001) by C, and the arc Ce by E, which we shall call the angular element of the crystal (see Chap, xnr, Art. 26), we have tan E = tan Ce = tan ZOe = c + a ............... (2). Again, the face e'(Oll) of the same pyramid occupies a similar position to q (Oil) in a prismatic crystal. But the faces e(101) and e'(Oll) of the pyramid are inclined to the plane XOY at the same angles, for the two faces are reciprocal reflexions in the vertical plane S. Hence, the angle It follows that a crystal of this class has only one element which varies with the substance. The element may be given either by the angle E, or by the ratio of the parameters c and a. When the numerical values of the parameters are introduced, it is usual to make a the unit of length. The relation between the angular element E and the linear element c is then given by c = tan J E t .............................. (2*). The parameter c may be greater or less than 1 ; and it can only be equal to unity under exceptional circumstances : even if at any temperature c = a, a change of temperature will alter the ratio of c : a ; for the coefficient of expansion along the principal axis differs from that along any line at right angles to it. 9. We have now exhausted all the possible types of the special forms in the class, which have parallel faces. It will be noticed that the parallelism of the faces in them is due to the faces being in each case parallel to one or other of the dyad axes. The sphenoid, K {hhl\. Suppose a face to be drawn through the line AA' of Figs. 173 and 180 to meet the principal axis at the point L ; then, since A A' is perpendicular to S, every face drawn through it, or parallel to it, is also perpendicular to S. Hence, if OA is a -H h and OL is c H- I, the symbol of the face is (h/d). Let this face be turned through 180 about OZ, the point L 232 TETRAGONAL SYSTEM, CLASS I. rehiains unchanged whilst the line A A' is transposed to the parallel line AA t , meeting OX and OF at equal distances in the negative direction. The new position of the face is given by the symbol (Mil). Similarly, a semi-revolution about OX leaves A unchanged but interchanges equal positive and negative lengths on Y and OZ. The face q 1 q 2 q 3 has, therefore, the symbol (hhl). By a similar rotation about OY the face is brought into the position of qq 2 q s , of which the symbol is (hhl). Repetition of the rotations gives no new faces. The form, there- fore, consists of the faces : hhl, hhl, hhl, hhl (f). Each of the faces is perpendicular to S or S'. Further, the edge qq l is parallel to A A' and AA t , and lies in the plane S'. This plane bisects the angle between the pair of faces (hhl), (hhl) which meet in qq l . Similarly, the plane S contains the opposite edge q z q 3 , and bisects the angle between the two faces (hhl), (hhl) which meet in the edge. The edges qq l , q 2 q s are therefore like and interchange- able edges, which are at right angles to one another and to the principal axis. It is also clear that the faces are all equal and similar isosceles triangles, for each face is bisected by a plane of symmetry ; for instance, the face qq 1 q 2 is bisected by S in the line Lq? The angles over the slanting edges passing through A, A', &c., are all equal ; but they are not equal to the angles over the horizon- tal edges. Again, a semi-revolution about a dyad axis, OX (say), changes only the ends q l and q 2 of the edge meeting it, but not the direction of the edge. The edge q l q 2 is therefore perpendicular to OX, and is truncated by the face (100), when the sphenoid and {100} occur together. The other slanting edges 5-5-2, &c., are all similarly truncated by the other faces of the tetragonal prism {100}. A complementary sphenoid is possible, the faces of which are parallel to those of the preceding sphenoid. Its symbol is K {hhl}, and it consists of the faces : hhl, hhl, hhl, hhl. Since the faces of the two forms are parallel, the faces will be equal and similar, and the angles over the corresponding edges will be equal. The two forms can be placed in similar positions by rotating one through 90 about the principal axis. Complementary forms connected by the fact that they can be brought to occupy the same space are said to be tautomorphous (p. 210). THE DISPHENOID. 10. The disphenoid, or scalenohedron, K {hkl}. If a plane is drawn through a line in the plane XOY, such as HK of Fig. 177, to meet the vertical axis in L at the distance c -=- 1, we obtain a face which has the symbol (hkl). Such a face meets the plane of symmetry S in a line joining L to M in the plane XOY, through which and H'K' a second face must pass. The new face H'K'L is symmetrical to the first with respect to S ; and has the inter- cepts a -4- k : a i H- h : c + l, and the symbol (khl). A semi-revolution about OZ brings the above two faces into positions in which they meet the plane XOY in the lines HK tl and H / K I of Fig. 177. The new faces have, therefore, the symbols (hkl), (khl). Again, as was proved in Art. 7, the pair of lines UK and H a K f are symmetrical with respect to S', and the faces through them and L must be so too. These faces must therefore intersect in an edge, Ln (say) of Fig. 181, lying in S'. For a similar reason, the line of intersection Ln' of (khl) and (hkl) lies in S'. If now the above faces are turned through 180 about OX, they are brought into positions in which they meet OZ at // /} where OL t = c-^l, and meet the plane XOY in the lines HK it , KH', &c., of Fig. 177. The new faces have, therefore, the symbols : hkl, khl, khl, hkl. These faces are necessarily symmetrical with respect to S and S', for the original set of four faces with which they change places are symmetrical with respect to the same planes. A semi-revolution about OY gives no new faces, for successive rotations of 180 about OZ and OX are together equivalent to a single rotation of 180 about OY. Hence, the disphenoid K.{hkl], Fig. 181, consists of: hkl, khl, khl, hkl} hki, khi, khi, hkl} (g) * FIG. 181. The faces are equal and similar scalene triangles, pairs of which intersecting in the sloping edges nt t , rit t , &c., are interchangeable by a semi-revolution about the axis to which these edges are perpen- dicular. The pairs of faces which meet in the edges Lt t , Ln, &c. are reciprocal reflexions in the planes of symmetry through these 234 TETRAGONAL SYSTEM, CLASS 1. edges, and are therefore antistrophic. The angles over the dissimilar edges of different faces are unequal, whilst those over similar edges bounding different faces are all equal. Hence each disphenoid has three different angles : (i) that over an edge, such as Lt t , lyiug in 8 or 8' ; (ii) that over a dissimilar edge, such as Ln, also lying in S or /S" ; and (iii) that over an edge, such as nt t , which is perpendicular to, and bisected by, one of the like dyad axes. 11. The planes through L t and the lines HK, H'K' of Fig. 177 are clearly possible faces; for they have rational indices and are parallel to a pair of faces of the disphenoid K {hkl}, although they do not belong to this form. Hence we obtain a complementary disphenoid K{hkl\, Fig. 182, the faces of which are parallel to those of the form discussed in Art. 10. The form has the following hkl, khl, khl, hkl) hkl, khl, khl, hkl} ..... ( >' The angles of the complementary di- sphenoids are necessarily equal, for pairs of parallel faces make equal angles with one another. Since also the lines of Fig. 177, in which the faces of both forms meet the plane XOY can be interchanged by a rotation of 90 about OZ, it follows that the same rotation must bring the faces of K {hkl} into the position of those of K {hkl}, and vice versd. Hence the comple- mentary forms are tautornorphous. Crystals showing sphenoids and disphenoids were said to be hemihedral with inclined faces, and this term is still used in descrip- tive works. 12. If a bipyramid {hOl} occurs on a crystal, measurement of one angle will enable us to determine the symbol, when the angular element E, or the parametral ratio c : a, is known ; or conversely, to determine the element when the indices h and I are given. Thus, if the angle over one of the edges MM t of Fig. 179 is measured, we know the angle LEO = /\LHL^2. But, if C is (001) and n is (hOl), then A Cn = A LEO. Hence, tan Cn = = ~ 7- . I h la = (rom (2))jtan.ff .............................. (3). FORMUUfc OF COMPUTATION. 235 The same expression can be easily obtained from the A.R. {CenA}, where e is (101) and A is (100). If, however, the measured angle is that marked F in Fig. 179, i.e. hOlhQhl, the angle Cn = OQlhhOl must be found from the right-angled spherical triangle having a right-angle at C, and arcs = CVi for the sides meeting at C. Hence, cosF=cos*Cn. But, sec* Cn = 1 + tan 2 Cn = (from (3)) 13. Should the only forms meeting the principal axis be sphenoids, one of them (the faces of which we shall denote by o) is selected as c{lll}, and the others have then the symbols K {Mil}. Now the face (hhl) meets the axes at a -=- h, a t -H h, c -=- 1. If displaced parallel to itself so as to pass through A and A' where OA = OA' = a the intercepts are a, o / , fa + l. Let Fig. 183 represent part of a section in the plane S, where Op is the normal (hhl), OC = /ic + l, and Om is the length intercepted by AA' on OM of Fig. 173. Hence, Om = OA cos 45 = a -r-^/2. But, since the angles at p and COm are right-angles, COp = 90 - pOm = CmO. .-. tan COp = tan CmO = ~ = . ^ ; Om la COB 46 When h = l, the sphenoid becomes o = K {111}, and we have tan Co --sec 45 = tan E sec 45 => /2 tan E ........................ (6). The parameter c, and the angular element E, can, therefore, be calculated. 14. In the discussion of the very simple problems, which have been so far solved, stereograms were not required ; but more general problems are best solved by their aid. In such diagrams the principal axis is always taken to be the diameter through the eye: the pole (7(001) therefore occupies the centre. On the primitive, arcs Am - mA' = ' over the horizontal edge of the sphenoid K {111} : this angle was determined by Haidinger to be 108 40'. Hence, C being (001), (7a> = <7o = 54 20'. .'. from equations (2*) and (6) c=i&nE= tan Co+J2 = tan 54 20' -f- ^2. L tan 54 20' = 10-14406 9-99355. .-. #=44 34-5'; and c= -98525. The parameter c differs from unity by an almost insignificant amount. The angle E is nearly 45, and oo"=a>o>' is not far removed from 109 28' (the angle of the regular tetrahedron). The crystals therefore have some- what the appearance of those of one of the classes of the cubic system ; and it was not until 1822 that the crystals were proved by Haidinger (Mem. Warn. Nat. Hist. Soc. Edin. iv, p. 1, 1822) to belong to the tetragonal system. We can now determine the angle ow = 2oe. For, in the right-angled triangle Coe, Fig. 188, we know Co = 54 20' and Ce = E = 44 34-5'. Hence, cos oe = cos Co -=- cos Ce, .-. o = 353-6' and ow=707'3'. From Fig. 187, it is seen that o meets z and z' in parallel edges ; and that the face u also meets z and z t in parallel edges. Hence, Fig. 188, z is the pole in which the zone-circles through o and w perpendicular to the planes of symmetry intersect. These circles pass through m, (110) and m(110), respec tively. The symbol of the pole z is, therefore, obtained by (23) of Chapter v The zone [wi,o] is [112] and [mw] is [112]; therefore z is (201), and the pole lies L. C. 16 242 TETRAGONAL SYSTEM, CLASS I. tan oz -r- tan os = in the zone [CA], and the face is that of a tetragonal pyramid. The angle zo can now be found from the right-angled spherical triangle Coz by the expression sin Co = tan oz cot 45 = tan oz. .: oz = 39 5-5', and m,z = mz = 90 - oz = 50 54-5'. Poles. A single measurement in the zone [zo] suffices to give the symbol of any pole lying in it, for we know the symbols of three faces and the angles between them. Thus, if os is 28 26-5' by measurement, and if s is (hkl), the A.B. {ozsm,} gives 111 110 201 ^ 201 _ftjjk m ' ilo ~*-*' hkl hkl L tan (oz= 39 5-5' ) = 9-90978 L tan (os = 28 26-5') = 9-73371 log 1-5 = -17607. h + k_3 dA _ 5 fc _ x But s lies in [omj = [112]. .-. h + k-2l=0; and 1=3. The pole s is therefore (513), and belongs to the disphenoid K {513}. Pole r. Knowing the symbol of the sphenoid r to be K {332}, we can find the angles. For, supposing the pole r to be inserted in Fig. 188, we have the A.K. {Com} in which A Cm =90. tan Cr 4- tan Co = 3 25-5', and rr" = 12851'. But Co = 5420'; therefore, by computation, Cr is the angle over the horizontal edge. The angle over one of the slanting edges, such as q^ of Fig. 180, can be now obtained from the right-angled spherical triangle Amr ; for the angles over such edges are bisected at the poles A, A', &c. Thus, cos Ar cos Am sin Cr = cos 45 sin 64 25-5'. .-. .4r = 50 22', and the angle rr, over the edge is 100 44'. The stereogram. Fig. 188 is constructed as follows. On the primitive, arcs Am=mA' = &c. = JZ- 1. But c changes by insensible increments as the temperature varies, which the integers I and k cannot do. Hence, X cannot be equal to v. 248 TETRAGONAL SYSTEM, CLASS II. Similarly, if \=n, c 2 = 2Z 2 4- (h - &) 2 , which cannot remain true as the temperature changes. Examples. 27. The crystals of the following substances belong to this class: cassiterite, SnO 2 ; rutile, TiO 2 ; anatase, TiO 2 ; zircon, ZrO 2 .SiO 2 ; thorite, ThO 2 .SiO 2 ; phosgenite, (PbCl) 2 CO 3 ; torbernite, CuO(U0 3 ) 2 P 2 5 .8H 2 0; apophyllite, H 7 KCa 4 (SiO 3 ) 8 .4iH 2 O ; vesu- vianite, (Ca, Fe) 6 Al(F, OH) A1 2 (SiO 4 ) 5 (?). Cassiterite. The crystals are usually short tetragonal prisms z{110}, the edges of which are truncated by A {100} and often also bevelled by narrow faces h (2 10}. The crystals are most commonly terminated by faces {111}, Fig. 195; and occasionally by faces of e{101} and 0{321}. These latter forms most frequently occur on a somewhat rare variety from Cornwall represented by Fig. 193. Crystals of the habit shown in Fig. 195 are generally much twinned, and are often hard to decipher. Optically, they are uuiaxal and positive, i.e. the ordinary refractive index, o>, is less than the principal extraordinary index, e. I? IG. 193. In crystals of the habit given in Fig. 195, the best measurements are obtained in the zones [s,4s], [sms,J\. The faces A, m, &c., in the zone [Am] are usually deeply striated parallel to the vertical axis, and the images given when this zone is adjusted on the reflecting goniometer consist of bands, such that the angles read are of little value. By measuring the angles in [s,As] and [A'ss'"], the student can prove the equality of the angles ss' and ss'", and that AJJ' = A*Cy = 90 ; from them A Cs can be calculated and will be found equal to that given by measurement of [sms /t ]. Such measurements enable him to place on the stereo- gram, Fig. 194, the poles A, m, s and e and their homologues. i. The face m is taken to be (110), then A is (100). A knowledge of mh=l8 26', or of Ah=2& 34', proves that ft is (210) (see table in Art. 7). Taking to be (111), we fix the para- meter c, and can then find it and the angular element E from equation (6). Thus, if s" = (lll A 111) is found by measurement to be 87 8', then Cs = 43 34', and tan E = c = tan (Cs = 43 M')-*-,J*. .: E = B3 55-5', and c = -6726. ii. Form z. The Cornish crystals in the Cambridge collection, which show the form z, are usually combinations of z, m and s ; but the forms h and e are occasionally slightly developed. To determine the symbol of the EXAMPLE (CASSITERITE). 249 form, two angles in different zones must be measured, unless the faces e are present and the zone [mse] can be measured. a. In this latter case we can determine either the angle mz or ze, and we know that the faces are tautozonal. Let z be (hkl), then, since it lies in [m] = [lll], h-k-l = 0. Again, let T be the possible pole midway between e and e t in which the zone [me] meets [Cm,]; then T has the symbol (ll2), and mr=90. Taking the A.R. [mzeT], we have tan mz -H tan me lin _ r+ 2k 101 If now the angles me and mz given by measurement are both fairly good, they may be introduced into this equation when the ratio of I + k is determined. Then, by the equation of the zone, the three indices can be found. If, however, the measured angles are not good, it will be best to use the theoretical value of me, which can be readily obtained when E = Ce is known. For, from the right- angled spherical triangle Ame, cos me = cos A m cos Ae = cos 45 sin Ce. If then either mz, or ze, given by measurement, is fairly good, the equation representing the A.R. gives the symbol of z. /3. But if e is not present, we are not at liberty to assume that z is in the zone [me], and the method given in Art. 16 must be employed. Suppose the angles mz = 25 0' and zz l = 20 54' to be obtained by measurement. Taking, as in that Article, p on [Cm] midway between z and z 1 to be (eep), where e = h + k and g = 21, we have from the right-angled spherical triangle mzp, cos mz = cos zp cos mp = cos zp sin Cp. .: sin Cp=cos25 0'-=-cos 10 27' ; and, by computation, Cp = 6T 9'6'. But, by equations (6) and (7), f h-t-k e - = ?= = tan Cp -r- tan Cs = tan 67 9-6'-=- tan 43 34 ; h + k 5 .: by computation -^- = ^ . /. h + k-5l=0 (IS). Knowing now Cp and zp, the angle pCz can be found; and this is equal to m/ of Art. 16=45-J/. The expression is cotj>Cz = cotzp sin Cp, .: cotm/=cot 10 27' sin 67 9-6'. /. mf=U 19', and ^/=3341'. Therefore, by the table given in Art 7, / is (320). Introducing the value 3 for h, and 2 for k in equation (15), we have 1 = 1 ; and the pole z is (321). The angle sz can, also, be now computed ; for cos sz = cos zp cos sp = cos 10 27' cos (Cp - Cs) = cos 10 27' cos 23 36-6'. A z=2541'. 250 TETRAGONAL SYSTEM, CLASS II. 7. To determine the symbol of z from measurement of mz = 250' and sz = 25 41'. We now know the three sides of the triangle smz, for ms = 46 26'. Hence, by the formula which gives the angles of a spherical triangle when the sides are known (McL. and P. Spher. Trig. i. p. 47), sin 2 7-5' sin 22 52-5' '33-5' sin 23 33-5" and A wz = 24 44-5'. From the right-angled triangle szp, the arcsps andpz can be now found; and the symbol of z is then found in the way given under case /3. For, tan#s = tan (sz=25 41') cos (msz = 24 44'5'), and sin pz = sin msz sin sz. Hence, by computation, ps = 23 35-6', &n&pz = W 27'. and C = 61 9-6'. iii. A stereogram, Fig. 194, of the poles is made as follows. The poles A , m, &c., of the tetragonal zone [AmA'] having been inserted in the primitive by a protractor, diametral zones [CA], [Cm], &c., are drawn. These zones coincide with the circles in which the planes of symmetry meet the sphere. On [AC] an arc Ce = 33 55-5', or on [Cm] an arc (78 = 43 34', is marked off by the con- struction of Chap, vn, Prob. 1. The zone-circles [Ae'], [A'e], [me], &c., are then easily described- The poles z are the points of intersection of zone-circles, such as [me] and [hs], and can therefore be easily placed. iv. The crystal is drawn as follows. The cubic axes being projected by the method of Mobs or Naumann (Chap, vi), lengths OC and 0(7, ( = -6726 04") are measured off on the vertical axis. The lines joining C and C t to the axial points A, A', &c., on the axes of X and Y give the polar edges of the pyramid s. The upper and lower pyramid are then separated by a prism {110}, the edges having the length which corresponds to the particular crystal. To intro- duce the faces {100}, cut off from the polar edges by proportional compasses equal lengths, such as will give, approximately, the relative dimensions of the two prisms {110} and {100}. The edge [As] is parallel to the polar edge [*'], and similarly for all the homologous edges; the vertical edges [Am] can then be drawn, and the figure completed. 28. Zircon is found in crystals resembling those of cassiterite. The parameter and symbols of the faces would be determined in the way described in Art. 27. The crystals from some localities have the habits represented in Figs. 196 and 197. Taking p to be (111), it is easy to verify the symbols of the faces represented in Fig. 197 from the following angles : ^# = 31 43', #^ = 29 57'. The element FIG. 195. Fi ZIRCON, ANATASE, APOPHYLLITE. 251 #=32 38' can also be found from the same data; and then c = -6404 can be computed. The crystals are uniaxal and positive. FIG. 197. FIG. 198. Anatase usually occurs in small crystals which are very acute pyramids. This pyramid is taken to be {111} and is generally indicated by the letter p as in Fig. 198. The angle of the pyramid over the face m is 43 24'. Therefore mp =-- 21 42' and Cp = 68 18'. Hence, by equation (6), E= 6038' ; and c=l-777. The forms shown in Fig. 198 are : m {110}, p {111}, r {115}, c {001}, u {105}, e {101} and q {201}. The crystals have a good cleavage parallel to c(001); and they are uniaxal and negative. Apophyllite occurs in simple prisms a {100} with the pyramid p {111}, Fig. 199. The angle ap being 52 0', the arc mp can be computed ; for cos ap = cos am cos mp. Hence, by equation (6), E is found to be 51 22-5', and c = 1-2515. In Figs. 200 and 201 crystals of different habits are shown ; the additional forms being c {001} and y {310}. There is a good cleavage with a remarkable pearly lustre parallel to (001). The crystals have very weak double refraction ; and are sometimes positive, sometimes negative. FIG. 199. FIG. 200. FIG. 201. 252 TETRAGONAL SYSTEM, CLASS II. Between crossed Nicols plates of positive crystals show a dark cross and dull purple rings on a white ground : some plates of negative crystals from Uto absorb most of the light and show a grey cross on a violet ground, whilst others show both phenomena in adjoining portions of the same plate, which otherwise seems to be homogeneous (Des Cloizeaux). 29. Vesuvianite (idocrase) is generally found in stout prismatic crystals of the habit shown in Fig. 205. The faces m are large, and their edges are truncated by a {100} ; the edges [ma] being often modified by faces of the forms {210} or {310} : all these faces are, as a rule, striated parallel to the principal axis. The faces of c {001} and of the pyramid w{lll} are generally well developed, and are frequently the only faces associated with the prisms ; but numerous small faces are often found to modify the edges and coigns. The stereogram, Fig. 202, from Brooke and Miller's Mineralogy gives the forms recognised in 1852. The double refraction is weak ; and the crystals are generally negative, although they are sometimes positive : the plates occasionally show segments which are irregularly biaxal. The poles shown in Fig. 202 are : a {010}, m{110}, A {130}, /{120}, c{001}, n{113}, ?/{112}, {!!!}, w?{221}, t{331}, r {441}, e{011}, g {021}, v{151}, x {141}, s{131}, z {121}, o{241}, i{132}. We shall now show how to find the angles in the principal zones, from a knowledge of the angle cu and the indices given. i. In the prism-zone, all the angles are fixed; and ah, af are given in Art. 7. ii The angle CM is given by Miller as 37 7', and by Dana as 37 13-5'. The angles of crystals from different localities vary, and no single value of CM fits all of them, adopt Miller's angle. Hence, from equation (6) Art. 18, c = tan=tan (cu=37 .-. JE = 28 9', and the parameter c = -5351. iii. Let p be any pole (hhl) in the zone [cum]; then, from the A.B. {cpum} we have tan cp -H tan cu = h -~ I, or tan cp = h tan cu -i- 1. FIG. 202. We shall in the following computations EXAMPLE (VESUVIANITE). 253 Assigning, in turn, to p the indices of the poles, n, y, Ac. we have : for n(113), tan en = tan 37 7'-r-3, .-. en = 14 9-5'; y (112), tancj/ = tan 37 7'-:-2, .-. cy = 20 43-5' ; t0(221), tan CM? = 2 tan 37 7', .-. cw = 56 32-75'; *(331), tan ct= 3 tan 37 7', /. ct = 66 13-6'; r(441), tan cr= 4 tan 37 7', .-. cr = 7143'. iv. We have already determined A ce= E to be 28 9', .-. for g (021), = 28 9'), .'. c# = 46 56-5'. v. Zone [avxsu]. From the right-angled triangle aum, we can find au ; and then, taking Q to be any pole in the zone [au], we can, from the A.B. {aQue'}, obtain a formula which enables us to find each of the angles. From A amu, cot au tan am = cos (mau = a'e') = sin ce'. But, tan (am = 45) = 1, .'. cotaM = sin(ce'=28 9'). From A. B. {aQue'}, tanaQ-i-tanau=h-t-k. Hence, for v (151), cot av = 5 sin 28 9', .'. av = 22 58-3' ; x (141), cot ax =4 sin 28 9', .-. as = 27 55'; s (131), cot as= 3 sin 28 9', /. as = 35 14-6' ; z(121), cot az = 2 sin 28 9', V. as = 46 39'6'; w(lll), cotau = sin289', .'. au = 64 44-5'. vi. Zone [mosgi"]. The angles in this zone can be obtained in an exactly similar manner. For, cos mg = cos am eosag = cos 45 sin (eg = 46 56-5'). .-. mg = 58 53-5'. For any pole P(hkl) in this zone, the A.B. {mPgu"} gives, since m"=90, tan mP -=- tan mg = __ _ k-l~h + l' the two latter equations being obtained by taking different pairs of columns. If the angles have been measured, the two equations give the symbols of o, , i". But taking the symbols given, we have: for o(241), tan mo = tan 58 53-5' -f 3, /. mo =28 55'; s (131), tan ms = tan 58 53-5' -r-2, /. ms = 39 39' ; i" (132) , tan mi" = 2 tan 58 53-5', /. mi" = 73 12-6'. vii. Drawing. To illustrate the method of drawing Fig. 205 we give Figs. 203 and 204. The unit cubic axes, projected in the way described in Chap, vi, Arts. 11 or 23, are pricked through from a permanent card. The vertical axis OA" is alone altered, by multiplying it by c = -5351. We thus get OC and OC t , as shown in Fig. 203. The pyramid is obtained by joining C and C ; to the points A, A', &o. From the edges CA, CA', &c., equal lengths Cd, 254 TETRAGONAL SYSTEM, CLASS II. Cd', &c., are cut off by proportional compasses, as shown in Fig. 204. Pairs of the points 55', &c., are joined and give the face c (001). A second set of equal FIG. 204. lengths Ay, A'y', &c., are cut off in a similar manner from the edges CA, CA', &c. Through 7 the lines ya, 701 are drawn parallel to CA' and CA, respectively to meet A A' and A A, in a and Oj. Similarly through 7 , and the corresponding point on CA,, lines y'p, &o., are drawn parallel to CA and CA. Through the points a, a lf /3, &c., vertical lines are drawn of any desired length so as to correspond with the development of the par- ticular crystal. Through a 2 , a 3 lines are drawn parallel to AA', AA, and to 701, ya. Through 7' the lower edge y'Z, of the pyramid is drawn parallel to C,A of Fig. 203, and Z t 8. 2 is cut off by proportional com- passes to represent the same length as Cd. The rest of the construction is obvious. The coigns can now be pricked through to a fresh paper and Fig. 205 produced. Should it be desired to introduce new faces, the directions of the edges must be determined, and lines parallel to them be drawn through points marked off by proportional compasses on the homologous edges already introduced. III. Acleistous (polar) tetragonal class; T {hkl}. 30. Crystals of this class have only a single tetrad axis T, and no other element of symmetry. The faces must, therefore, occur in sets of four, which change places with one another after each rotation of 90, or any multiple of 90, about the tetrad axis. There is one exception to this rule in the case of the possible face (Chap, ix, Prop. 3, Cor. 1) which is perpendicular to the axis. This face cannot be repeated, and constitutes a pedion which is frequently ACLEISTOUS TETRAGONAL CLASS. 255 called the base. The general form is an acleistous tetragonal pyra- mid which cannot by itself enclose a finite portion of space. The class may therefore be called the acleistous tetragonal class. No crystal of the class can consist of a single form, but must be a combination of two or more forms. The possible crystal shown in Fig. 206 consists of the tetragonal pyramid T {hOl} with the pedion T {001}. The axis of symmetry is uniterminal and the crystals are hemi- morphic ; they should show pyro-electric phenomena. 31. The tetrad axis is taken as the axis of Z, and any pair of edges perpendicular to it which are at right angles to one another give the directions of the axes of X and T. These latter axes are parallel to the edges in which the pedion is met by the pyramid to which the symbol r{101}, or r{hOl\, is assigned. Fig. 206, representing the pyramid T {hOl} and the pedion r{001}, gives the directions of the axes of X, Y and Z. But we have already seen that the plane MLM' through opposite polar edges of the pyramid is parallel to a possible face ; and MM t MM', therefore the line MM' makes 45 with the axes of X and Y. The possible vertical face through X parallel to M t M' must therefore meet OF at a point such that OY=OX. Hence, we can take equal parameters, on these axes ; and such parameters are clearly the most convenient. The parameters may be given by a : a t : c, or by 1 : 1 : c. If there are a number of tetragonal pyramids in different azimuths, i.e. such that they meet the pedion in lines inclined at various angles to one another, one of the pyramids is taken to give the directions of the axes, and the others are then assigned symbols which correspond with their positions. There is nothing, however, to limit the choice of the pyramid selected to give the axes, and the crystallographer is influenced chiefly by a desire to give the simplest indices to the faces. But in every case the possible prism-face equally inclined to the axial planes XOZ and YOZ is taken to be (110), so that equal parameters are measured on OX and OY. 32. The tetragonal pyramid, r {hOl}. If the face parallel to OY in Fig. 206 is taken to 'be (hOl), then the homologous face, obtained 256 TETRAGONAL SYSTEM, CLASS III. by a rotation of 90, passes through MM' and L, and has the symbol (Ohl). The remaining two faces are (hOl) and (Ohl). Hence, the form T {hOl} consists of the faces : hOl, Ohl, hOl, Ohl (k). Again, if a section of this crystal is taken in the plane LOX, we have a right-angled triangle LXO, in which OX--=a + h, and OL = c + l. Hence, tan LXO = OL+OX = ^-. I a If n is the pole of this face, and C is the pole (001), then A Cn= f\LXO. .'. tanCVz, = T -. I a If we take the particular case of the pyramid in which h and I are both unity, and if e is the pole (101), then tan Ce = c -=- a = tan E. We, therefore, have the same formulae for determining the element c-^a, or E, as those given in (2) and (3) of Arts. 8 and 12. 33. The special forms are : (1} pedions ; (2) tetragonal prisms, {110}, {100}, T {MO}. 1. The pedions are the faces perpendicular to the tetrad axis ; and if both faces are present, they form complementary pedions and have different characters. Since they are parallel to the axes of X and Y, the one has the symbol r{001}, and the other the symbol r{OOT}. 2. {110}. We have already had in Art. 31 a prism-face (110) parallel to M t LM' of Fig. 206. This face must be associated with three others, the positions of which are obtained by rotations of 90, 180 and 270 about OZ. The prism is geometrically the same as {110} in the two preceding classes, and the Greek prefix may be omitted. The prism {110} has the faces : 110, TlO, HO, ITO. {100}. Again, faces through the similar and interchangeable edges M t M, MM', &c. of Fig. 206 can be drawn parallel to OZ. Since the faces are parallel to two of the axes, the symbols are 100, 010, 100, 010. They constitute a prism similar to that which has its faces parallel to the vertical axial planes in the two preceding classes. The symbol may therefore be written {100}. ACLEISTOUS PRISMS AND PYRAMIDS. 257 The tetragonal prism, T {MO}. If a prism occur in any general azimuth, one of its faces may be supposed to be drawn through the line HMK of Fig. 177. By rotations of 90, 180 and 270 this line is brought successively into the positions K'H^, H t K tj , K t H '. For we saw in Art. 7 that alternate sides of the ditetragon of this figure are at 90 to one another. Hence the. prism r {MO} consists of the faces : MO, MO, MO, MO (1). A similar prism T {MO} can be formed by drawing the faces through the remaining edges of Fig. 177 ; but it has no necessary connection with T {MO}, though the faces can be placed in a similar geometrical position by rotation through 180 about the line OM or the line OM t . But such a rotation interchanges different ends of the tetrad axis ; and the facial development and physical characters at the two ends are dissimilar. The two prisms cannot therefore be brought into similar positions without disturbing the arrangement of parts having the same physical characters. 34. The tetragonal pyramid, r {hhl}. Through the lines A A', AA t , &c., of Fig. 173 in which the faces of the prism {110} meet the plane XO Y, faces can be drawn to meet the tetrad axis at a finite distance on either side of the origin. One of the faces which meet at an apex L has the intercepts OA : OA' : OL, or a : a t : he -f- 1, if OL = hc + 1. The face has therefore the symbol (hhl). The three other faces all pass through L, and the signs of the first two indices are alone changed. Hence T {hhl} consists of : hhl, hhl, hhl, hhl (m). Again, if o is used to represent -the faces of the particular case in which h = I, then the pyramid oisT{lll}; it comprises the faces : in, in, ni, in. The inclinations of these two pyramids to the pedion, or to the plane XOY, are given by equations (5) and (6) of Art. 13. 35. The tetragonal pyramid, r{hkl}. Similarly, through the alternate edges HK, K'H ti , H t K ti , K t H' of Fig. 177 faces can be drawn to meet the tetrad axis at a point L, where OL = c + l. These four faces are interchangeable by rotation about the tetrad axis and constitute a pyramid T {hkl}, and have the symbols : hkl, khl, hkl, khl (n). In all the preceding pyramids the apex L may be on either side L. c. 17 258 TETRAGONAL .SYSTEM, CLASS III. of the origin ; and if we remember that I may represent either a positive or a negative number, the above expressions include all cases which can occur. 36. We have therefore pyramids r {hQl}, T {hhl\, T {hkl} which may be said to belong to three series. In the first the edges of the base are parallel to the axes, in the second they are at 45 to the axes, and in the third they may occupy any azimuth which is limited by the equation tan XOG = k-h. But we have already stated that any pyramid may be taken to give the directions of X and Y. There can therefore be no essential distinction between pyramids of the different series. The only difference which will be caused by changing the series, to which the pyramid giving the element belongs, is to alter the value of the element (whether it be E or c -=- a) ; and this new value can be determined when we know the pyramids which have been changed and the element corre- sponding to one of them. Thus, let e(101) and o(lll) be the faces of two pyramids in the first representation, and let E or cn-a be the corresponding element ; and in the second representation let o be made (101), so that Co is E' and the corresponding parametral ratio is c' : a'. Now, by equation (2), tan E = tan Ce = c + a. And by equation (6), tan Co = cJ2 ^a = f j2 tan E. But Co = E\ and tan E' = c' + a'. If a and a' are both taken to be unity, then c = tan E' = tan Co = J2 tan E = c^/2. The new parameter c' is therefore the original c multiplied by V2. From equation (9) we can find E" and c" the values of the elements when a face t of the pyramid r {hkl} is taken to be (101). For Ct - E", and c" - tan E" when a" = 1 . .-. c" = tan E" = ^ -^ tan E = ^ Jtf + tf. Thus, if (hkl) is (211), c" = cJ5. 37. If, however, the crystals are regarded as merohedral, the pyramids T {hQl}, T {hhl}, have each one-half the faces in the bi- pyramids {hOl} and {hhl} of Class II, which has the greatest sym- metry possible in the system, and the forms of which are regarded PRINCIPLE OF MEROHEDRISM. 259 as holohedral. The pyramids T {hQl} and T {hhl} are therefore hemi- hedral. But the pyramid r {hkl} has only one-fourth the faces of the ditetragonal bipyramid {hkl} of Class II, and is therefore tetartohedral. As, however, it is a matter of choice on the part of the crystallographer which of the series of pyramids is selected to give the axes, the different series cannot be some of them hemi- hedral and the others tetartohedral. For by taking (hkl) to give the element c" we change a tetartohedral form into a hemihedral one; or rather, a form which was in the first representation re- garded as tetartohedral has in the second case to be regarded as hemihedral. Similarly, the prisms {100}, {110} are regarded as holohedral, whilst T {MO} is taken to be hemihedral ; and by a variation in the axes of reference OX and OY, we can vary the manner in which these prisms are regarded. We see, therefore, that the views underlying the idea of rnero- hedrism lead to inconsistencies, and to representations of the crystals which are not in accordance with the facts. These difficulties are avoided when each class is treated as consisting of a group of crystals having definite elements of symmetry, of which the facial development and the physical characters are the consequence. 38. It is clear that analytically the method of determining the symbols of the faces and the parametral ratio from the measured angles, or of determining the angles from the symbols and element, must in this class be exactly the same as that given in preceding sections. 39. Crystals of toulfenite (PbMo0 4 ) have been described which show forms of this kind. The crystal shown in Fig. 207 (after Naumann) includes the forms: n=r{lll}, n = r{lll}, e=r{101}, 8=r{432}, X=T {3ll}. The element ^=57 37 '3', and c = 1-5771. The principal axis being one of uniterminal symmetry should be a pyro-electric axis : this has not been established in wulfenite; but in barium- antimonyl dextrotartrate, Ba(SbO) 2 (C 4 H 4 6 ) 2 . H 2 0, the crystals of which are held to belong also to this class, the tetrad axis has been found to be a pyro-electric axis. Crystals of these two substances do not rotate p IO 207. the plane of polarization. 172 260 TETRAGONAL SYSTEM, CLASS IV. IV. Diplohedral tetragonal class ; tr {hkl}. 40. Crystals of this class possess a tetrad axis, a centre of symmetry, and therefore, by Chap, ix, Prop. 4, a plane of symmetry IT, perpendicular to the tetrad axis. The forms can, therefore, be deduced from those of the preceding class by the addition of faces parallel to those given in Arts. 32 35. Or the new faces can be obtained by regarding the plane LT as a mirror in which the faces are reciprocal reflexions. 41. We can clearly take axes having to one another the same relation as the axes of crystals of class III ; i.e. OZ is the tetrad axis, OX and OY are parallel to the horizontal edges of any tetragonal bipyramid, and equal parameters are taken on them. The element c, or J, is obtained from the possible or actual pyramid to which the symbol {101} is given. The analytical relations of the element, angles and face-indices, and the methods of solution of the problems which may occur, are identical with those of crystals belonging to the preceding classes. 42. The prisms given in class III remain unaltered, for the faces, being parallel to a tetrad axis, occur in pairs which are parallel to one another. Hence : {100} consists of 100, 010, TOO, 010. {110} 110, T10, TTO, 1TO. ir{hkO} = r{hkO} MO, khO, hkO, MO. The faces perpendicular to the tetrad axis both occur together and form the pinakoid {001}. 43. The tetragonal pyramids become diplohedral, and each of them encloses a finite portion of space. They can, therefore, occur alone on crystals of this class. The bipyramid {hOl}, Fig. 179, having its faces parallel to the axes of X and T, is geometrically identical with that described in Arts. 8 and 22 as a possible form in each of classes I and II. The bipyramid includes the eight faces given in table e. The bipyramid {hhl}, which has its horizontal edges parallel to the diagonals of the square formed by bipyramids of the first series, is geometrically identical with {hhl} of Art. 23 and may be represented TETRAGONAL BIPYRAMIDS. 261 by Fig. 190 : it comprises the faces given in table i. The tetragonal bipyramids of these two series are therefore apparently holohedral. The bipyramid TT [hkl\ consists of the eight faces of table j, Art. 24, which pass through the sides of the square formed by the alternate sides of the ditetragon of Fig. 177, and have the symbols : hkl, khl, hkl, khl, hkl, khl, hkl, khl (o). The general appearance of the form is the same as that of each of the bipyramids {hOl\ and {hhl}; but, as it has only one-half the faces of the bipyramid of Art. 24, it was regarded as being hemi- hedral with parallel faces. 44. Since a bipyramid of any series may be selected arbitrarily to give the axes of X and Y, and the element, there is no essential distinction between the tetragonal bipyramids of the different series. Also, when a change is made in the axes and element, by selecting a bipyramid of a different series to give them, the values of the element in the two representations are connected by the equations given in Art. 36. It is clearly inconsistent with the symmetry of the crystals, of which these several forms are the immediate consequence, to regard two of the series as holohedral and the others as hemihedral. Different crystallographers might, by a mere change in the series to which the bipyramid selected as {101} belongs, make a definite bipyramid holohedral in the one case and hemihedral in the other. The same remarks apply to prisms {100}, {110} which are geometrically identical with those of the three previous classes, whilst TT {MO} has only half the faces present in {MO} of classes I and II. 45. Crystals of scapolite (wernerite), /??Ca 4 Al^O^ + /iNa^l^i^d ; of scheelite, CaW0 4 ; of stolzite, PbW0 4 ; of erythroglucine, C 4 H 10 4 ; belong to this class. Fig. 208 represents a crystal of meionite, in which variety of scapolite m : /? ranges from 1:0 to 3:1. The forms are : a {100}, m {110}, r {111}, and Z=TT {311}. The angle mr=58 9', and from equation (6) c = '43925. The double refraction is weak and negative, and it diminishes as the amount of sodium increases. Crystals of scheelite are optically positive. Fig. 209 shows a crystal having the forms: e{101}, o{lll}, h=n {313}, s=n {131}. The faces o are usually present and are sometimes those which predominate : as a rule, Fio. 208. TETRAGONAL SYSTEM, CLASS IV. they are bright and smooth, and give good reflexions. The faces e{101| are striated parallel to the edges [eh], and give poor reflexions. Hence the element is computed from measurement of the angle 00,,= 49 27'. By equation (6), =65 16-5') = 56 55-6', c = 1-5356. Zone [ehos]. Measurement of the angles in the zone [eh] enables us to determine the symbols of the faces; or, vice versa, knowing the symbols and the element, we can determine the angles which h, o and s make with e. For, as shown in Fig. 210, the poles lie in the zone [A'eA,]. Hence, if T is taken to be a pole (hkh) lying in this zone, we have from the A.B. {A' Toe} But, from the right-angled triangle A'om, we have cos A'o = cos (4'm=45) cos (mo = 24 43-5'), .-. A'o=50 2-25'. Hence, if T is taken to be * (131), .-. tan A's = tan 50 2-25' -f-3 ; and ^'s = 21 41-5'; and if to be h (313), .-. tan A'h = 3 tan 50 2-25', .-. A '/i = 74 23-6'. The construction of the stereograni, Fig. 210, presents no difficulties. The poles A , TO, 0}. i. If the face (qpQ) truncates the edge ns of Fig. 218, it lies in the zone [khl, hkl]. Hence by Weiss's zone-law, an equation which gives the ratio of q : p. L. c. 18 274 TETRAGONAL SYSTEM, CLASS VII. ii. Let OF be the normal to (qpQ), and OG' be the normal to (MO). Then XOG' = 45 + XOF. But, by equation (1), tan XOF = p -=- q ; and tan XOG' = h + k. tan XOG' - tan 45 If tan X0(?' is replaced by AH- k and tan 45 by 1, we have p _h+k-l _h-k which is the same result as before. A zone, in which a face inclined at 45 to any face of the zone is possible, will be called tetragonal (Chap, ix, Art. 12). 64. Some crystallographers give the relations of the faces of the sphenoid r a {khl} in the following manner. A face (khl) of the form is taken and rotated about the principal axis OZ through 90. The transposed face now occupies a position which is the reflexion in a mirror, situated in XOY, of a second face of the form. This face must therefore slope in a direction opposite to that of the first face so as to meet OZ at L t . The second face meets the plane XOY in HK lt of Fig. 217; and its symbol is (hkl). On a second rotation of 90 about OZ in the same direction the same operation is repeated : the face (hkl) being, after rotation, reflected in the plane XO Y, the homologous face (khl) must meet the principal axis in the same point L as the first face. The joint result of the two operations is equivalent to a simple rotation of 180 about OZ. When the operation is, again, repeated, a face (hkl) meeting OZ at L t is obtained ; and a third repetition of the operation brings the face to its original position. The lines H'K', HK ti , &c., in which the transposed face meets the plane XOY after each rotation, necessarily form a square. The special forms derived from this principle are : (1) a pinakoid {001}, (2) tetragonal prisms r {MO}, &c., such as have already been described. CHAPTER XV. THE CUBIC SYSTEM. 1. IN Chapter x, the cubic system was stated to include all crystals having four triad axes, the directions of which are given by the diagonals of a cube : they are therefore similarly situated with respect to three rectangular axes, parallel to the edges of the cube, which are either tetrad or dyad axes. The crystals fall into the five following classes : I. The plagihedral (pentagonal-icositetrahedral) class, in which the rectangular axes are tetrad axes, and the opposite ends of the triad axes are similar and interchangeable. These seven axes involve the presence of six like dyad axes which are parallel to the face- diagonals of the cube of which the triad axes are the diagonals. II. The hexakis-octahedral (diplohedral ditrigonal, holohedral) class, the crystals of which have a centre of symmetry associated with the thirteen axes of symmetry described under class I. It follows that the crystals have also nine planes of symmetry three cubic planes, each perpendicular to a tetrad axis ; and six dodecahedral planes, each perpendicular to one of the dyad axes. III. The tetrahedral (polar trigonal, tetrahedral-pentagonal- dodecahedral, teta/rtohedral) class, in which the rectangular axes, parallel to the edges of the cube, are dyad axes, and the triad axes are uniterminal and hemimorphic. The crystals have no other element of symmetry. IV. The dyakis-dodecahedral (diplohedral trigonal, parallel- faced hemiliedral) class, in which the rectangular axes are dyad axes and are associated with a centre of symmetry and with three planes of symmetry, each perpendicular to one of the dyad axes : the distribution of faces about opposite ends of each triad axis is similar, but the ends are not interchangeable. 182 276 CUBIC SYSTEM. V. The hexakis-tetrahedral (polar ditrigonal, hemihedral with inclined faces) class, in which the three dyad and four triad axes of symmetry characteristic of class III, are associated with six dodeca- hedral planes of symmetry intersecting in sets of three in each of the triad axes and in pairs in each of the dyad axes. 2. We shall begin with the two classes of crystals having three tetrad axes at right angles to one another. These axes are neces- sarily like and interchangeable ; for rotation through 90 about any one of them interchanges the two others, and therefore any faces similarly situated with respect to each of them. In Chap, ix, Prop. 12, ii (c), it was shown that the above is the only possible arrangement of several tetrad axes ; but it is easy to establish the proposition independently. For assume two tetrad axes to be inclined to one another at an angle a < 90, and let their directions be given by the radii of a sphere emerging at T and T l of Fig. 221 : further, let a be the least angle possible between any pair of such axes. If the sphere and crystal supposed to be rigidly connected together are turned about the diameter T through 90, then T^ is brought to TS on the great circle CT, where CTT^ = 90, and r\TTz=l\TT l . Then the diameter through T z must be the direction of a tetrad axis similar to that emerging at T t . Similarly, by a rotation of 90 about the diameter through T 1} T is brought to T s on the great circle CT l ; and the diameter through T 3 must be the direction of a tetrad axis similar to that emerging at T: also A T 3 T 1 = f\TT^. But the point C, in which the great circles CT, CT l intersect, is the pole of the great circle TT U for the angles at T and 7\ are right angles. It follows, therefore, that T t T 9 is less than TT^ This contravenes the limitation imposed on a, viz. that it is the least angle between tetrad axes. It is also clear that, if T 2 and T a are taken as the initial pair of axes, we can in a similar manner find new tetrad axes inclined to one another at a still smaller angle ; and that the process can be continued indefinitely. But in a crystalline substance, bounded by planes inclined to one another at finite angles, there cannot be an infinite number of tetrad axes making indefinitely small angles with one another. Hence a cannot be less than 90. If, however, T and T 7 , are 90 from one another, the points T THREE TETRAD AND FOUR TRIAD AXES. 277 and 7 T 3 of 'Fig. 221 obtained by a rotation of 90 about T and T^ re- spectively coincide in the point C, the pole of the great circle TT r . There can therefore be three tetrad axes mutually at right angles to one another; and, as a rotation of 90 about any one of them interchanges the remaining two, they are all like and interchangeable tetrad axes ; i.e. the distribution and arrangement of faces about each of them are similar, and the faces at each end of the axes are interchangeable. 3. It was shown in Chap, ix, Prop. 12, ii (c), that the three tetrad axes are associated with four like triad axes, which are the diagonals of the cube having its edges- parallel to the three tetrad axes. This proposition can also be established independently. a. Let, in Fig. 222, the three tetrad axes meet the sphere at the points T, T', T", and construct ac- cording to Euler's theorem (Chap, ix, Prop. 7), the triangle TpT', where l\pTT'= A P rr=45. Then p is the extremity of a diameter, rotation about which is equivalent to successive rotations of 90 about the diameters through T and T'. The spherical triangle TpT' is isosceles; and the great circle T"p8" bisects the side FT', and meets it at 90 in the point 8". Hence in the righE-angled spherical triangle Tp8", we have cos Tp&" = sin pTT' cos T8" = sin 45 cos 45 = 1 -=- 2 .'. A T P 8" = 60, and A TpT' = 2T P 8" = 120. The external angle 8pT' of the triangle at p is 60, and the least angle of rotation about the diameter through p is 120: the dia- meter is therefore a triad axis. The same point p and the same angle of rotation are obtained, if successive rotations of 90 clockwise about the diameters through T" and T' are taken. Hence all the angles at p are 60 ; and A Tp = A F'p = A T"p. The diameter through p is therefore a diagonal of the cube having its edges parallel to the tetrad axes. Similarly, the points p', p", p'" in each of the remaining octants above the paper are the extremities of diameters which are triad axes similar to, and interchangeable with, Op : they also are diagonals of the cube. 278 CUBIC SYSTEM. ft. We can also prove the proposition in the following way. In Fig. 223 (a), the sphere, with the crystal rigidly attached to it, is in the initial position; and the tetrad axes emerge at T, T' and T". The sphere is then turned through 90 about the diameter through T" in the direction of the arrow. The point T is trans- posed to T, and T to T, when the sphere is given by Fig. 223 (ft). Fio. 223 (a) FIG. 223 (b) FIG. 223 (c). The sphere is now turned through 90 about the right-and-left diameter, which has its extremity at T of Fig. 223 (b) ; the direction of rotation being indicated by the arrow at T. The positions of the several axes after this second rotation are shown in Fig. 223 (c) ; and are exactly the same as if the crystal had been turned once through 120 in the direction of the arrow about the diameter through p, which is equally inclined to the diameters through T, T' and T". In the same way it may be shown that the diameters through p, p", p" of Fig. 223 (a) are also triad axes equally inclined to the adjacent tetrad axes. It is also clear that successive rotations of 90 about the diameter through T" brings the axis p into the position of each of the other similar triad axes p, p", p". 4. Successive rotations about a tetrad axis T and a triad axis p, Fig. 223 (a), are, by Euler's theorem, equivalent to a single rotation of 180 about an axis emerging at 8", the apex of the triangle P T8". For the angle Tp8" = 60, and A pTS" = 45, each being one-half the angle of rotation about the respective axis : the angle at 8" is also a right angle and the equivalent angle of rotation about 08" is 180. Hence 8" is the extremity of a dyad axis. Similarly, the other points 8, 8', &c., bisecting each of the quadrants T'T", T"T, &c., are also the extremities of similar dyad axes interchangeable with 8" and each other. THIRTEEN AXES OF SYMMETRY. 279 5. We have therefore thirteen axes of symmetry three tetrad, four triad, and six dyad axes which meet the sphere in the points T, p and 8 of Fig. 222, respectively; and which are also shown in Fig. 224. In this latter diagram the tetrad axes are parallel to the FIG. 224. edges of the cube, and are indicated by lines of strokes and four dots ; the triad axes are the diagonals of the cube, and are indicated by lines of strokes and three dots ; and the dyad axes are parallel to the face-diagonals, and are given by lines of strokes and two dots. It is now necessary to prove that successive rotations about any pair of these axes give rise to no new axes. The combinations which remain to be examined are those in which pairs of dyad axes, pairs of triad axes, or a dyad and triad axis together or each with a tetrad axis, are those of successive rotation. 6. If two dyad axes are taken as those of successive rotations, we have two cases to consider : (i) that in which the plane of the dyad axes contains a pair of tetrad axes ; (ii) that in which their plane does not contain a tetrad axis. i. Suppose the pair to be 8", 8"', those in the plane of the primitive of Fig. 225. Then Enter's construction gives the triangle 8"T"8"'; and OT" is the equivalent axis of rotation. But, since the angle 8"T"8"' = 90, the external angle at T" is also 90 ; and the angle of rotation about OT" is 180. Hence successive rotations of 180 about the pair of dyad axes are equivalent to a single rotation of 180 about the tetrad axis OT". No new axis, and no new rotation has been introduced ; for 180 = 2x90 is made up of two rotations of 90 about the tetrad axis. ii. Suppose 8* and S 5 of Fig. 225 to be the pair of dyad axes. Now 280 CUBIC SYSTEM. 8 4 and 8' are in the same plane with T" and at 45 from it : they are therefore at 90 from one another. The axis S 4 is also at 90 from T' : it is therefore at 90 from every point in the great circle T'p8', and the arc p8 4 = 90. Similarly, 8 5 is at 90 from both 8 and T, and from every point in the great circle Tp8: hence arc p8 5 = 90. In the same way it can be proved that p8"'=p8 /// =90. It follows that p is the pole of the great circle 8 4 8 5 which passes through 8"'8 ttl . Further, every great circle through p must meet the great circle 8"'S 4 8 5 at right angles. But to find the axis equivalent to successive rotations about the axes S 4 and 8 5 , we have, by Euler's theorem, to draw great circles through 8 4 and 8 5 at right angles to the great circle 8 4 8 6 : these circles meet at p, which is, therefore, the extremity of the equivalent axis. Again, from the right-angled triangle T cos S 4 ^ = cos T"8* cos T'W = cos 2 45 = 1 -i- 2. .-. A 8 4 8 5 = 60. Similarly, A8'"8 4 =AS 5 8 /// = 60. Hence, we have three dyad axes in the plane of the great circle 8'"8 4 8 5 inclined to one another at angles of 60. By Chap, ix, Prop. 11, the diameter Op perpendicular to their plane is a triad axis. Or we can esta- blish this independently as follows. The arc 8 4 8 5 = A 8 4 p8 6 subtended at the pole of the great circle 8 4 8 6 . Hence, the external angle of the triangle 8 4 p8 5 at p is 120. The angle of rotation about the equivalent axis Op is therefore 2xl20 = 360-120. Hence, after rotations of 180 about 08* and OS 5 , the crystal is in exactly the same position as if it had been turned through 120 about Op in a direction opposite to that required by Euler's theorem. The least angle of rotation being 120, the axis is a triad axis. In the same way it may be proved that the directions of any other pair of dyad axes (not in a plane with a tetrad axis) are at 60 to one another and perpendicular to one or other of the triad axes. Hence, successive rotations about any pair are equivalent to a rotation of 120 about the triad axis perpendicular to both the dyad axes. 7. Successive rotations about a pair of triad axes give no new axis. For if p and p are taken, the triangle formed according to Euler's theorem is pT"p', or p7"p' according to the order and direction of the rotations. The angles pT"p and p^'p' are each 90. Hence the two rotations are equivalent to a rotation of 180 about T" or T'. These are the same as two rotations of 90 about each axis ; and hence we have no new axis and no new rotation. ANGLES BETWEEN THE AXES OF SYMMETRY. 281 If the two axes p and p" of Fig. 225 are taken, the triangle formed by Euler's construction is pp'p" or pp'"p" according to the order and direction of the first rotations. But the angles pp'p", pp'"p" are each 120. The angle of rotation about the equivalent axis p' or p"' is therefore 120, i.e. that about a triad axis. 8. Successive rotations about a tetrad axis T" and an adjacent dyad axis 8 are equivalent to a rotation of 240 about the adjacent triad axis p or p'. But this is the same as a rotation of 120 in the opposite direction about the same axis. Similarly, a combination of an adjacent p and 8 gives one of the adjacent tetrad axes. 9. The only remaining combinations are those in which a tetrad or triad axis is combined with a dyad axis inclined to it at 90. Rotations about a tetrad axis T" and a dyad axis 8" are equivalent to one of 180 about T or T'. Hence we have no new axis or rotation. Similarly, successive rotations about p and 8* are equivalent to a single rotation of 180 about 8'" or 8 5 . For according to Euler's theorem we have to construct a spherical triangle p8 4 8'" or p8 4 8 5 , in which the angles at p are 60 and those at 8 4 are 90. Hence, the apex is at 8"' or 8 6 , and the angle of rotation is 2 x p8"'8 4 = 2p8 5 8 4 = 180. 10. The angles between the several axes of symmetry are clearly fixed and constant, for the triad and dyad axes are a necessary consequence of the coexistence of three tetrad axes, and it was proved that these latter must be at 90. The angles can be all determined without difficulty. The triad axes Op, &c., are all equally inclined to each of the tetrad axes; hence, in Fig. 225, Tp T'p = T"p. Further, the planes containing each a tetrad and a triad axis, such as T"p, bisects the angle between the two other tetrad axes. Hence, T8" = &'T' = &c. = 45. But the arc TB" is equal to the angle 7T"8", for T" is the pole of the great circle TT'. The angles P T"T = pT"T' = pTT' = &c. = 45. Hence, from the right-angled triangle T"p8, we have But A T"8 = A 8T"p = 45, and cot 45 = 1. .*. tan p8 = sin 45 ; and Ap8 = 3516'. Hence App' = 2p8 = 70 32' = Ap'p" = &c. also arc Tp = 90 - P 8 = 54 44' = pT' = pT". :. App" = 2pT" = 109 28' = /\p'p" = &c. It has already been shown (Art. 6, ii.) that AS 4 S 5 =A8'"S 4 = &c. = 60. 282 CUBIC SYSTEM. 11. It is clear that each octant in Fig. 225 included between tetrad axes is made up of six equal triangles having a common apex at a point p, where the triad axis emerges. Rotation through 120 about Op will necessarily interchange three of these triangles. Thus, taking the first octant TT'T", we have the three equal tri- angles Tp&", T'p8, T"p8', interchangeable by rotation about p. We have also the three similar equal triangles T'pb", T"p8, Tp8' which are interchangeable with one another but not with any one of the first set. If now the sphere is turned through 90 about a tetrad axis, such as T", each of the above sets of three triangles is interchanged with a similar set in either of the adjacent octants. Further, if the sphere is turned through 180 about a dyad axis, 8, say, lying in the plane dividing the octants, we must clearly interchange similar sets of three triangles. It is easily seen that the sets interchanged by the latter rotation are identical with the sets interchanged by a rotation of 90 about T", although the individual triangles of the two sets which change places with one another are different. Hence the sphere is divided into forty-eight equal triangles, each of which has its angles at one of the points T, p, 8, &c., respectively. These triangles are divisible into two sets of twenty- four, such that the triangles of one set are interchangeable with one another, but not with those of the other set. 12. We shall in each class of the system adopt as axes of reference the three axes parallel to the edges of the cube, for they are either dyad, or tetrad, axes which remain at right angles to one another at all temperatures. When the axes of reference are tetrad axes, the lines bisecting the angles between each pair are dyad axes, and therefore necessarily possible zone-axes (Chap, ix, Prop. 2). When the axes of reference are only dyad axes, the face-diagonals of the cube are the directions of zone-axes but not of dyad axes : for each of them is the inter- section of a face of the cube with the possible plane containing the two triad axes which join the extremities of the face-diagonal to the centre of the cube. But in Art. 6, ii, it was shown that the radii to 8"', 8 4 , 8 6 all made 90 with that through p. Hence the diagonal Op of the cube is perpendicular to the possible face parallel to the radii through 8'", 8 4 and 8 5 : this face is always taken as the para- metral plane (111). But by a rotation of 120 about Op the axes of THE EQUATIONS OF THE NORMAL. 283 reference are interchanged, whilst the direction of the plane (111) remains the same. Hence it must meet the axes at equal distances from the origin ; and o = b = c (1). This may also be proved from the equations to the normal Op of the parametral plane, which are a cos Tp = b cos T'p = c cos T"p. But it was shown that Tp = T'p = T"p. Hence cos Tp = cos T'p = cos T" P . We shall denote the pole (100) by A, (010) by A' and (001) by A" : they coincide with T, T', T" of Fig. 225, respectively. The points p are the positions of the poles of the faces of the octahedron, which we shall denote by the letter o. Further, we shall denote by a t a length measured on T equal to a on OX, and by a lt the same length measured on OZ. 13. It follows that in the cubic system no element of the crystal varies with the substance, and that the angles between faces with known symbols must be fixed and constant. This can also be proved from the equations of the normal; for, if P is the pole (hkT)^ the equations are a cos A P a CosA'P a /t cosA"P h ~~k I ' But the equal lengths, a, a,, a lt can be each taken as unity or cancelled, when the equations become cos^P cos^'P cos^l"P 1 IT The last term is obtained as follows. Let each of the first three terms = t ; then ht = cosAP, and hW = cos?AP; kt = cosA'P, W = cotfA'P; It = cos A" P, W = co&A"P. Adding the squared equations together, we have i 2 (A 2 + k 2 + Z 4 ) = cos 2 A P + cos 2 A'P + cos 2 A"P = 1 ; for, the axes being rectangular, cos 2 AP + cos*A'P + cos 2 A"P=1. 284 CUBIC SYSTEM, CLASS I. It is clear therefore that the arcs AP, A'P, A"P can be computed for any values of h, k and I introduced into equations (2). The angles between any two faces whether of the same or of different forms, must consequently be fixed and constant. For, taking P (hkl) and Q (pqr) to be the poles of any two faces, we have for P the equations (2), and for Q the similar equations : cos -4$ cos^'^ cos A"Q 1 pqr ~ Jj? + q> + r>' But the general expression for the angle between two poles, given in Chap, xui, Art. 36, is in this system cos PQ = cosAP cos AQ + cos A'P cos A'Q + cos A"P cos A"Q. Introducing the value of cos^lP, cos AQ, &c., from the equations of the two normals, we have hp + kq + Ir cos PQ = .(3). It is therefore only necessary to introduce on the right side of (3) the numerical values of the face-indices to obtain the cosine of the angle between any pair of faces. From equations (2) and (3) all the angles corresponding to any particular cases can be deduced, as will be shown in later articles. I. Plagihedral class; a {hkl}. 14. The class having three tetrad axes, four triad axes and six dyad axes, but no other element of symmetry, we shall call the plagihedral class of the cubic system ; and we shall use the symbol FIG. 226 THE CUBE AND OCTAHEDRON. 285 a {hkl} to denote the general form which is a pentagonal icositetra- hedron. This form is the only one which is peculiar to the class, and the Greek prefix will be omitted before the symbols of the special forms which are common to this and the next class. 15. The cube, {100}. The simplest form is that in which each face is parallel to two of the tetrad axes and therefore perpendicular to the third. If the vertical face (100) is turned through 90 three times successively about OT" of Fig. 226, it is brought into the position of three other faces which have the symbols (010), (TOO), (010), respectively. By similar rotations of 90 about OT', it is brought to the positions in which the faces have the symbols (001), (TOO), (OOT), respectively. We have already seen (Art. 3) that successive rotations about OT' and OT" are equivalent to a single rotation about one of the triad axes. Hence no other faces can belong to the form, which may be denoted by the symbol {100}, and includes the six faces: 100 010 001 TOO OTO OOT (a). The edges, parallel to the tetrad axes, meet in sets of three at coigns p on one or other of the triad axes, and are each bisected at right angles by one of the dyad axes. In theoretical discussions the faces are, except when the contrary is expressly stated, supposed to be of equal dimensions. Such an ideal figure is shown in Fig. 226 : since it serves as the basis for the construction of several important forms of the system, we shall, for convenience, denote its faces, edges and coigns as cubic faces, cubic edges and cubic coigns, respectively. The method of drawing the cube was described in Chap, vi, Arts. 9-12 and 22, 23. The faces are usually denoted by the letter a ; and the poles by A, A', A", &c. 16. The octahedron, {111}. The form {111}, of which the parametral plane is a face, is a regular octa- hedron, Fig. 227, with eight equal faces, each of which is an equilateral triangle : 111 1T1 TTl Till 11T ITT TTT TIT/" The coigns are at the points T, T', T", &c., in Fig. 226 where the tetrad axes meet the cubic faces, so that the figure is readily drawn, when the cube has been projected by FIG. 227. 286 CUBIC SYSTEM, CLASS I. either of the methods given in Chap. vi. The triad axes are per- pendicular to pairs of parallel faces, and pass through their middle points ; and the dyad axes are parallel to the edges. When the form is equably developed, each face is an equilateral triangle. In Nature the faces are apt, owing to the accidents attending the deposition of the crystalline matter, to be very unequally developed, as is shown in Fig. 228. The angles over the edges of adjacent faces of the regular figure are clearly all equal to the least angle between the triad axes, i.e. to 70 32' ; for it was proved in Art. 10 that pp = p'p" = &c. = 70 32'. This angle can also be found from the equations (2) of the normal, which become for o (111) : FIG. 228. cos Ao = cos A'o = cos A"o = 1 -r- ^3 ............... (4). .'. tan 2 .4o = sec 2 Ao-l = 3- 1 = 2 '. tan Ao= J2 44', and Aoo'= , ............ (5). 2^o = 70 32'. 17. The rhombic dodecahedron, {110}. Another important form is that which has each of its faces perpendicular to one of the dyad axes. Each face is therefore parallel to two triad axes ; for it was proved in Art. 6 that S 4 of Fig. 225 is at 90 from p, and, similarly, it is at 90 from p". The face of which S 4 is the pole is also parallel to the tetrad axis OT' and the dyad axis 08', for the arcs &T' and 8 4 8' are also 90 : the face is equally inclined to the two other tetrad axes, for f\&T= f\8*T" ; it there- fore makes equal intercepts on these axes of reference, and its symbol is (T01). The same rela- tions hold for the faces perpen- dicular to each of the other dyad axes, and the form must consist of twelve similar and interchange- able faces, pairs of which are parallel, for they are parallel to both a tetrad and dyad axis. THE RHOMBIC DODECAHEDRON. 287 The form, Fig. 229, is most easily drawn by first constructing the cube, and taking the axes OA, OA', &c., through its middle point. On these axes points D y D', &c., are marked off; where OD = 20A, OD' = 20A', &c. These points are then joined to the four adjacent cubic coigns p; and give the edges of the form. In the diagram the front edges of the cube are shown by lines of interrupted strokes. It is clear that a face, such as DpD'p ti contains the cubic edge 'pp ti , parallel to the tetrad axis OD" ; and that DD' lies in the face and bisects the line PPii at 8". For A'D' = OA' = OD' H- 2, and AT = pp -=- 2 = OD -=- 2. Hence A'D' : AT = OD' : OD. The point 8" therefore lies on DD' and bisects it. Also, since OD is equal and parallel to pp', the plane figure ODpp is a parallelogram. Therefore Dp is equal and parallel to Op. The edges of the rhombic dodecahedron are therefore equal and parallel to the triad axes joining the centre to the coigns of the cube. The form includes the twelve faces : 110 110 110 110\ on on on on I 101 TOT TOI IOT (c). The faces having their symbols in a row are parallel to and interchangeable about the tetrad axes OZ, OX and Y, respectively. Those which have their symbols in a column meet at the alternate coigns p, p tii , p", p t , respectively. Again, the angles between ad- jacent faces which meet in edges, such as Dp, parallel to the triad axis are each of them 60 (Art. 6) ; the angles between alternate faces meeting at tetragonal coigns, D, D', - Again, in Fig. 232, the arc G between two adjacent poles, such as g and g 6 , is the hypothenuse of an isosceles right-angled triangle gAg 5 . Hence, by Napier's rules, cos G = cos gg & = cos Ag cos Ag 6 = cos 2 Ag. But .'.cos G The values of cos F and cos G can be easily found from the general expression (3). The following are the angles for a few of the tetrakis-hexahedra of most frequent occurrence : {210} {310} {410} 26 34' 18 26 14 2 F=hkQf\khQ 36 52' 53 8 61 56 36 52' 25 51 19 45 CYCLICAL ORDER. 291 19. The cyclical orders employed in Art. 18 can be explained by Figs. 233 (a) and (6). In the former a small circle may be FIG. 233. (&) supposed to be described through the extremities X, Y, Z of the axes with p (the extremity of the triad axis) as centre. The indices h, k and I of the first face of a triad, interchangeable by rotations of 120 about Op, are attached to X, Y and Z. When now the crystal is rotated with the arrow through 120 about Op, any length on OX is transferred to the axis OY; and, similarly, a length on OY is transferred to OZ, and one on OZ to OX. The axes of reference themselves retain fixed directions in space, but the faces of the crystal and the axes of symmetry are moved into homologous positions. Hence the length a -=- h on OX is after rotation measured as a t -rh on OY; and, similarly, a^k on OF as a tl -^-k on OZ, and ' a /t -T-l on OZ as a-^-l on OX. The symbol of the face in its new position is therefore (IhK). The change in the order is indicated in Fig. 233 (a) by writing the indices against the corresponding axes outside the circle and in different type. A second rotation in the same direction produces similar changes ; and the length a -=- 1 on OX is now transferred to OY, a t + h on OF is now measured on OZ, and a,, 4- A; on OZ as a + k on OX. The symbol of the new face is (kill) ; and the change is shown in the diagram by letters of different type placed inside the circle. A third repetition brings the face back to its original position ; and the reader can easily verify that a repetition of the process on the last arrangement of the indices . gives the original set. The three faces connected together and interchangeable by rotation about a triad axis are hkl, Ihk, and klh, the symbols of which have one cyclical order. If the indices in the symbols are taken in the reverse order, we have a second triad of faces interchangeable by rotation successively through 120 about the triad axis, the symbols of which are said to be in the reverse, or opposite, cyclical order. 192 292 CUBIC SYSTEM, CLASS I. If the indices alone are written on the circumference of the circle, Fig. 233 (6), and if, starting from each index in turn, they are taken in the direction of the feathered arrow, we obtain the symbols of the triad of faces in one cyclical order ; viz. hkl, Mh, Ihk. If, however, starting from each index in turn, we take them in the reverse order, given by the unfeathered arrow, we obtain the triad of faces, the symbols of which are in the reverse, or opposite, cyclical order to the first ; viz. Ikh, khl, hlk. The cyclical orders of the above symbols are a necessary consequence of the fact that rotation through 120 about a triad axis interchanges the axes OX, OY and OZ. They are of con- siderable assistance in verifying the correctness of the symbols of the faces of a form. Thus the symbols in each of the columns of (d) are those of triads, the faces of each of which can be interchanged by rotation about Op. The faces can be easily rearranged in triads such that the faces of each triad are inter- changeable about any of the other triad axes ; and the faces of each triad are in cyclical order. A coign formed by three faces having their symbols in cyclical order is called a trigonal coign ; one formed by two triads of faces the indices of which are the same numbers taken in opposite cyclical orders is called a ditrigonal coign. 20. If the faces of the form are parallel each to only one dyad axis, i.e. to an edge of the octahedron, we have two forms with parallel faces which can be constructed as follows. Two planes are drawn through each edge of the octahedron to meet on opposite sides of the origin that tetrad axis which is perpendicular to the octahedral edge at distances ha + l. The number of octahedral edges being twelve, the forms have twenty-four faces : they differ in angles and in the shapes of the faces according as h < I. The triakis-octahedron, {hhl}, h>l. Let, in Fig. 234, H"AA', H t/ AA' be two faces drawn through the octahedral edge A A' to meet OZ in H", H a , where OH" = OH it = ha^ + l the points H being more remote from the origin than the points A. Through the similar edges A'A", A"A t , &c., pairs of like planes can be drawn to H and H on OX; and through A" A, AA /t , &c., pairs of similar planes can be drawn to H' and H t on Y, where OH=OH'=&c. =/ta + l. The faces AA'H", A A" II' pass through A, and meet the plane THE TRIAKIS-OCTAHEDRON. 293 YOZ in the lines A'H", A"H', respectively. These lines inter- sect at 8 on the dyad axis 08, and the two faces meet in the edge A8. Similarly, the faces AA'H", A'A"H have A' in common, and meet the plane ZOX in the lines AH", A"H, respectively. The edge of intersection is therefore A'8', 8' being a point on the dyad axis 08' in which AH" and A"H intersect. The two edges A8 and A'8' are common to the face AA'H", and therefore meet in a point p which must lie on the triad axis ; for the three faces are interchanged by fr, ' A* rotations of 120 about this axis, and must therefore meet at a point on it. Similarly, the two faces A'A"H and A" AH' meet in the line A"8" which also passes through p. Flo 2341. Again, rotation about the dyad axis 08" must interchange X and Y, and equal positive and negative lengths on OZ. We therefore have the three faces meeting in the edges Ap /it A'p ti , A^p^; which can be drawn in the same way as those meeting at p. The edge AA' is at right angles to 08" and is common to both the faces AA'H", AA'H lt . The triads meeting at p" and p t are obtained in a similar manner; and the four triads can be interchanged by successive rotations of 90 about OX. Four similar triads, having A for common ditetra- gonal coign, occupy the four octants behind the paper. The figure may be regarded as the result of placing on each face of the octahedron a like trigonal pyramid having its base congruent with the octahedral face on which it is set. The form is known as the triakis-octahedron. In Fig. 235 the face p" drawn through AA'H" of Fig. 234 has the intercepts a : a t : 1ta it -=- 1. When transposed parallel to itself, the face may be given by a H- h : a t H- h : a,, -H I, and by the symbol 1 To avoid complicating the diagram the lines of construction are not all shown. Several of the points on the dyad axes are indicated by crosses ; only one dyad axis is drawn, and the tetrad axes are not continued through the origin. 294 CUBIC SYSTEM, CLASS I. (hhl). Similarly, the face p is A'A"H of Fig. 234 and intercepts lengths on the axes in the ratios a -=- 1 : a, + h : a tl -=- h ; it is (Ihh). In the same way it may be shown that p' is (hlh). The indices in the three symbols are in cyclical order ; and, since two of the indices are equal, there is only one such arrangement. The triads in the other octants only differ from the preceding triad inasmuch as one or more of the axes of reference are met on the negative side of the origin ; consequently one or more of the indices change signs. The form consists of the following faces : Ihh hlh hhl Ihh hlh hhl\ Ihh hhl hlh Ihh hhl hlh I ihh hlh hhl ihh hlh hhi I ^' Ihh hhl hlh Ihh 7ihl hlh } As particular instances we have {221}, {331}, {332}, {443}. In the stereogram, Fig. 236, the poles lie on the zone-circles in Ihk FIG. 236. which the planes containing two triad axes, one tetrad and one dyad axis meet the sphere. From the construction it is clear that pairs of the faces are tautozonal with pairs of octahedral faces, and that a dyad axis is parallel to each of the edges which lie in the axial planes. It is clear also that a combination of the octahedron {111} with {hhl}, h> I, is obtained by drawing through points on Ap, A'p, &c., equally distant from p, p', , l, Art. 20. 6. The icositetrahedron {hhl\, h,, = 54 44'. THE TETRAHEDRON. 313 33. The dyad axes, being at right angles to one another, are the most convenient axes of reference, and we shall take OX to coincide with 08, OY with 08', and OZ with 08". We shall also take the face p'p'"p t , of Fig. 264 to be the parametral plane (111); and, as it is perpendicular to pOr, it will retain the same direction after each rotation of 120 about this axis. But a rotation of 120 about pOr interchanges 08, 08' and 08", which must therefore be equal lengths. Hence the lengths intercepted by (111) on the axes are equal ; and a - b = c. These equalities can also be proved from the equations of the normal Or to the face (111), which are a cos XOr _ b cos YOr _ c cos ZOr ~~Y~ ~1~~ ~T~ But XOr = YOr = ZOr; :. a = b = c. We shall, as before, use A to represent the pole (100), A' (010) and A" (001) ; and we shall denote a length a by a t when it is measured on OY, and by a lt when it is measured on OZ. As the above axes of symmetry with their consequent rotations exist in all cubic crystals (Art. 1), being the least assemblage of axes of symmetry which can occur in any crystal having more than one triad axis, it follows that the same axes and parameters can be taken for crystals of all classes. Further, the equations of the pole P (hkl) are cos AP _ cos A'P _ cos A"P = 1 ~~h k I ~ V/^+^ + J 2 ' equations already established in Art. 13. It follows also that expression (3) holds for the cosine of the angle between any two poles P and Q, whichever the class to which the crystal belongs. 34. The tetrahedron, T{111}. This form consists of the four faces: 111 111 Til III (j). The form is most easily drawn by first constructing the cube by one of the methods given in Chap. vi. Opposite corners of each face, such as p, p" of Fig. 226, are then joined so that three edges bounding each face cut off portions of the cube at alternate coigns- By tracing Fig. 226 on thin paper, and joining the pairs of points 314 CUBIC SYSTEM, CLASS III. p', p'"; p"', p ti ; p ti , p'; &,c., the student \vill form a tetrahedron T {111} similar and similarly placed to Fig. 264. From the method of construction it is clear that each face is an equilateral triangle, for the edges are all diagonals of the faces of the cube. If the pairs of coigns p, p"; p, p/, &c., of Fig. 226 are joined, we obtain the complementary tetrahedron r{llT}. This is tauto- morphous with r{lll}, for a rotation through 90 about one of the dyad axes parallel to the edges of the cube interchanges the two diagonals of the cubic face to which the axis is perpendicular, and adjacent coigns of the cube. As was shown in Chap, m, Art. 2, the tetrahedron can be derived from the regular octahedron by omitting alternate faces of the latter, and extending the faces retained to intersect one another as shown in Fig. 266. Further, a geometrically equal and similar tetrahedron can be formed from the same octahedron by retaining the faces omitted in the first ; and the faces of the one are parallel to those of riG. 2bo. the other. Hence the complementary tetra- hedron can be represented either as T{llT} or by T{ITI}; and comprises the faces : 111 111 111 III. We shall use the letters o to denote the faces of T{111}, and generally the letters w to denote those of the complementary form 35. Tlie cube, {100}. The cube, the faces of which are given in table a, Art. 15, is a possible form of this class. The faces are each parallel to two dyad axes, and must therefore occur in pairs of parallel faces which truncate opposite edges of the tetrahedron. For each face is perpendicular to a dyad axis, which bisects the angle between the normals to the two tetrahedral faces meeting in that edge through the middle point of which the axis passes. Figs. 267 and 268 represent two possible crystals in which the cube is combined with a tetrahedron. In the first the tetrahedron o = r{lll} is the predominant form, and the faces of {100} are narrow planes truncating the edges. In the second the cubic faces are large, and those of the tetrahedron r{lll} are small. In drawing such combinations the cube should be first constructed. THE RHOMBIC DODECAHEDRON. 315 Equal lengths, measured from alternate coigns, are then cut off by proportional compasses on each cubic edge ; and the pairs of points - FIG. 268. which give edges parallel to the face-diagonals are afterwards joined, and complete the figure. 36. The, r/tombic dodecahedron, {110}. If a face is drawn through an edge of the cube making equal angles with the cubic faces forming the edge, it must meet the two dyad axes perpendicular to that edge at equal distances from the origin. Suppose the face to be drawn through the upper horizontal cubic edge parallel to OF; then its symbol is (101). By a semi-revolution about OX this face is brought into the position of one through the lower horizontal cubic edge and meeting OZ at the same distance from the origin as the first, but on the negative side; its symbol is therefore (101). Again, a semi-revolution about Y brings these two faces into the positions of parallel faces. We therefore have the four tautozonal faces : 101, 10T, TOT, T01 ; and the angles they make with one another are 90. By rotations of 1 20 about one of the triad axes, the axis OY is brought in succes- sion into the position of OZ and OX, and the four faces just given must be repeated in two sets of four similar faces passing through the cubic edges parallel to the axes of OZ and OX. The form so con- D t structed is the rhombic dodeca- hedron {110}, which was fully de- scribed in Art. 17. As was there shown the edges are parallel to the triad axes; and Fig. 269 is most easily drawn by determining the p IO 2 69. 316 CUBIC SYSTEM, CLASS III. points D, D', &c., on the axes at distance 2a from the origin, and joining these points to the four nearest coigns of the cube. As the elements of symmetry characteristic of this class occur in all classes of the cubic system, it follows that the cube and rhombic dodecahedron are common to all classes of the system. 37. The dihedral pentagonal dodecahedron, T {MO}. When a face, drawn through a cubic edge is unequally inclined to the cubic faces through that edge, it meets the axes perpendicular to the edge at unequal distances from the origin. Thus, let the face be Hpp tl of Fig. 270 : its symbol is (A&O), where h is greater than k. A semi-revolution about OX brings the face into the position Hp'"p,, in which it meets Y at the same distance on the negative side of the origin ; and the new face is (MO). Again, a semi-revolu- tion about the dyad axis OZ, parallel to the two faces, brings them into positions parallel to their original ones. The new pair of faces therefore have the symbols (MO), (MO). Owing to the triad axes, this set of tautozonal faces is repeated in two other similar sets of four faces parallel, respectively, to OX and OY. We have also seen in Art. 19 that the faces inter- changeable by rotations of 120 about a triad axis have their symbols in cyclical order. Hence (MO), (Ohk), (kQh) constitute a triad meeting at an apex on Op, and the two new faces must belong to the form r {MO} ; which therefore consists of the twelve faces : MO MO MO MO Ohk Ohk Ohk Ohk kOh kOh kOh kOh (k). On comparing this table with (d) of Art. 18, it will be seen that twelve of the faces of the tetrakis-hexahedron have been retained, viz. those which have their symbols in the same cyclical order. The form, Fig. 270, can be most easily drawn from Fig. 231. Of the four faces meeting at a tetragonal coign II, where AHA'H' = &,c.ka-^h, two only are retained which intersect in an edge e'e t , parallel to the vertical axis. Similarly, through H', H", &c., horizontal edges have to be drawn parallel respectively to the axes of X and Y. The above edges are often called the cubic edges, for they are parallel to those of the cube. The cubic coigns p remain coigns of r {MO}, for three faces of Fig. 231 still meet at these coigns THE DIHEDRAL PENTAGONAL DODECAHEDRON. 317 in the new figure. It is, therefore, only necessary to find the points e, e', &c., to be able to complete the figure. The trace #8" in the plane XOY of the face (MO), drawn to the middle point 8" of pp (/ , is produced in Fig. 270 to meet the horizontal cubic edge through H' in the point marked e". Similarly, H'8 is prolonged to meet the cubic edge through H" in e, and H"&' to meet the vertical cubic edge through H in e" ; and so on for all the homologous points e. Each of these points is then joined to the two nearest coigns of the cube, and the figure is com- pleted. Each face is a similar and equal pentagon, four sides of which are like edges, whilst the fifth the cubic edge differs from the others. The form is generally called the pentagonal dodecahedron ; but, when it has to be distinguished from the less common form described in Art. 43, it may be called the dihedral pentagonal dodecahedron ; for the faces occur in pairs meeting in cubic edges, and also in pairs which are parallel. By a rotation of 120 about Op, the three faces meeting at p are interchanged ; and the edges pe, pe', pe" must be similar, and the angles over them equal. Again, by rotation through 180 about the dyad axis OX, the coigns p and p t are interchanged ; and the triad of edges meeting at p t must be similar and equal to those meeting at p, and the angles over both sets of edges must be all equal. But the form consists of pairs of parallel faces : hence, the edges at the coign p' must be similar and parallel to those at p,, and the angles over them must be equal. Hence, the edges at p are similar to those at p, and the angles over them are all equal. Therefore p'e = pe pe" = &c. The same is true of each of the other faces. We shall denote the angle between pairs of faces meeting in an edge, such as pe, by U ; and that over a cubic edge, such as e'He^ by D. The angles D and U can never be equal, as can be easily proved from expressions (17) and (18) (p. 318), which can be obtained either from the general expression (3), or from the geometry of the figure. Thus, in Fig. 270, A'H' =ka + h, and A'8 = a (see Art. 18) ; .'. tan OH'8 = A'Z + A'H' = h + k. 318 CUBIC SYSTEM, CLASS III. cos 2 AP- sin 2 AP But if A and P of Fig. 271 are the poles (100) and (MO), .'. AP = 90 - OH*" = 90 - OH' '8, .*. tan AP = cot OH'S = k + h. This is the same expression as was shown in Chap, xiv, Art. 7, to hold for the angle Ag in a tetragonal zone. But cos D = cos 2^P = cos 2 AP - sin 2 AP since cos 2 AP + sin 2 AP = 1 . Dividing the numerator and denomi- nator of the last ratio by cos 2 JP, and replacing tan A P by its value k -=- h, we have cosZ? Again, in Fig. 271, which shows the poles P of the form T {MO}, we have the right-angled triangle PA'P' ; and cos U= cos P/" = cos A'P' cos ,4'P = cos AP sin ^P ; since A'P' = AP, and 4T = 90 - AP. Transforming the equation, replacing sec 2 JP by 1 4- taivMP, and tan AP by k + h, we have TT cos 6- The following are the angles for common occurrence : r{310} T {210} r {320} r{430} few of the forms of most 18 26' 26 34 33 41-4 36 52 36 52' 53 8 67 22-8 73 44 72 32-5' 66 25 62 31 61 19. 38. Suppose the cubic edges through ff, H', H" to be drawn parallel to the axes OY, OZ and OX respectively, that is, in directions at right angles to those in the form T {MO}. Let also new lines of construction US', H'S", &c., be each prolonged to meet the cubic edges in the same axial plane in new points which may be denoted by e', &c. ; and let these points be joined, as before, to the nearest coigns p of the cube. The drawing represents the comple- mentary form T {MO}, the faces of which have their symbols in the THE TRIAKIS-TETRAHEDRON. 319 reverse cyclical order to those of T {hkO\. By rotation through 90 about one of the dyad axes the form T {MO} can be brought into a position geometrically identical with T {hkO}, and the two comple- mentary forms are tautomorphous. The angles over corresponding edges are therefore equal for the same values of h and k. 39. We shall now show that the regular pentagonal dodecahedron of geometry is not an admissible form of this kind. For, in the regular polyhedron, the edges are all equal, and it may be regarded as the particular case of r{hlcQ} in which the angles over all the edges become equal. Hence, DU; and cosZ)=cosZ7. A-ff hk _ " t 2 a value involving a surd. The indices are therefore not rational ; and the form is inadmissible as a particular case of this type of pentagonal dodecahedron. 40. By drawing pairs of faces through each edge of the tetrahedron to meet the two dyad axes, which do not intersect that edge, at distances la + h from the origin, we obtain special forms, corresponding to those similarly derived in Arts. 20 and 21 from the octahedron. The forms are known as the triakis-tetrahedron T {hhl}, when h < 1; and as the deltoid dodecahedron T {hhl}, h>l. The triakis-tetrahedron, T {hhl}, h I. This form can be constructed from the auxiliary tetrahedron r{lll} in a manner similar to that employed in Art. 40 to give the triakis-tetrahedron. Let, in Fig. 274, the auxiliary tetrahedron be p'p"'p,,p ; and on the axes OA, OA', OA", &c., let the points L, L', L", &c., be taken at distances la + h from the origin. Since these points and the edges of the figure lie within the auxiliary tetrahedron, the axes and several other lines of construction are omitted, as they would un- duly complicate the diagram. The line L'L" is parallel to A' A", and the latter is parallel to p,,p'". Similarly, LL', LL", being klhf\hkl, and T being hkl/\hkl. Expressions for the cosines of these angles can be easily obtained from expression (3), and are : (24). 44. From a {Ikh} of Art. 24 we can also obtain two similar pentagonal dodecahedra, T {Ikh} and r{hkl}', of which the latter, shown R~^ //' in Fig. 278, has the faces: hkl hkl hkl hkl ' Ihk Ihk Ihk Ihk (O). klh klh klh klh The two dodecahedra, T {Ikh} and T {hkl}, are tautomorphous, for each of them consists geometrically of those faces of a {Ikh} which are situated cos r = cos (W AHA) = 9^ ~ , , TTv ^A/ A?' til co$ = cos(klh/\hkl) = M~"W" cos T 7 = cos (AJW A hkl) = h ' ** P 326 CUBIC SYSTEM, CLASS III. in alternate octants; and all the faces of a {Ikh} can be inter- changed by successive rotations of 90 about any of the axes of reference. But neither T {hkl} nor T {Ikh} can be brought into a position of congruence with T {hkl} or T {Ikh}. The dodecahedron T {hkl}, Fig. 278, may be said to be the inverse of T {hkl} of Fig. 276; and vice versa. For, the numerical values of the indices being the same, the faces of the one are parallel to those of the other; and parallel faces can only be interchanged by the axes of symmetry of this class, when the former are parallel to one of the dyad axes. Again, T {hkl} and T {hkl}, are enantiomorphous, for the faces of the one are reflexions of those of the other in the axial planes. Crystals of this class may be expected to rotate the plane of polariza- tion of a beam of plane-polarised light traversing them : this has been established for crystals of sodium chlorate and sodium bromate, but not for all the substances placed in the class. 45. If the regular pentagonal dodecahedron of geometry can be a particular case of this form, the angles r, < and T must be all equal. Equating the expressions for cos r and cos $ given in (24), we have hk-kl-lh; This equation is satisfied if 1 = 0, or if h+k = 0. The first indicates that the symbol must be r{MO}, a case already discussed in Art. 39. When A+&=0, the form has the symbol r{hhl}, and is a triakis-tetra- hedron (Art. 40) or a deltoid dodecahedron (Art. 41), and the faces are not pentagons. It follows that the regular pentagonal dodecahedron is not admissible as a form of this type. 46. Crystals of the following substances belong to this class : Substance. Chemical composition. Forms and combinations. Barium nitrate Ba(NO 3 ) 2 oa>, ao, ocoa. Lead nitrate Pb(N0 3 ) 2 oa>, o ca ae = r{210}, Sodium chlorate NaC10 3 <%j = T{210},Fig.280(a); adop= r{120},Fig. 280(6) ; o. Sodium bromate NaBr0 3 ao&d. Sodium uranyl acetate NaUO 2 (C 2 H 3 2 ) 3 od. Sodium sulphantimonate Na 3 SbS 4 . 9H 2 O ocade, ocode^r {120} . Sodium strontium arsenate NaSrAs0 4 . 9H 2 aa>de, aa>de, . BARIUM NITRATE, SODIUM CHLORATE. 327 Barium nitrate. Fig. 279 represents a plan on the plane YOZ of a crystal showing an exceptionally large number of different forms ; namely : tf . Fio. 279. Dr. Wulff (Groth's Zeitsch. f. Kryst. u. Min., iv, 122, 1879) has exhaustively studied the crystals of barium, strontium and lead nitrates. He observed that when crystals of barium nitrate are deposited from pure aqueous solutions the cube predominates. The cubic coigns are modified by faces of the complementary tetrahedra of unequal size, and of which the largest were taken tobe7-{lll}: the crystals also showed sometimes faces of r {120}, and sometimes elongated triangular faces of a tetrahedral pentagonal dodecahedron X. The positions of the faces X could not be fully determined, but they approached those of r {421}. When sodium nitrate is present in the solution, crystals are deposited which resemble regular octahedra, having their coigns modified by faces of the cube {100} and by very small faces of the forms X and r{120}. From solutions containing also potassium nitrate and sugar, he ob tei ned more complicated crystals in which the faces of the cube and tetrahedra were absent: the predominant form was r{211}, and the forms X, 7-{120} and r{221} appeared as small faces modifying the acute coigns of the triakis-tetrahedron. In all the above crystals r{120} and the tetrahedral pentagonal dodecahedron X occupied the same relative positions, so that the X faces were nearer in position to the faces of the complementary form r{210} than to those of r{120}. The crystals manifest a weak, but anomalous, double refraction, and no rotation of the plane of polarization has been established. Crystals of strontium nitrate and of lead nitrate have very nearly the same characters : they also manifest anomalous double refraction but no rotation of the plane of polarization. Sodium chlorate. When obtained from aqueous solutions at ordinary temperatures, the crystals are cubes h {100}, having their edges and coigns modified : (i) by narrow faces d{110}, jo = i-{210} and tetrahedra O=T{I!I}, Fig. 280 (a); or (ii) by d{110}, jt> = r{120}, and o = r{lll}, Fig. 280(6). The complementary pentagonal dodecahedra, r{210} and r{120}, never occur together on the same crystal. The crystals represented by Fig. 280 (a), rotate the plane of polarization of a plane-polarised beam to the left ; i.e. to an observer receiving the light transmitted by a plate of the crystal the rotation is counter-clockwise. The crystals shown in Fig. 280 (b) rotate the plane of polarization to the right ; i.e. under similar circumstances to those of the first case the rotation is clockwise. This rotatory power is 328 CUBIC SYSTEM, CLASS IV. possessed by all simple (i.e. untwinned) crystals of the substance, and is invariably associated with the arrangement of the faces given above. FIG. 280. The rule connecting the direction of rotation with the facial develop- ment may be given as follows. The cube being placed in the usual position with the tetrahedron as r{lll}, then in a laevogyral crystal the pentagonal dedecahedron is r{210}, and its vertical face (210) is to the left of (110) ; in a dextrogyral crystal the form is r{120}, and its vertical face (120) is to the right of (110). General forms, i.e. forms having finite unequal indices, have never been observed. But we may anticipate that, if such forms are discovered, laevogyral crystals will have r{hkl} and r{hJct} in which h>k>l ; for in these forms the limit, when 1=0, is a pentagonal dodecahedron, r{hkQ}, having the vertical face to the left of (110). Dextrogyral crystals may, on the other hand, be expected to show r {khl} and r {kht}, the limiting form r (MO) having its vertical face to the right of (110). The triad axes in untwinned crystals have been found to be pyro-electric axes (Friedel and Curie ; Bull. Soc.fran$. de Min. vi, p. 191, 1883). IV. Dyakis-dodecahedral class ; IT {hkl}. 47. If a centre of symmetry is added to the assemblage of axes of symmetry characteristic of the last class, there must also be three planes of symmetry, each perpendicular to one of the dyad axes. These planes are parallel to the faces of the cube having its edges parallel to the dyad axes, and are called the cubic planes of symmetry, II. The elements of symmetry of the class may be represented by : 4 P , 38, C, 311. The arrangement of axes and parametral plane (HI) of the last class is clearly not altered ; and, as before, The general equations (2) and (3) of Art. 13 will therefore hold for the angular relations of faces in this class. THE DYAKIS-DODECAHEDRAL CLASS. 329 48. The forms of Class III having each of their faces parallel to one or two dyad axes belong also to this class, for they have parallel faces. Hence, the cube {100}, the rhombic dodecahedron {110}, the pentagonal dodecahedron r{hkO}, have the same facial development, whether they appear in crystals of this class or of Class III. The reader will, on referring to Arts. 15, 17, and 37, perceive that the faces of these forms are symmetrically placed with respect to the axial planes, and that groups of them are bisected at right angles by these planes. The dihedral pentagonal dodeca- hedron, of which good instances are observed in pyrites, is denoted by the symbol TT {MO}, when it is regarded as a form of this class : the symbols of the faces are given in table k. 49. The remaining forms of Class III are all modified by the addition of parallel faces. Thus, from the tetrahedron T{111}, we obtain the octahedron {111}, the faces of which are given in table b of Art. 16. From the triakis-tetrahedron r {hhl}, h < I, we obtain the icositetrahedron {hhl}, identical geometrically with that de- scribed in Art. 21 and represented in Fig. 238; and from the deltoid dodecahedron T {hhl}, h > I, we have the triakis-octahedron {1M}, described in Art. 20 and represented in Fig. 235. No Greek prefix is placed before the brackets in the symbols of these forms ; for geometrically they are the same as the similar forms of Class II. The forms, having been fully described in Arts. 16, 20 and 2 1 , need no fresh discussion ; but they can be derived inde- pendently from the elements of symmetry characteristic of this class in a manner identical with that given in the Articles quoted. Thus we can draw through any edge of the octahedron a pair of faces equally inclined to the cubic plane of symmetry through this edge ; the faces meeting the axis of reference perpendicular to the cubic plane at points L at equal distances on opposite sides of the origin, where OL = ha + l. This pair of faces must be associated with a pair of similar faces through the same points L and the opposite edge of the octahedron ; but the four faces are those in the similar construction of Arts. 20 and 21 ; and the forms are therefore geometrically identical. The symbols of the faces are given in tables e and f ; and the forms are common to Classes I, II and IV. 50. Tlie dyakis-dodecahedron, Tr{hkl\. The face (hkl), the indices h, k and I being finite and unequal, is associated with two other like faces (Ihk), (klh), the symbols of which are in the same cyclical 330 CUBIC SYSTEM, CLASS IV. order (Art. 19). Again, since the axes of reference are dyad axes, there must be similar triads, having their symbols in the same cyclical order, in each of the alternate octants. The twelve faces so connected constitute the form T {hkl\ of Art. 43. But, since the crystals in the present class are centro-symmetrical, the above twelve faces are associated with a set of twelve faces, each of which is parallel to one of the first set ; the second set of faces constitut- ing T {hkl} of Art. 44. The form IT {hkl}, Figs. 281 and 282, consists therefore of the twenty-four following faces : hkl hkl hkl\ Ihk Ihk Ihk hkl Ihk klh hkl Ihk klh klh hkl Ihk klh klh hkl Ihk klh klh m Ihk .(p). The faces in the upper half of each column are symmetrical in pairs to those in the lower half of the same column with respect to the horizontal plane of symmetry II". The faces can also be arranged in pairs, such as (hkl) and (hkl), symmetrical with respect to one or other of the planes of symmetry II' and II. Since twelve faces, consisting of the triads in alternate octants, are interchangeable, and the remaining twelve are parallel and opposite each to a face of the first set, it follows that the faces are all similar and equal trapezia. Two of the edges of each face meeting at a trigonal coign are also equal, for they change places when the THE DYAKIS-DODECAHEDRON. 331 crystal is turned through 120 about the axis through the coign. The four edges meeting at a point H on an axis of reference consist of two dissimilar pairs, one pair of like edges lying in the same axial plane. Hence the crystal has three different angles between ad- jacent pairs of faces ; viz. T = hkl A klh over edges converging to a triad coign, D = hkl A hkl over the longer edges lying in the axial planes, and W-hkl f\ hkl over the shorter edges lying in the axial planes. Expressions (25) for the cosines of these angles are easily obtained from (3) ; they are : /i 11 77v hk + kl + lh cos T = cos (hkl A klh) = - - , cos D = cos (hkl A hkl) = cos W = cos (hkl h hkl) = .(25). The values of the angles for some of the forms most commonly met with are : rftfi} ,r{321} 7T{421} w {531} =IW f\hkl 36 42' 29 12 32 19 r 38 13' 48 11 48 55 D W 31 0' 64 37' 25 13 51 45 19 28 60 56 51. By the law of rational indices, the faces having their symbols in the reverse cyclical order to those in ?r {hkl} are possible, but they cannot occur as a part of this form. They however constitute a complementary form ir{lkh}, Fig. 283, which can by a rotation of 90 about any one of the axes of reference be brought into a position identical with that of TT {hkl} ; for it was seen, in Art. 24, that a rotation of 90 about an axis of reference inter- changes faces having their symbols in opposite cyclical orders. It follows, therefore, that the two forms are tautomorphous; and that the angles over corresponding edges are, for the same values of the indices, equal. 332 CUBIC SYSTEM. CLASS IV. 52. Crystals of the following substances belong to this class : Substance. Chemical composition. Forms and combinations. Pyrites Cobaltine Smaltine Sperrylite Skutterudite Hauerite Potash Alum FeS 2 CoAsS CoAs 2 PtAs 2 CoAs 3 MnS 2 K 2 A1 2 (S0 4 ) 4 .24H 2 also the isomorphous salts of ammonium, &c. Stannic iodide SnI 4 oe. a, o, e = 7r{210}, {331}, a*=7r{321}, [aes, &c. a, o, e, ae, oe, aoe. a, ao, forms rr {hkO} rare, a, ao, aeo. on = {2 11}, ond. o, oe. o, a, oa, oda, oe. FIG. 284. Pyrites. This very common mineral is found in crystals which are gene- rally single forms or simple combinations. The cube is common ; and the faces are often striated, as shown in Fig. 284. The striaa on adjacent faces meeting in a coign are parallel respectively to the edges meeting in it, so that a rotation of 120 about the triad axis interchanges the striae. Another common form is the octahedron {111}, the faces of which are usually smooth and bright: occasionally they each have three sets of striae, parallel to the lines in which the face would intersect the adjacent faces of TT {210}. The pentagonal dodecahedron n {210} is fairly common ; and the faces are usually striated : the crystals can be divided into two groups, in one of which the striae are parallel to the cubic edges, in the other they are perpendicular to these edges. Rose discovered that the crystals of the two groups are thermo-electrically active; and J. Curie (Bull. Soc.frang. de Min. viu, 127, 1885) has shown that the crystals striated parallel to the cubic edges are more positive than ^^/ : -^-a^ooi ...^^N antimony, whilst those striated perpen- dicular to the cubic edges are about as negative as bismuth. Different portions of the same face are sometimes striated in directions at right angles to one another, and these portions are thermo-electrically different. A combination of the cube and n {210}, which is fairly common, is shown in Fig. 285 : in such crystals each face of the cube is usually striated parallel to those faces of TT {210} which are inclined to it at the least angle; vertically, i.e. parallel to [ae"]. FIG. 285. thus (100) would be striated The relative dimensions of the two forms EXAMPLES (PYRITES). 333 vary much. In drawing such a combination, the form ir {210} should be first completed, when the faces of the cube are easily introduced. The combination of the octahedron {111} with TT {210} (slightly developed) shown in Fig. 286 is sometimes found in crystals of pyrites, but it is more frequent in crystals of cobaltine. The crystals of this latter mineral are also often found with the habit given in Fig. 285. Fro. 287. Fig. 287 represents a combination not infrequently observed in crystals of pyrites from Elba. The forms are a {100}, o{lll}, e=ir {210}, and s = n {321}. The dyakis-dodecahedron ?r{321} also occurs as a single form in crystals from Elba. In drawing the combination it is best to construct TT {321} first, the octahedral faces can then be introduced ; and the edges [se] are parallel to adjacent edges [so] (o, s, e being tautozonal), so that the figure is quickly completed. V. Hexakis-tetrahedral class ; /u. {hkl}. 53. To the assemblage of four triad and three dyad axes of Class III six planes of symmetry, 2, may be added, which intersect in sets of three in each of the triad axes and in pairs in each of the dyad axes. The planes are parallel to pairs of faces of the rhombic dodecahedron, and are called the dodecahedral planes of symmetry. The forms are not centro-symmetrical, for each of the planes of symmetry would in that case be associated with a dyad axis perpendicular to it, and we should have four axes of symmetry in a plane, inclined to one another at angles of 45. The axis of symmetry perpendicular to this plane would then be a tetrad axis (Chap, ix, Prop. 11) ; and the elements of symmetry would be those characteristic of Class II. The triad axes are uniterminal, and have been shown to be pyro-electric axes in crystals of blende and boracite : the pyro-electricity can be tested by a simple method invented by MM. Friedel and Curie (Bull. Soc.franq. de Min. vi, p. 191, 1883). The elements of symmetry of the class may be represented by : 4 P , 38, 62. 334 CUBIC SYSTEM, CLASS V. 54. The cube, the rhombic dodecahedron and the tetrahedron of Class III belong also to this class. It is easy to see that these forms are geometrically symmetrical with respect to the planes of symmetry. As, moreover, the planes of symmetry are parallel to possible faces, viz. to those of the rhombic dodecahedron, we can in this class prove that the triad axes are possible zone-axes ; for they are parallel to the edges of this form. The faces of the tetrahedron are also possible, for the edges are parallel to the normals of the planes of symmetry; and, as we saw in Chap, ix, Prop. 1, these normals are possible zone-axes. The dyad axes are selected as axes of reference, and a face of the tetrahedron as parametral plane (111). Hence, as before, a = b = c; and the analytical relations established in Art. 13 hold for crystals of this class. 55. Again, the triakis-tetrahedron and the deltoid dodeca- hedron of Arts. 40 and 41 belong to this class. The student will perceive that Figs. 272, 273 and 274 are symmetrical with respect to planes passing each through an axis of reference and a triad axis. For instance, the plane through OL and Or of Fig. 288, identical geometrically with Fig. 272, bisects the planes p,,rp", p tt r t p", and contains the edges rp, r t p formed by faces which are reciprocal reflexions in the plane. It is clear that the plane of symmetry repeats the point L' in L", and vice versa. Hence, the one face being (hlh), the other is (hhl). We have therefore the triad (hhl), (Ihh), (hlh) in the first octant, symmetrical in pairs with respect to the planes of symmetry through Or and each of the axes OL, OL', OL". Further, the second plane of symmetry through OL also passes through p /t p", and the above triad of faces is repeated over the plane in a similar triad meeting at r t . But this is the triad obtained by a rotation of 180 about the axis OX. Hence no new faces are introduced. The figures, having been fully discussed in Arts. 40 and 41, need no fresh description. When they are forms of this class, the symbols p. {hhl} are used. The symbols of the faces in /*. {hhl} are given in THE HEXAKIS-TETRAHEDRON. 335 table 1, p. 320 : for the triakis- tetrahedron h < I, and for the deltoid dodecahedron h > I. 56. The tetrakis-hexahedron, {MO}. In Chap, xiv, Art. 7 it was shown that a face (hkQ) parallel to a dyad axis, in which two planes of symmetry intersect at right angles, these angles being bisected by two dyad axes perpendicular to the first axis, is repeated in seven other faces. Their traces in the axial plane perpendicular to the first dyad axis are given by the lines forming the di- tetragon of _Fig. 177. Hence the faces, hkO, MO, MO, MO, MO, MO, MO, hkO, occur together in any form which has such an arrangement of elements of symmetry. The triad axes interchange the axes OX, OY and OZ ; and at the same time any faces symmetri- cally related to them. Hence, there must be two other similar sets of eight faces parallel, respectively, to OX and OY and symmetrical with respect to the pairs of planes 2 through these axes. We, therefore, obtain a twenty-four faced figure, Fig. 289, geometrically identical with Fig. 231 described in Art. 18. The planes of symmetry pass each through a set of edges, such as pp', Hp, H Pi of that figure. The Greek prefix is dropped before the symbol {MO} of the form, which includes the faces given in table d. The form is therefore common to Classes I, II and V. 57. The hexakis-tetrafadron, p. {hkl}. A face (hkl) having any general position, i.e. h, k and I being finite and unequal numbers, is associated with five other faces, the symbols of which are obtained by taking the indices in the two opposite cyclical orders. For rotations of 1 20 about the triad axis bring the face to the positions given by (Ihk) and (klh). Again, the faces are arranged in pairs equally inclined to each of the three planes of symmetry traversing the octant which contains the face. Thus (hkl) being one of the faces, (khl) is its reciprocal reflexion in the plane of symmetry through OZ and bisecting the angle between the axes of X and Y : similarly, (Ihk) and (hlk), (klh) and (Ikh) are pairs of faces which are reciprocal reflexions in the same plane. But (khl), (hlk), (Ikh) FIG. 289. 336 CUBIC SYSTEM, CLASS V. are in cyclical order that opposed to the first ; and there can be no other homologous faces meeting at the same coign on the triad axis. The six faces meeting at a ditrigonal coign in one octant are : hkl Ihk klh khl Ikh hlk. These six faces are repeated in a similar set of six faces by each of the dyad axes forming the edges of the octant ; and the four sets occupy alternate octants. Now rotation of 180 about an axis of reference changes the signs of the indices on the two other axes but not their order. The form p. {hkl}, Fig. 290, includes therefore the faces : hkl Ihk klh Ikh hlk khl hkl Ihk klh Ikh hlk khl hkl ihk kih Ikh hik khi hki ihk kih ikh hlk khi (a). The poles of the faces lie in sets of six on small circles FIG. 290. FIG. 291. surrounding the tetrahedral poles o situated in alternate octants : they are shown in the stereogram Fig. 291. The form may be regarded as the result of placing on each face of the tetrahedron a similar ditrigonal pyramid, and hence it is called the hexakis-tetrahedron. The faces are all equal and similar scalene triangles, and the form has three different angles, viz. those over the dissimilar edges of each face : we shall denote them by THE HEXAKIS-TETRAHEDRON. 337 the letters, F, G and T. Expressions for their cosines are obtained from (3) ; they are cos /^= cos (hid A khl) = K ' h? + k 2 + P cos G = cos (hkl A hlk) = =g For the following particular cases, the angles are : n{hkl} F G T p {321} 21 47' 21 47' 69 5' ^{421} 17 45 35 57 55 9 /*{531} 27 40 27 40 57 7 58. By raising similar ditrigonal pyramids, one on each face of the tetrahedron u. { 1 1 1 }, we obtain a complementary hexakis-tetra- hedron p. {hkl}, Fig. 292, the faces of which are parallel respectively to those of n {hid}. The two forms are tautomorphous, for a rotation of 90 about one of the axes of reference interchanges faces in adjacent octants. The angles over the corresponding edges are equal for the same numerical values of the indices. 59. Crystals of the following substances belong to this class : Substance. Diamond Blende Chemical composition. c ZnS Fahlerz 3(Cu 2 , Fe)S . (Sb, As) 2 S 3 Boracite Mg 7 B 16 O 30 Cl 2 Helvine (Be, Mn, Fe) 7 Si 3 12 S Haiiyne Na,Ca(NaSO 4 Al)AlSi s O 12 Eulytine Bi 4 Si 3 12 Diamond. This mineral is sometimes found in apparently regular octahedra, which have their edges replaced by grooves in the way shown in Fig. 293. Another apparently regular octahedron is shown in L. c. 22 Forms and combinations. o, dao, am, dm, &c. o, oa, od, n, on, &c. o,oa, aou>, aoa>c? o, ou>d, ond, ons. om, da. n, nan p. 338 CUBIC SYSTEM, CLASS V. Fig. 294, the edges of which are also replaced by grooves ; but in this case the sides of the grooves are made by rounded faces belonging to an FIG. 293. Fro. 294. indeterminate form p. {hkl}. A few crystals have been observed which are hexakis-tetrahedra, the faces being too rounded for the angles to be measured : occasionally the obtuse coigns of these hexakis-tetrahedra are modified by faces of the tetrahedron. Blende. The crystals are sometimes combinations of 0=^(111} with o> = /*{lll}, the faces of the latter being slightly developed, and often dull and pitted, whilst the faces o are large, bright, and striated. These two forms are often associated with the cube, the faces of which appear as narrow truncations of the edges of /i {111} when this form predominates. But the most common and prominent form is the rhombic dodecahedron ; its faces being parallel to perfect cleavages. This form is usually associated with faces of the cube and of the tetrahedron /i{lll}: it is also frequently associated with elongated triangular and rounded faces of a form fi {hhl}, hk>L Then eight faces, having symbols in which h occupies the same place, meet in each point H. Pairs of lines are drawn cross- wise in each axial plane joining a point L on one axis to a point K on the other; e.g. L'K" and K'L" in Fig. 301. These pairs intersect in points 8, 8', 8", &c., on the dyad axes lying in their plane. The lines H8, H'o', H"o", &c., give the edges Hp, H'p, &c., of Fig. 302. For the face (hkl) passes through HK'L", and (hlk) through HL'K"; and similarly for other pairs of the faces. Ik. Ihk DRAWING THE GENERAL FORMS. 341 The coigns p on the triad axes being now determined, it only remains to find the points d at which two pairs of faces, such as (hkl) and (khl), (hkl) and (khl), meet the respective dyad axis. These points are found by joining crosswise in pairs the points H on one axis to the points K' and K" on the others, and the points K to the points H', H", &c. Thus, in Fig. 301, H'K" and K'H" intersect in d on the dyad axis in their plane, and the line Ld coincides with the edge pd of Fig. 302. The figure is now com- pleted by joining the points H to the adjacent points d. 61. p. {hkl}. Geometrically this form consists of the faces of {hkl} which occupy alternate octants in sets of six. Hence the coigns p, p", p^ p ///} and the edges pH, plf, pd, pd', kc., remain coigns and edges of p {hkl}. But at the points d only two faces meet, and the edges Lpd, L'pd', L"pd", &c., have to be all prolonged to meet the triad axis traversing the adja- cent octants at the points R, R'", &c., of Fig. 303. The figure is com- pleted by joining each of the points H to the adjacent coigns R. 62. TT {hkl}. All the coigns H and p of ,{hkl}, Fig. 302, remain coigns of the dyakis-dodecahedron IT {hkl} for four faces meeting at each coign H are common to both forms, and of the six faces, meeting at each ditrigonal coign of the former, a triad having their symbols in cyclical order remain in the new form. Further, one pair of the edges in each axial plane meeting at coigns H remain edges of TT {hkl}. Thus, the edge H'd of Fig. 302 and the corresponding edge joining H' to a point K tt on OZ t remain edges of Fig. 304 : they have to be prolonged to meet the edges [klh, klh] and [klh, klh] at new coigns e. These new edges are easily constructed for they are 342 CUBIC SYSTEM. the lines joining the pairs of points H ", L' ; &c. The homologous points e, e, e", &c., being deter- mined, fix the directions of the edges [pe], [pe'], &c. The complementary f orm ir{lkh}, Fig. 305, is drawn in a similar manner ; the edges Hd', H"d, H'd", &c., being prolonged . to meet new edges H"L = [Ikh, Ikh], &c., in homologous points e r e tl , &c. These are then joined to the adja- cent points p, so completing the drawing. F IG . 305. 63. a {hkl}. The coigns // and p of Fig. 302 remain coigns of a {hkl}, and for the same reasons as were given in Art. 62 for the retention of these coigns in -rr {hkl} ; but the edges of Fig. 302 have all to be replaced by new ones. The triads of faces of a {hkl}, Fig. 306, meeting at p, p", p t and p itl in the first 1 and alternate octants are the same as those of TT {hkl\ which lie in the same octants. Hence the edges pe, pe', pe" of Fig. 304, and their homologues in the alternate octants are common to both forms. In the adjacent octants the faces of a {hkl} are the same as those of TT {Ikh} which meet at the coigns p', p'", p t/ , p of Fig. 305; and the edges in these octants are the edges of Fig. 306 which meet at these coigns. The edges of a {hkl} meeting at a tetragonal coign If are found by joining H to four points o, which are the intersections of alternate sides of the ditetragon formed by joining the points K and L in the axial plane perpendicular to the axis through H. Thus, in Fig. 301, the face (Jikl) meets the plane YOZ in the trace K'L" the face (hlk) meets the same plane in a trace L'K lt , where OK ti is a^ + k measured on OZ ',. These traces meet at a', and the edge [hkl, hlk] is the line joining H to a'. Similarly, the edge 1 By first octant is meant that in which all the indices of a face are positive. DRAWING THE GENERAL FORMS. 343 [hkl, hlk] is the line joining // to a", where a" is the intersection of K'L" and L t K". The construction of these edges is the same as that of the polar edges of a {hkl} of Chap, xiv, Art. 52. Similar points are found in each of the other axial planes and joined to a point // on the perpendicular axis. The lines Ha, Ha.", &c., meet the edges from p /t , p, die., which bound the same faces at new coigns; and the adjacent pairs of these coigns, being now joined, give the edges which are bisected at right angles by the dyad axes at the points d", d, d', &c. 64. r {hkl}. This form consists geometrically of twelve of the faces of each of the three last forms, viz. of those triads which have their symbols in one cyclical order and lie in alternate octants. Hence the edges pe, pff, &c., situated in alternate octants of Figs. 304 and 306, are common also to the pentagonal dodecahedron, Fig. 307. Through the point H, common to (hkl) and (hkl), a line is drawn parallel to the trace K'L", Fig. 301, in the perpendicular axial plane ; and the similar edges through each of the points H', H", 90 and with C an angle < 90. Hence there must be a point r between m and C which is at 90 from Y. The point r is also at 90 from Z; since, for any point r in [Cm], t\rZ= t\rY. The pole r(100) lies therefore between m and C, and on the same side of C as the axial point X. Similarly, r lies near Y between C and m' ; and r" near Z between C and m". Again r, r, r" are nearer to C, or more remote from it, than the axial points X, 7, Z according as A XY= A YZ = I\ZX are each of them greater or less than 90. For it is clear that XY>rY, when r is, as in Fig. 310, nearer than X to C ; and rY being 90, XY> 90. 5. The angle pOr, or the arc Cr, between the triad axis and the normal to one of the axial planes we shall denote as the angular element D of a rhombohedral crystal. We shall now show how from the angular element D the angle between the poles r, r of two of the axial planes, and the angle XO Y between any pair of the axes can be found ; and, vice versa, how the angle D can be determined when either of the angles rr' or XOYis given. The spherical triangles rCr', XCY are both isosceles, and have the common angle rCr 120. The sides rr and XY of the triangles are also bisected at right angles by the great circle ZCy at the points e", y. Hence, sin re" = sin (rCe" = 60) sin Cr ; .'. sin rr' = sin 60 sin D (1). Similarly, sin XY= sin 60 sin GX (2). Again, since Zr = Zr' = 90, Z is at 90 from every point in the great circle rr'. .'. Ze' = 90, and CZ= 90 - Ce". RELATIONS OF CRYSTAL-ELEMENT. 349 Also CZ=CY= CX; and from the spherical triangle rCe", cos 60 = tan Ce" cot Cr; (3). FIG. 311. 6. We shall denote by c a length on the triad axis, such as Op of Fig. 311, which is the distance of the apex of the axial pyramid (often called the fundamental pyramid) from the pedion XYZ ; and we shall denote by 3 the length of one of the sides of the equilateral triangle XYZ; the factor 3 being introduced to avoid the re- currence of fractions, and to conform to the usage of the crystallographers who take for element the ratio c : a. If we take a to be unity, we may call c the linear element of a rhombo- hedral crystal. We shall now find the relation between c-f-a and D from the geometry of Fig. 311. Since A ZGX=W, and ZG bisects the angle XZY, A ^=30 ; and ZG = Jr^cos30 = 3acos30 .................. (4). Again, by the well-known properties of the centre of gravity of a triangle, Zp = 2pG, and ZG = 3pG. :. P = acos30 ........................ (4*). The angular element D is the angle ZGO, the inclination of a face of the fundamental pyramid to the pedion. Hence tanZ> = 0p^ p = c^acos30 ............... (5). .'. whena=l, c = tan D cos 30 .............. (6). The reader must not confuse the a= XY+- 3 of this Article with the length OX which is used in Art. 3 as the Millerian parameter. The two are connected by the following relation. OX sin XOp = X P =Yp = Zp = 2a cos 30. .'. OXsmCX=2acoa3Q" ..................... (7). If D is known, the angle CX is found from (3). 350 RHOMBOHEDRAL SYSTEM, CLASS I. I. Adeislous trigonal class; T {hkl}. 7. The crystals of this class having a single triad axis and no other element of symmetry, the general form consists of a trigonal pyramid such as was employed in the preceding articles to give the axes of reference of any rhombohedral crystal. Any pyramid possible on a crystal of the class may be selected as the fundamental, or axial, pyramid r{100}. That pyramid which is most frequently met with, and has the largest faces, is usually taken; but, when cleavages are discovered parallel to the faces of a pyramid, it is most convenient to take the intersections of these cleavages for the axes of reference. There is, however, no essential difference between the pyramid r{100} and that having the general symbol r{hkl], for the choice has been guided by a desire for simplicity of symbols and facility of identification. The pedions r{lll} and T{III} are two distinct possible special forms. They give equal parameters on the axes. 8. The trigonal pyramid, T {hkl}. From the facts that the axes are interchangeable in order and the parameters are equal, it follows that the form T {hkl} consists of the three faces : hkl, lhk,klh (a); in which the indices are taken in cyclical order. This cyclical order was explained in Chap, xv, Art. 19, and was there shown to be a consequence of the interchangeability of lengths on axes of reference similarly placed with respect to a triad axis. The fact that the axes are no longer axes of symmetry at 90 to each other in no way affects the cyclical order. If a plane (hkl), intercepting on the axes the lengths a -f- h, a^k, a it -^l (a being OX], is a possible face, then the parallel plane (hkl) is also a possible face ; for the indices are rational and only differ in sign from those of the first face. The second face does not belong to the form T {hkl}, but to a complementary form T {hkl} which includes the faces : hki, Ihk, kih. Since the faces are parallel to those of r {hkl}, the angles over the polar edges (p. 112) of the two forms are equal. Again, the face (hkl) being possible, then (hlk) is also a possible face. From the equal inclination of the axes to the vertical, and from their lying in vertical planes at 1'20 to one another, it follows THE TRIGONAL PRISM, T {Oil}. 351 that the two faces are equally inclined to the horizon and to the vertical plane XOp ; i.e. (hkl) and (hlk) are reciprocal reflexions in the plane XOp. But (hlk} belongs to a pyramid T {hlk} consisting of the triad of faces : hlk, khl, Ikh ; the symbols of which are in the reverse cyclical order to those of T {hkl}. Since the faces of the two pyramids are equally inclined to the horizon, they are geometrically similar ; and the angles over the polar edges are equal. But the two pyramids are distinct and separate forms ; and they can only be brought into similar positions by turning one about the principal axis through an angle which varies with h, k and I. We have also a fourth form r{hlk}, the faces of which are parallel to those last discussed. These four forms being geo- metrically similar were, according to the principle of merohedrism, formerly regarded as tetartohedral forms derived from the general form of class III. 9. The trigonal prism, r{011}. Since the triad axis Op and the edges of the pyramid selected to give the axes of reference are possible zone-axes, a plane containing Op and one of the axes, OX (say), is parallel to a possible face. Since the face is parallel to OX the first index is zero. Now the plane containing OX and Op passes through the line XpE of Fig. 309, which is the trace of the plane on the pedion ; and since the triangle formed by the points JT, Y, Z, where the axes meet the pedion, is equilateral, it is clear that YE = EZ. If the plane OXpE is transposed, remaining parallel to its original position, until it passes through Z, it must in its new position meet the plane YOZ in a line ZY ] parallel to OE, as shown in Fig. 312, which gives the lines in the plane YOZ; for parallel planes meet a third plane z E__ y in parallel straight lines (Euclid xi, 16): therefore the triangles ZYY f , EYO are similar. Hence (Euclid vi. 4), 77: YO = ZY:EY= < 2,:\ :, YY = and OY = OY. FIG. 312. Hence the plane in its transposed position meets the axes at distances OX ^ 0, OY + T, OZ+ 1. But the equal lengths OX, OY and OZ can be taken as the parameters. Therefore the face has the symbol (Oil). 352 RHOMBOHEDRAL SYSTEM, CLASS I. If the plane OXpE is transposed in the opposite direction so as to pass through Y, it is easy to prove that it will meet OZ at Z t where OZ t =- OZ. The symbol of this new face is therefore (OlT). The two parallel faces (Oil) and (OlT) are not associated together in crystals of this class, but belong to separate trigonal prisms r{OTl} and r{OlT}, which are complementary. Again, rotation through 120 about the triad axis Op brings the plane OXpE into the position OYpF, and the face (Oil) into a position in which it meets OX at X, and OZ at Z t where OZ t = - OZ. The new face has therefore the symbol (10T). A second rotation in the same direction brings the plane through Op into the position OZpG, and the parallel face to meet OX at X t and OF at Y : its symbol is (TlO). The trigonal prism r{OTl}, Fig. 313, includes the faces : Oil, 10T, TlO (b). The angles between the faces are 120 ; and the poles are those marked a, a and a" in Fig. 316. The complementary trigonal prism, r{OlT}, having its faces also parallel each to the triad axis and to one of the axes of reference, consists of the faces Oil, 101, 110. The poles are indicated by a, a /} and a ti . Besides differing in signs from those of the first triad (b), the symbols of this last triad of faces are also in the reverse cyclical order to those in the first ; hence there can be only two comple- mentary forms having these symbols. 10. We shall now establish a relation existing between the indices h, k, I of a face N when it is parallel to the triad axis, i.e. is one of the faces of a trigonal prism. From two of the faces of the prism r{011} the zone-symbol can be determined, for these faces are not parallel to one another. By the rule (Chap, v, Art. 4) the zone-symbol is [OTl, 101] = [111] ; and by Weiss's law, h+k+l=0 (8). Hence a face is parallel to the triad axis when the sum of its indices is zero : some of the indices must necessarily be negative. The three faces of the prism r [hkl\ have the symbols : hkl, Ihk, klh. THE TRIGONAL PRISM, T {112}. 353 An important trigonal prism FIG. 314. 11. The trigonal prism, r {112}. is that of which each of the faces passes through one edge of the equilateral triangle in which the axial pyramid meets the pedion. The symbols of the faces can be found from the geometry of Fig. 314. Let a plane be drawn through XGY parallel to the triad axis Op, and meet OZ in L t . Then a plane through OZ and Op intersects the prism plane XYL t in the line GL it which must be parallel to Op. Therefore, in the plane ZGL t , the two triangles ZOp, ZLG are similar; for they have the common angle GZL t , and the third sides Op and GL t , are parallel. Hence (Euclid vi, 4), OL t : OZ = pG :pZ. But (Art. 6), pZ = 2pG /. OZ=WL t . Hence, since L t lies on the negative side of the origin, the prism-face XYL t intercepts on the axes the lengths OX, OY, OZ+2. The symbol is therefore (112). It can also be found from (8) and the fact that the face lies in the zone [111, 001]. Rotation through 120 about the triad axis brings the face into positions in which it passes through the other edges of the triangle XYZ. The prism T { 1 1 2} consists therefore of the faces : 112, 211, 121 (C). The complementary trigonal prism T{2lT}, Fig. 315, has each of its faces parallel to a face of the preceding prism, and its faces can be drawn through points, such as X it Y t ,L of Fig. 315. Its faces and poles will always be denoted by the letters m, m', m". The prisms in Figs. 313 and 315 are terminated by the complementary pedions T {111} and T {III}. 'FIG. 315. 12. The position of the normal P to any face (hkl) can be given by the general equations (1) of Chap. iv. ; but, since the parameters L. C. 23 354 RHOMBOHEDRAL SYSTEM, CLASS I. are equal, the equations can be simplified by taking the parameter to be unity. Hence, for the normal P to a face (hkl), we have ^ p _ cos XP _ cos YP _ cos ZP _ cos XP + cos YP+ cos ZP ~~h~ ~T~ ~T~ h + k + l the last term being obtained by adding the numerators and denominators of the three preceding ratios to form a new ratio equal to each of the preceding terms. 13. Equation (8) connecting the indices h, k and I of a prism-face N can also be found from equations (9). In Fig. 316, let the triad axis, emerging at the pole (7(111), be the diameter through the eye; and let X, Y and Z be the axial points. Then the poles N, N' and N" of the trigonal prism r {hkl} lie in the primitive at 120 from one another. Describe the great circles XN, YN, ZN ; then, from the right-angled tri- angles XmN, Ym'N, Zm"N, we have cos XN- cos Xm cos mN - sin OX cos mN, cos YN= cos Ym' cos m'N = sin CFcos m'N, cos ZN - cos Zm" cos m"N= sin GZ cos m"N FIG. 316. ..(10). Therefore adding equations (10), we have cos XN + cos YN + cos ZN= sin CX (cos mN + cos m'N + cos m'N). But m'N= 120 - mN, and m"N= 120 + mN; .'. cos m'N + cos m'N = cos ( 1 20 - mN) + cos (1 20 + mN) = 2 cos 1 20 cos mN = - cos mN ; since 2 cos 1 20 = -1. /. cosXN + cos YN + cosZN=sin CX(cos mN- cosmN) = O...(ll). The numerator of the last term of (9) therefore vanishes when the pole (hkl) lies in the primitive (i.e. when the face is that of a prism) : the denominator must consequently vanish ; for each of the terms of (9) is equal to a finite length OP, that of the perpendicular on the face from the origin. Hence, the sum of the indices h, k and I of any prism-face is zero, and EQUATIONS OF A NORMAL. 355 14. The numerator of the last term of equations (9) can now be expressed in a simple manner whatever may be the position of the pole P(hkl). Let Fig. 317 be a projection of the poles P of a pyramid r {hkl}, and let the diametral zone [CP] meet the primitive at N. Describe the great circles PX, PT and PZ. Then, from the spherical triangles GXP, CYP, CZP, we have (McCl. and P. Spher. Trig. I, p. 36) cos XP = cos CX cos CP + sin CX sin CP cos (XCP = mtf), cos YP = cos C Y cos CP + sin C Y sin CP cos ( YCP = m'N}, cos ZP= cos CZ cos CP + sin CZ sin CP cos (ZCP = m"N} But CZ= CY= CX; .'. by addition, cos XP + cos YP + cos ZP = 3 cos CXcos CP + sin CXsin CP (cos mN + cos m'N + cos m"iV). But it was proved in Art. 13 that, for any pole N lying in the primitive, cos mN + cos m'N + cos m"N = 0. FIG. 317. .(12). .*. cos XP + cos YP + cos ZP= 3 cos CX cos CP. Hence, equations (9) become cos XP _ cos YP _ cos ZP _ 3 cos CXcos CP h k~ I h+k+l (13). 15. Since C, the point at which the triad axis meets the sphere, Fig. 317, is the pole (111) of the parametral plane, it follows that N, the point of intersection of the zone-circle [CP] with the primitive, is a possible pole to which we shall give the symbol (efg). But we have already seen in Art. 10 that the primitive has the zone-symbol [111]; and by Chap, v, Art. 4, the symbol of [CP] is [k - I, I h, h k]. Hence the indices of N are given in terms of h, k and I by the following equations (Chap, v, table 23) : = M-e ...... (14), 232 356 RHOMBOHEDRAL SYSTEM, CLASS I. where 6 = h + k + I an abbreviation which we shall use in cussion of rhombohedral and hexagonal crystals to denote the sum of h, k and I. But, since there are three poles N, P, C in the zone [CP], and two of them, If and (7, are at 90 from one another, it follows that Q is also a pos- sible pole, where A CQ = A CP (Chap. ix, Art. 2); and the H. R. {tfPCQ} = l + 2. Let the symbol of Q be (pqr) : it is required to find the relations con- necting the indices of P and Q. Now, H. R. {NPCQ} gives ,4ft smNP . sinQP _ 1 sin NC ' sin QC ~ 2 the dis- FIG. 317. hkl 111 pqr hkl pqr 111 (15)- Employing the first two columns in the above expression, we have 1 _ ek -fh pk - qh i""r7*vjHT/ from (11) I*-?*. o*(3*-g)-M3*-g)_ 2*. '' - --- ' The equality of the third ratio involving r and I to the others in (16) is inferred from the symmetrical manner in which the several indices enter into the expression (15) for the harmonic ratio. It can be proved by taking together either the second and third columns of the right side of (15), or the first and third. We have now to establish that the signs of the equivalents for (pqr) given by (16) are correct. To prove this, we add the numerators and denominators together, when we find each ratio = P + q + r = P+_V + r (17) From the equations of the normals to P and Q, Art. 14, we have cosTP cosZP 3cosCXcosCP h cosZQ k cosYQ cosZQ _ 3 cos C X cos (CQ = r p+q+r DIRHOMBOHEDUAL FORMS. 357 The numerators of the last ratios in the equations of the two normals are equal. Hence the denominators, 6 a,udp + q + r, must be both positive when P and Q are above the primitive; and both negative when the two poles are below the primitive. Hence, expression (17) is positive. The pole Q given by the ratios (16) is therefore on the same hemisphere as P, as is required by Fig. 317. The relations between the indices of P and Q can also be given as follows : p = 20-3h = -h + 2k + 2l,\ ? = 20-3& = 2h- k + 2l,( (18). r=20-3Z = 2h + 2k- I) The sum p + q + r = 3 (7t + k + I) ; hence the sums of the indices are both positive or both negative at the same time. 16. From equations (16) or (18) it is easy to obtain expressions for h, k and I in terms of p, q and r ; so that, Q being the known pole, P (the unknown one) is determined from Q. For, adding together equations (18), we have p + q + r = 60 - 3 (h + k + f) = 3 (h + k + I) = 30. Hence, 9A = 2 (p + q + r) - Zp,\ 9k = 2(p + q + r)-3q\ (19). 91 = 2 (p + q + r) - 3r } The common factor is cancelled, when the indices are introduced into the symbol. Poles, such as P and Q, connected together by the relations given in (16) and (19) will be called dirJiombohedral poles. They belong to separate forms, which will be called dirhombo/iedral ; for the forms are geometrically similar in each class of the system, and the angles between adjacent faces of the one form are equal to those between corresponding faces of the other. If the forms occur together equably developed, they will compose hexagonal or dihexa- gonal pyramids according to the class to which the crystal belongs. But the forms are independent, and when both are present it is often easy to distinguish by their physical characters the faces of the one from those of the other. 17. The student should bear in mind that the indices of the faces of a general form need not be all positive, and that there may be several forms on a crystal in which one, two, or all three, of 358 RHOMBOHEDRAL SYSTEM, CLASS I. the indices may be negative. Thus in the prism T {Oil}, one of the indices is always negative; in TJ2TT}, two are always negative. It is therefore often necessary to determine on which hemisphere of the sphere of projection a pole lies ; or, what is the same thing, to determine whether a face, having a known symbol (hkl), meets the triad axis at the end which is above or below the plane of the primitive. For this purpose the last term in equations (13), viz. 3 cos (7Z cos OP h+k+l ' affords a ready test ; for the ratio must be always positive. But, since A CX is less than 90, the sign of the numerator depends on the value of A CP. But cos CP is positive when CP is less than 90 ; the denominator must then be also positive. Hence, h + k + I > 0, for any pole lying above the paper in the stereogram, Fig. 317 ; and for any face which meets the triad axis on the side of the origin directed upwards. But, if CP is greater than 90, then cos CP is negative ; and h + k + l<0. Hence the sum of the indices of a face, meeting the triad axis at the end directed downwards, is negative. The positions of the poles with reference to the zone-circles [r'r"], [r"r], [rr], are, similarly, determined from the second, third, and fourth terms of equations (13) respectively. For, if h is positive, cos XP is positive, and XP is less than 90. If h = 0, cos XP = ; and f\XP= 90: the pole P then lies on [r'r"]. Hence, CP being less than 90, P lies on the same side of [r'r"] as C when h is positive : and, when h is negative, /\XP>$0, and P lies on the side of [r'r"] remote from C. Similarly, when k is positive, P lies on the same side of [r"r] as C ; and when k is negative, P and C lie on opposite sides of [r"r]. In the same way I is positive when C and P are on the same side of [rr'] ; and I is negative when C and P are on opposite sides of [rr']. These relations are general, and hold for all classes of the system. 18. It was shown in Art. 8 that, for any particular values of h, k and /, two similar pyramids, r {hkl} and T {hlk}, are possible, the faces (hkl) and (hlk) of which are reciprocal reflexions in the vertical plane XOp. Similarly, the other faces of the two pyramids are, in pairs, reciprocal reflexions in XOp ; viz. (Ihk) and (Ikh), (klh) and EXAMPLE (SODIUM PERIODATE). 359 (khl). The pyramids are therefore enantiomorphous ; and the crystals should rotate the plane of polarization of a beam of plane- polarized light transmitted along the triad axis. This has been established in crystals of sodium periodate. The enantiomorphism can only be recognised geometrically, when pyramids are present having their normals in azimuths differing from 60 ; thus, in Fig. 318, the angle between the azimuths containing the normals (010) and (T85) is 16 6', that between the zone-axes [111, 100] and [111, 504] in Fig. 319 is 49 6'. The triad axis is uniterminal, and should be a pyro-electric axis : this does not seem to have been proved for crystals of sodium periodate. 19. Crystals of sodium periodate, NaI0 4 . 3H 2 O, belong to this class. Figs. 318 and 319 represent plans on the pedion T {ill} of two crystals described by Professor Groth (Pogg. Ann. cxxxvn, p. 436, 1869) ; the Fia. 318. FIG. 319. forms being: r = r{100} and c,=T{lll} (both largely developed), d=r {Oil}, e=T{lll}, z=r{504}, and t = r{l85}. On such crystals the angle c,r, or the angle rr' t may be measured. In the former case, the angular element Z) = 180 e,r=5137'6' is obtained by direct observation. Measurement of A f = 100 A 010 gives 85 31 -5', from which D is calculated by equa- tion (1). The angle D being known, the height c of the apex of r above the pedion is fouud from equation (5) in terms of the sides of the triangular base : it is 1-0937. The crystals rotate the plane of polarization of a beam traversing them in the direction of the triad axis ; and sometimes the rotation is to the right, sometimes to the left: those shown in the figures are laevogyral. Composite crystals are sometimes obtained, which show in convergent polarised light Airy's spirals : such crystals are generally regarded as twins of a dextro- and a laevo-gyral crystal united together along a common pedion ; the bases c, of both being parallel and directed outwards . 360 RHOMBOHEDRAL SYSTEM, CLASS II. II. Diplohedral trigonal class; TT {hkl}. 20. When a centre of symmetry is associated with a triad axis, no other element of symmetry is necessarily involved. The class having these two elements of symmetry we shall call the diplohedral trigonal class ; it has been known as the parallel-faced hemihedral class of the system. FIG. 320. Owing to the triad axis, any face r, Fig. 320, inclined at a finite angle (other than 90) to the triad axis is repeated in two similar faces r and r", which have to one another the relations of the three faces forming a pyramid in the last class : but, since the crystals are centro-symmetrical, the above three faces are associated with three similar faces r, r t , r tt , each parallel to one of the first set. The first set of faces meeting at an apex F, the second set may be drawn through an apex V t at an equal distance from the origin. But, since the face r (F/^/A/A,,) meets the parallel faces r" and r tl , the edges Vp. / and /A/A /7 must be parallel (Euclid xi, 16) : and, since the face r meets the parallel faces r and r t , the edges Vp. fl and /A, /A must also be parallel. Hence the face Vfijtp^ is a parallelogram. It is also a rhombus, for the two sides Vp. tl and I 7 /*,, meeting at the apex, are interchangeable when the crystal is turned through 120 about the triad axis. Similarly, every other face can be shown to be an equal and similar rhombus. Hence the form is a rhombohedron; and it has given the name to the system. We shall call the like and interchangeable edges which meet at the same apex, such as V, co-polar edges', those, like /A,/A, /A/A,,, &c., which occupy a middle position the median edges. THE FUNDAMENTAL RHOMBOHEDRON. 361 21. Since the three faces meeting at one of the apices are similar to the faces of the pyramid selected in Art. 3 to give the axial planes, we may take the axes of X, Y and Z to be three lines through the middle point parallel, respectively, to the co-polar edges V t p., V t p.', V t p." of some conspicuous rhombohedron. This rhombohedron we shall therefore call the fundamental rhombo- hedron ; and we shall denote its faces, and their poles, by the letters r, r', r", &c. We shall also take the upper face of the pinakoid perpendicular to the triad axis for the parametral plane (HI). The parameters are therefore equal ; and may be taken to be unity, or any three equal lengths on lines parallel to the axes such as V t M, V t M' VM t , and V, M" of Fig. 320. The formulae of computation and the rela- tions between the several lines of the axial and parametral planes established in preceding Articles hold for crystals of this class; and need not therefore be repeated. The fundamental rhombohedron r{100}, Fig. 320, includes the following faces : 100 010 001 TOO OTO GOT ...................... (d). 22. Any face (hkl) inclined to the triad axis at a finite angle (other than 90) gives rise to a rhombohedron TT {hkl}, which includes the faces : hkl Ihk klh hkl Ihk klh ........................ (e). The only limits to the relative magnitudes of h, k, and I are : (1) that they cannot be all equal, and (2) that h + k + l>0. 1. If h-k = l, the face belongs to the pinakoid {111}, which includes the faces (111) and (111), both perpendicular to the triad 2. When h + k + l = 0, the face is parallel to the triad axis and the form tr {hkl} consists of a hexagonal prism, adjacent faces of which are inclined to one another at angles of 60. The symbols of the faces of the prism TT {hkl} are given in table e, the difference between the symbols of the general form and of a prism arising from the particular relation between the indices; thus, {101}, {2ll}, ir {321}, 7r{43l} are hexagonal prisms, ir{421}, ir{421) are rhombo- hedra. The rhombohedron {100} is not a special form: it only differs from other rhombohedra inasmuch as the axes of reference have 362 RHOMBOHEDRAL SYSTEM, CLASS II. been taken parallel to its three co-polar edges. Similarly, the parti- cular cases of the hexagonal prisms {Oil}, {2lT} differ in no essential respect from the hexagonal prism IT {hkl} : they include six faces which are geometrically identical with those of the corresponding pairs of complementary prisms of the last class, and in the case of the two former with the six faces given in tables f and g of class III. Crystallographers have not always agreed as to the rhombohedron to be selected to give the axes, i.e. as {100}. By comparing the values of D or c, given at the beginning of the description of the crystals, it is easy to see whether the same fundamental rhombohe- dron has been selected or not. Thus, in dioptase, CuH 2 Si0 4 , Miller's {100} is Dana's {Til}. The faces are those labelled s in Fig. 321. Dana does not indeed adopt a set of axes such as that described in Art. 3, but employs four axes which will be explained in the next chapter. But in trans- forming from Dana's representation to Miller's, the fundamental rhombohedron is not the same. Miller took s, the rhombohedron most conspicu- ously developed on the crystals, to be {100}; Dana selected as axial rhombohedron that which is parallel to the cleavages and truncates the polar edges of s, which then becomes {Til}. Hence Miller's Z> = (111A100) = 5039'; whilst Dana's Z>=(0001 AOTll) = 3140', and this latter angle is Miller's (111 A Oil). The two angles do not quite accord ; for, if Dana's value is accepted as correct, then Miller's angle should be 50 58'. The Greek prefix IT is omitted before the brackets in the symbol of the form, whenever two of the indices are equal or the forms are one or other of the hexagonal prisms {Oil} and {2ll}; for in these cases the forms are geometrically identical with the corresponding forms of the next class, which was by Miller regarded as the holohedral class of the rhombohedral system. We shall find that one or other of these forms belongs also to other classes of the system. 23. Crystals of dioptase, CuH 2 SiO 4 , and phenakite, Be 2 SiO 4 , belong to this class. Fig. 321 represents a crystal of dioptase in which the faces s are {111}, z = tr {443}, a; = 7r {221} and a{OTl}; the above symbols being obtained from Dana's fundamental rhombohedron. Measurement EXAMPLE (PHENAKITE). 363 of one of the zones [aoca,,*'] suffices to determine the symbols of all the faces by means of the A. R. of four tautozonal faces ; for the symbols of s'(lll), a y , (110) and (111) are easily determined from the assumption of the fundamental rhombohedron. The angles are : a n x = 28 48', a tl z = 39 31', a,,s = a,/ - 47 43'. Two crystals of phenaMte from Colorado, described by Prof. Penfield, are represented in Figs. 322 and 323. The poles indicated by the same FIG. 322. FIG. 323. letters -are represented in Fig. 324. The forms are: r{100}, ^^^.,, p (on the right of rf, Fig L 323) = -{201}, p' (being p on the left of d) 7r{102}, z= {T22},#=rr{2Tl}, rr{3Tl}, = 7r{20l}, a = It should be noticed that r and z y p and p', are dirhombohedral forms ; and that the indices of the faces of the pairs p and p', x and #,, are in reverse cyclical orders. The faces of the latter pairs of forms are associated together in class III ; and the members of each pair may therefore be regarded as com- plementary forms. The symbols of the forms can be determined by obser- vation of, and measurement of the angles in, a few of the more important zones ; for in most of them the indices of three or more poles, such as a, ,/', 2, are immediately found when r is selected for the fundamental rhom- bohedron {100}. Thus, /' lies in [r'm'] and [ar], and is (111) ; also z' is (2l2), and x is in [ar] and [wr"] : it is therefore (2ll). Similarly, o, p and d are in pairs of zones, the symbols of which are readily determined, and therefore the indices of the faces. 364 RHOMBOHEDRAL SYSTEM, CLASS II. We can determine the element D, and the symbols of p', p and s from measurement of the zone [a,,rdps]. For, adopting as measured angles, a"s 28 21', a"r' = 58 18', a"p = 18 22', a"d = 90, a"p'= 101 38' ; then, from the right-angled triangle Udr (G being the central pole (111): as the pinakoid is not actually present, the pole is not shown) sin dr = sin 60 sin D (see Art. 5). But dr = 90-a // r = 3142'; .-. D = 37 21-3'. Again, d being the pole between p and p' in which the zones [a"r] and [r"m tl ] intersect, then d is (110) ; and since it is at 90 from a", the A. K. |a"/sd} gives tan a"r' 110 010 110 M-0 110 010 110 //A-0 k-h k + h' L tan (a'V = 58 18') = 10-20928 Ltan(a" =28 21')= 9-73205 47723 = log 3. .-. j^ = 3, .'. h = I, fc = 2; k + h and the symbol of s is (120). Similarly, for p in the same zone we have from the A. B. (a"r'pd) h + k tan(a>=7822') ' by com P utatlon ' .-. h=l and k-2; &ndp is (120). Being given ax = 62 17' and ao = 70 42', we can, in a similar manner, find the symbols of x and o; for ar = 90, and af can be easily found from the right-angled spherical triangle m t af in which A am, and A ram,= Arm are both known. Or, conversely, knowing x to be (2ll), we can find the angle ax. Thus, tan a/' = tan(m / a = 30)-f-cos(raa/' = 7Mr) = tan30-7-siu 37 21 '3' ; /. by computation, af =43 34-6'. And, since ar=90, we have from the A. B. {af'xr} tan ax tan (af = 43 34-6') Oil 211 joli 111 .-. by computation, ax= 62 16-75'. The stereogram, Fig. 324, is made as follows. The primitive being described with any convenient radius, arcs of 30 are measured off on it, and diametral zones through these points are then drawn. The alternate points at 60 from one another are the poles a {Oil}, the other alternate points are the poles m{211). On the radius through m an arc rm = 90-37 21-3' is marked off (Chap, vn, Prob. 1) ; and the homologous poles r', r" as well as the dirhombo- hedral poles z, z', z" are then found at the same distance from the centre. The zone-circles [ar], [a"r], [mz'\, [m"r] are then described, and fix the posi- tions of all the poles. THE FUNDAMENTAL RHOMBOHEDRON. 365 24. III. Scalenohedral class; {hkl}. This class, the most important one of the system, may be derived from class II by the introduction of a dyad axis, or plane of symmetry. Suppose a dyad axis to be added to the elements of symmetry of class II ; it must be at right angles to the triad axis, or it will introduce other triad axes, thus contravening the definition of the system, viz. that the crystals have only one triad axis : the cases in which there are several triad axes have been dis- cussed in Chap. xv. But a dyad axis perpendicular to the triad axis must be associated with two other like and interchangeable dyad axes, both of them at 90 to the principal axis and at 120 to the first dyad axis and to one another. Again, a centro-symmetrical crystal must have a plane of symmetry, 2, perpendicular to each dyad axis. There must therefore be three like and interchangeable planes of symmetry intersecting in the triad axis at angles of 60. The crystals of this class have therefore the following elements of symmetry : p, 38, C, 32. The arrangement of the planes and axes of symmetry is shown in Fig. 325. The triad axis is a possible zone-axis, for it is the line of inter- section of three planes 2, which, by Chap, ix, Prop. 1, are parallel to possible faces. It is also perpendicular to a possible face that parallel to the three dyad axes. The central plane parallel to this face will, in this and the hexagonal systems, be called the equa- torial plane : it is not a plane of symmetry in crystals of this class. FIG. 325. 25. The rhombohedron, {100}. Since a dyad axis is a possible zone-axis, a form is possible having a face parallel to this axis, and perpendicular there- fore to the plane of symmetry of which the dyad axis is the normal. When the face, as, for instance, V^ t p.p. ti of Fig. 326, is inclined to the triad axis at a finite angle, other than 90, there will be three such faces meeting at an apex in the triad axis, each parallel to one of the dyad axes and perpendicular to the corresponding plane of symmetry. Again, since the crystal is centro-symmetrical, there will be three like faces meeting the triad axis at an opposite apex, each of them being parallel to a dyad axis and perpendicular to a plane 2. But a face parallel to a dyad axis is, by a rotation 366 RHOMBOHEDRAL SYSTEM, CLASS III. of 180 about this axis, brought into the position of the parallel face. Thus the parallel faces F/A, /*/*, F^'/Z/A" are parallel to OS and interchangeable by a semi-revolution about it. Again, the faces F/x ( //'/Z and F,/A/A'/A" change places when the crystal is turned through 180 about the axis OS bisecting the edge p.^" at right angles. The faces meeting at opposite apices are therefore sym- metrical with respect to the dyad axes ; and no new faces are introduced by them. Similarly, no new faces are introduced by the planes of symmetry, each of them being perpendicular to a pair of parallel faces, and bisecting the angles between the pairs of other faces meeting at an apex. The figure is geometrically similar to that described in Art. 20, and is a rhombohedron. In Fig. 326 the dyad axes are the lines OS, OS', OS" : they pass each through the middle points of opposite median edges, e.g. ft,//' and /A'/*,,, kc., and are perpendicular to the edges which they bisect. The planes 2 pass each through a pair of parallel polar edges, and also through the polar diagonals of the pair of faces to which they are respectively perpendicular. Thus 2 passes through the polar edges F/Z, F,/A, and through the polar diagonals F/x and V^L. 26. As before, the axes of X, Y and Z are taken parallel to the three co-polar edges of any possible rhombohedron, which is then called the fundamental rhombohedron : its faces are denoted by r, r', &c. The axes lie each in one of the planes of symmetry, and we shall consider XX, of Fig. 309 to liejn 2, YY t in 2' and ZZ t in 2": they are parallel, respectively, to VM, VM t and VM it . The points S and M of Fig. 326 are projected in the manner described in Chap, vi, Art. 19 ; 08' being placed in the prolongation PROJECTIONS OF RHOMBOHEDRAL AXES. 367 of D'C of Fig. 51 : the rhombohedral axis IT which is perpendicular to OS' is consequently in the plane CyA', and this plane coincides with 2'. The positive direction of the rhombohedral axis OX projects to the front and right in a vertical plane through Cy inclined to D'Cy at an angle of 30. Similarly, the negative direction of OZ lies to the front and left in a vertical plane through Cy inclined to D'Cy at an angle of 30. This arrangement is also adopted in Chap, xvn in the representation of hexagonal crystals by rhombohedral axes. Or we may, as is the case in most of the drawings of calcite, place 08 ti in the back-and-fore axis CA' of the cube projected in Fig. 51, when the axis ZZ t lies in the plane D'Cy which now coincides with 2". The rhombohedral axes OX and OY lie in vertical planes inclined to D'Cy at angles of 60 on opposite sides of it. The vertical planes containing the axes of reference are now in azimuths inclined at 30 to those of the first position. Cy is the unit of length on the triad axis, and OV=cxCy; the linear element c being connected with D by equation (6). The method of finding the rhombohedral axes from Naumann's projection of the cubic axes is the same as that described for the case where Mohs' axes serve as basis. The parametral plane (111) is taken to be the face perpendicular to the triad axis and parallel therefore to the dyad axes 08, 08', 08". The parameters may therefore be taken to be any three equal lengths which may be convenient; and in theoretical ex- pressions, such as, for instance, the equations of a normal, may be taken to be unity. The expressions established in Arts. 5, 6, 12 16, hold therefore for crystals of this class. The angular element D is the inclination to the equatorial plane of each face of the fundamental rhombohedron, and is the angle Cr = 111 A 100. It is connected with the angle between the faces r, r' by equation (I), with c by equation (6), and with the inclination of the axes of reference to the principal axis by equation (3). 27. Other special forms are the pinakoid, and hexagonal and dihexagonal prisms. 1. The pinakoid, {111}, consists of the faces (111) and (111), both perpendicular to the triad axis. 2. The hexagonal prism, {HO}. When a face is parallel to one of the planes of symmetry 2 (say), it is parallel to the axis of reference XX in this plane : the corresponding index is therefore zero. 368 RHOMBOHEDRAL SYSTEM, CLASS III. oTi i ! \tTo\ \ a, a,, ^' y FIG. 327. Thus, in the particular instance of the face being parallel to XX /5 the first index is zero. The relation between the two other indices can be obtained from the geometry of Fig. 309 in the manner described in Art. 9. Hence, the pair of faces parallel to 2 and to XX t are (Oil) and (OlT) : they are perpendicular to 08. By rotation about the triad axis through 120, the two faces are brought successively into positions in which they are parallel to YY t and ZZ t and are respectively perpendicular to 08' and 08". The form, Fig. 327, is therefore a hexagonal prism having the faces : Oil 101 T10 OlT T01 1TO (f). The angles between adjacent faces are clearly all 60. The prism is constructed by drawing lines through the points fL, ju.,, p.", &c., of Fig. 326 parallel to the principal axis. 3. The hexagonal prism, {2TI}. When a face is parallel to the triad axis and to a dyad axis, we also get a hexagonal prism, the angles between adjacent faces of which are 60. A face of the prism may be drawn through one edge of the triangle XYZ of Fig. 309 ; for the sides of this triangle are parallel each to a dyad axis, and they are perpendicular each to a plane of symmetry. Hence, the symbol of the face XYL t can be obtained from the geometry of the figure in the way employed in Art. 11. The face has therefore the symbol (112); and the form {211}, Fig. 328, includes the six faces : 21 T T2T TT2 211 121 112 (g). The symbol of the face XYL / can also be obtained from the relations of the two zones to which it is common : viz. the zones [Oil, 10T] = [111] and [111, 001] = [1TO]. Hence, if the face is taken to be (hkl), we have from the first zone h + k + I = 0, and from the second, h-k =0. FIG. 329. .'. h = k=l, andZ=-2; or h = k = - 1, and 1 = 2. The parallel faces have therefore the symbols (112) and (112). FIG. 328. DIHEXAGONAL PRISMS. .(h). 132 X31 321 312 The prism is constructed by drawing lines through the points 8, 8 lt , 8', &c., of Fig. 326 parallel to the triad axis. The faces of {211} will be denoted by the letters m as indicated in Figs. 328 and 329, the faces of {011} by a. The faces of the one prism truncate the edges of the other ; and adjacent faces a and m are inclined to one another at angles of 30. A combination of the two prisms and the pinakoid {111} is shown in Fig. 329. The poles of the two prisms are given by the points marked m and a in Fig. 331. 4- The dihexayonal prism, {hkl}, where h 4- k + I = 0, has its faces arranged in pairs which are interchangeable by rotation about the dyad axes S, and in pairs which are symmetrical about the planes 2. The faces can also be arranged in triads inter- changeable by rotation about the triad axis. Hence the form consists of : hkl Ihk klh Ikh hlk khl\ hkl Ihk kih ikh Ilk khi} The symbols of the faces in each of the four triads making up the form are con- nected together by cyclical order, and Fio. 330. the faces of a triad change places with one another on rotation about the triad axis. Again, the two faces in each column are parallel. The particular case {312} is shown in Fig. 330. It remains to prove that triads in opposite cyclical orders must coexist. This follows from the fact that an axis OX (say) lies in a plane 2 with respect to which the two other axes OY and OZ are reciprocal reflexions, for 2 is the plane OXpE of Fig. 309. Hence if two faces are symmetrical with respect to 2, they must meet OX at the same point, whilst the intercepts on OY and OZ of the two faces are reciprocal reflexions. If therefore (hkl) is one of the faces, the other is (hlk), in which the intercepts on OY and OZ have changed places; the symbols being in opposite cyclical order. Owing to the triad axis, (hkl) is repeated in (klh) and (Ihk), and (hlk) in (kid) and (Ikh) ; and pairs of the new faces, one from each of the triads, are symmetrical with respect to 2. The six faces are also symmetrical in pairs to 2' and 2", for rotations of 120" about the triad axis bring 2 to 2' and 2" in succession. Again, the dyad axis OB it is parallel to the side XY of the base in Fig. 309, and is perpendicular to 2" and OZ. Hence, it is equally inclined to OX and OY, and bisects the angle XOY r A L. c. 24 370 RHOMBOHEDRAL SYSTEM, CLASS III. rotation of 180 about 08 /t interchanges equal positive and negative lengths on OZ ; and changes a positive length on OX with a negative one on OY, and vice versa. A face (hkl) is therefore by this rotation brought into a position given by the symbol (khl). This face is parallel to one of the six faces already obtained. The form having the faces given in (h) is therefore complete. The angles between pairs of adjacent faces are constant, and those over two adjacent edges are in all cases unequal, whilst the angles between alternate faces are 60. For the angle over an edge lying in 2, such as hkl A hlk, is double the angle either face makes with 2. Similarly, the angle, such as hkl A Ikh over the adjacent edge meeting 08' is double the angle which (/tkl) makes with OS', the extremity of which coincides with the pole a' in Fig. 331. But 2 and the adjacent dyad axis 08' are at 30 to one another. Hence the angles hkl A hlk and hkl A Ikh are together equal to 60. Further, if the two angles were equal, the zone would be divided isogonally by poles of possible faces into arcs of 15 and 45"; and it was shown in Chap, ix, Art. 12, that angles of 45 cannot occur in a zone containing angles of 30 or 60 repeated in succession between possible faces. The distribution of the poles of the prism {hkl} on the primitive is shown in Fig. 331 : from it we can, by the A.R. of four tautozonal poles, find the angles mN and hkl A hlk, or the angles a'N and hkl A Ikh, and prove that they are con- stant and independent of the crystal- element D which varies with the substance. The angles can indeed be calculated for any particular values of the indices, and apply to all crys- tals of the system in which faces having the same symbols occur. Let, in Fig. 331, a', the pole of the plane 2,, be (10T), N be (hkl), m (211) and ra, (121), where ma' 30 and m t af = 90. Then, from the A.R. {a'Nmin}, we have hlk klh FIG. 331. tan a,'N+ tan a'm = - k -=- =- = - 101 121 hkl hkl 101 121 2TT 211 3k (20). DIHEXAGONAL PRISMS. 371 In the figure, and also in the A.R., the pole N is taken to lie somewhere between a' and m y ; hence k is negative and 2h must be numerically greater than k : the expression on the right of (20) is therefore positive, as is required by the way in which the poles have been taken in the A.R. The angle a'N can therefore be computed when the indices h and k are given ; and all the angles of any prism can then be found. We have the following angles for the prisms given in the first column : {hkl} tsMa'N a' A hkl hkl /\ Ikh hkl f\ hlk {312} '- {413} f <' {523} ^ 19 6-4' 13 54 10 53-6 23 24-8 38 13' 27 48 21 47 46 49-6 21 47' 32 12 38 13 13 10-4. 28. For drawing the dihexagonal prism and other forms of the rhombohedral system, Fig. 332 is useful. In it the paper coin- cides with the equatorial plane perpendicular to the triad axis ; A A, A'A t and A"A it are the dyad axes; OM, OM', OM" are the traces of the planes of sym- metry ; M, M', M" are the points in which the lower polar edges V t p., V t p.', V t /A" of the fun- damental rhombohedron, Fig. 326, meet the equatorial plane. The apex V t is at a distance c below 0. The lengths V f M, V,M', V f M", on edges parallel to the axes OX, OY and OZ, may be taken to be the Millerian para- meters : they are equal and parallel to OX, OY and OZ of Fig. 309. We shall first determine the relations between the lengths OM, OA, &c., and the lengths of the lines in the triangle XYZ of Fig. 309. The triangle MM'M" is equal and similarly placed to XYZ of Fig. 309 ; for the three rhombohedral faces through F are parallel to the axial planes through 0, and OV t Op. Hence MM' = M'M" = M"M=3a of Art. 6. 242 372 RHOMBOHEDRAL SYSTEM, CLASS III. In Art. 25 the faces of the rhombohedron {100} are shown to be parallel each to one of the dyad axes; the positions of which are therefore OA, OA', OA", parallel respectively to M' M", M"M and MM', and to YZ, ZX and XY of Pig. 309. Since is the centre of gravity of the triangle MM'M", we have 0^OB I = JI = OM^2. Also OA : EM" = OM : BM = 2 : 3, /. OA = 2BM" - 3 = M 'M" -=- 3 = a. The length a = JTFn-3 is therefore an arbitrary length measured on the dyad axis, with which the linear element c is connected by equation (5). We shall, whenever it is necessary, denote a length a measured on OA' by a t , and on OA" by a tl ; and similarly, we shall denote lengths = OB mea- sured on OM, OM' and OM" by b, b t and b it , respectively. Again, AOA ti is an equi- lateral triangle, for its sides are parallel to those of MM'M"; and OB t bisects the angle AOA ti , and is perpendicular to AA /t . Hence OB t = OA cos 30. /. OB = OB t = OB ti = b = a cos 30 ; and OM = OM' = OM" = 20B = 2b = 2a cos 30 (21). We shall now find the lengths cut off on the lines of Fig. 332 by the prism-face (hkl); and show how to derive the symbols of the other prism-faces, one method having already been given in Art. 27. The dihexagon H8'H, r .. is the trace on the equatorial plane of the prism {hkl} ; the face (hkl) meeting the axes and the parallel polar edges at distances V t M + h, V t M' + k, V f M"^-l. But in projections by means of parallel rays, lengths on any line are projected in lengths having to one another the same ratios. Hence, OM being the projection on the equatorial plane of V t M, and Off that of V,M+h, it follows that OH=OM+h. Similarly, OM' and OK=OM' + ka,re the projections of V,M' and V,M' + k; and OM" and OL = OM" -r I are the projections of V M" and DIHEXA.GONAL PRISMS. 373 V t M" + l. The trace HKL of the prism-face (hkl) is given by OM+h, OM' + k, OM" + l. The planes 2, 2,, S,, pass respectively through OM, OM', OM", and are perpendicular to the paper. Hence the face (hkl) is repeated by 2 in a face which meets the paper in the trace K t HL t , where the angle OHK t = A OHL. But A HOL = A HOK t = 60. The tri- angles HOL and HOK t are therefore equal, and OK i = OL = OM' + l; and A OLH= A OKH. Again, the triangles OKL and OK I L I are equal, for they have the common angle KOL = \2Q, and A OLK= A OKL t . Also the side OL = the side OK t . The remaining sides are therefore equal, and OL, = OK= OM" -=- k. Hence the trace K I HL I is given by OH=OM+h, OK = OM' + l, OL t = OM" -=rk\ and these lengths are the projections on the equa- torial plane of lengths V t M + h, V t M' -r 1, V t M" -f- on the axes. The new face is, as before (p. 369), (Jdk). In the figure, OH is the shortest length, and is measured on the same side as M : the corresponding index h is positive and numeri- cally the greatest. The points K and L are measured away from M' and M", and the corresponding indices k and I are negative. In no case can the indices be all positive, for h + k + 1 = 0. By a semi-revolution about the dyad axis A t A' the trace K1IL is brought to GB'H^K', where A 08'#,= A 08' H. The angle 8'OH it = A 8'OH = 30 ; . '. OH tl = OH = OM" - h ; since OH tl is measured on the side of away from M". Similarly, OG=OL = OM+l; for G is on the same side of as M, whilst L and M" are on opposite sides of : the sign of the index I has therefore to be changed. Again, OK' = OK, the two being measured in opposite directions on a line perpendicular to the dyad axis : the signs of the indices are therefore changed. Hence OK' = - OK = OM' -H k. The trace GU H K' is given by OM+l, OM' ^k, OM" ^h; and the intercepts made by the prism-face through GH ti K' are V t M+l, VM' + k, V f M" + h. The symbol of the face is therefore (Ikh). The rule connecting a pair of prism-faces interchanged by a semi- revolution about a dyad axis is, therefore, (a) that all the signs of the indices are changed, and (/3) that the indices referring to the two axes inclined to the axis of rotation change places, whilst the index referring to the perpendicular axis changes sign only : the symbols are therefore in reverse cyclical orders. 374 RHOMBOHEDRAL SYSTEM, CLASS III. The above rules connecting the symbols of consecutive faces symmetrical with respect to a plane 2 and to a dyad axis have been established for the dihexagonal prism only ; but they will be shown to hold generally whatever be the position of the face (hkV). The symbols of the remaining faces of the prism are readily obtained by repeating the above operations. Thus the face through H tl K H is (klh), the next through H' is (khl) and that through H'8" is (Ihk). To make an orthographic or clinographic drawing, all that is needed is to describe, in the plane of the horizontal cubic axes of Chap, vi, a figure similar to Fig. 332. The vertical lines through the projected points corresponding to ff, 8', Jf t/ , &c. are the edges of the dihexagonal prism. 29. We can now determine the lengths 08', Od lt and OE in which the trace HKL of the prism-face (hkl), Fig. 333, meets the three dyad axes. For, since the triangles H08' and 8'OL make up the triangle HOL, we have OH . 08' sin 30 + 08' . OL sin 30 = OH.OL sin 60. Dividing by OH . OL . 08' and by sin 30 = cos 60, we have tan 60 1 J_ 08' ~ OH*~OL' In this expression OH and OL are treated as positive lengths. Hence, when they are replaced by their equivalents in terms of OM and the indices, attention must be paid to the signs of the latter. Thus for the face taken, h is positive whilst I is negative ; and in the above expression OM 4- 1 must be taken as the length OL. Hence, since tan 60 = ^3 and, from (21), OM=aJZ; \ FIG. 333. 08' OM I OM h-l X/3 .-. 08'=~ (22). In a similar manner Od tl (d /t being the point in which HKL cuts 0-4,,) can be found from the triangles H0d /it d n OK and HOK. M -=i < 23 >- THE FUNDAMENTAL RHOMBOHEDRON. 375 The length OE (E being the point in which 11 KL meets the axis AA) is obtained from the triangles EOS, 8'0d lt and E0d it , the angles E08', 8'0d it being both 60. Thus OE . Od tf sin 120 = OE . 08' sin 60 + 08' . Od it sin 60. But sin 120 = sin 60 : dividing therefore by OE . 08' . Od ti sin 60, 1 1 1 we have -^, = 55-=- + , :. OE = 3a (24). 30. By the aid of Fig. 333, we can give an independent proof of the relation, h + k + l = Q, holding between the indices of a prism-face. For the triangle KOL contains the two triangles KOH and HOL. :. OK. OL sin I 20 = OH . OK sin 60 + OH . OL sin 60. Dividing by OH . OK. OL sin 60, we have 1 OH 1 1 OL + OK in which the lengths are all positive. We must therefore write OM+k for OK, and OM-^-l for OL ; since the face taken meets the axes of Y and Z on the negative side of the origin. Hence, h = -k-l, and The Rhombohedron. 31. When the angle D, or the angle r'r of the fundamental rhornbohedron {100}, is known, we can find the distance c of its apex V from the origin, 08' (Fig. 335) being the unit of length a on the dyad axis. Let Fig. 334 represent a section through the plane EOX of Fig. 309 and V,OM of Fig. 335, so that the point p of Fig. 309 coincides with the apex V, and XV^ZVK. Now OX is parallel to V,M, OE to fj.V [the polar face-diagonal of (100)] ; and the normal Or lies in 2 and meets p V at r. The inclination of the face to the equatorial plane is therefore /> = A VBO=I\OEX=I\ VOr. Since OV = V, and OX is parallel to V,M, .'. from (21), OM= VX = 2a cos 30. 376 RHOMBOHEDRAL SYSTEM, CLASS III. Also O8= VE= 7X-2 = 0J/-2=:acos30 (25). Hence, c= 0V = OB tan D = a cos 30 tan D (5*), and tan VOX = tan OV,M = OM '-=- OV t = WB + c = 2 cot D. Therefore C being the point at which the triad axis meets the sphere aud X being the axial point, tan 02: tan/) = 2 (3*). Fig. 336 represents a section of Fig. 335 in the plane VV M" exactly similar to, and on the same scale as, Fig. 334. We can now draw the rhombohedron {100}. Taking 08 t to be the unit of length CD' of Fig. 51, or 07, of Fig. 60, then the apex V is found by marking off on Cy or OA" of the same figures lengths equal respectively to Cy cos 30 tan D, or OA" cos 30 tan D: in Fig. 335 07 has been taken to be OA" x M. Again, for the cleavage-rhombohedron r of calcite, r'r = 74 55' ; and, by computation from (1), Z> = 4436'6'. Therefore OV=OV. = OA" cos 30 tan 44 36-6' = OA" x -8543 : also JT07= 707=^07= 63 44 -75'. The hexagon 8,88,,... and the points M' ', M t , M, &c., of Fig. 57, being projected in the horizontal plane by the method described in Chap, vi, Art. 19, the lower polar edges join 7, to the points M, M', M" ; and the upper polar edges join V to the DRAWING THE RHOMBOHEDRON. 377 opposite points M, M t , M it . The median edges p.p. t , p., p.", &c., are then drawn through the points 8 y/ , 8, " = //'- 2 = F//'-4 ; since the dia- gonals of a rhombus bisect one another. Again, from Fig. 336, 0V : Ot= VB" : B"p." = 3 : 1. Hence the median coigns lie on the lines parallel to OM, OM t , fec., drawn through the points t and t f which trisect the length FF y . Again, F/Z is parallel to OX, and Vp t to OY; also fjis - AA t =OA=a, since OAA f , Fig. 332, is an equilateral triangle. Hence tan \XO Y = tan s F/A, = ps -5- Vs - 3a - 2 VB". But in Fig. 336, OB" - VB" cos J), and OB" = a cos 30. (26). 33. The direct and inverse rhombohedra, mR and mR. The method of drawing the rhombohedron {100} applies to all rhombohedra ; for any rhombohedron possible on the crystal may be selected to give the directions of the axes : relations equivalent to those given in Arts. 31 and 32 for {100} hold between the several DIRECT AND INVERSE RHOMBOHEDRA. 379 lines, coigns and angles of every rhombohedron. Further, by the law of rational indices, faces belonging to different forms of the same crystal will, if drawn through the same point on any zone- axis, meet other zone-axes at distances measured from the origin, which are to one another in commensurable ratios. Hence three faces of all rhombohedra belonging to the same substance can be drawn through the lines MM', M'M", M"M of Fig. 57 or Fig. 332 to meet the triad axis at apices V m or V m , where OV m - OV m = mc, and m is a commensurable number. Two rhombohedra can be obtained for each value of m, according as the points M, M', M" of Fig. 57 are joined (i) to the lower apex V m in the manner adopted in constructing Fig. 335, or (ii) to the upper apex V m . i. In the first case the upper polar edges are given by V m M, V m M t , V m M ti ; and the lower polar edges by V m Af, &c. The figure is completed in the manner described in Art. 31 by finding the points of trisection A, Fig. 338, of the polar edges nearest to the equatorial plane. This rhombohedron is called by Nau- mann the direct rhombohedron, and is denoted by the symbol mR. Its poles g lie in the diametral zone- circles Cr, Or', Cr" of Fig. 342, p. 382, and on the same side of C as r, r' and r". When m = 1 , the rhombo- hedron is R, and is the same as Miller's {100}. Fig. 338 is a vertical section of R and mR showing the lower polar edges V t M and V m M, and the upper polar face-diagonals J r /t and V m \. The lines /i< 7 , AT, are drawn through the median coigns p, and A parallel to OM ; and meet the triad axis at t t and T,. Hence, ^ : OM = J> : VM = 2 : 3; and since Or t = V m H- 3, XT, : OM= V n \: V m M = 2 : 3. .'. XT = fJ .t i ^20M : 3 -(from (21)) 2a^ ^3 (27). ii. The second rhombohedron, called the inverse r/wmbohedron, and denoted by mR, is obtained by joining the points M, M', M" to V m , and the opposite points M, M t , M it to V m ; i.e. the points in 380 RHOMBOHEDRAL SYSTEM, CLASS III. the equatorial plane are joined to the apices in the reverse order to that adopted in the first case, and this is indicated by making m negative. The mode of construction is the same as that given for {100} and mJR. Fig. 339 represents the inverse rhom- bohedron - R, or {1 22}, the faces of which will be denoted by the letter z: it corresponds exactly (except that the scale has been reduced in the ratio of 3 : 4) with the direct rhombohedron R, or {100}, shown in Fig. 335. FlG - The following are a few instances of rhombohedra which are frequently observed in crystals of this class : %R, R, 2R, 3R, -R, - R, 2R, - 3R. The faces of the direct and inverse rhombohedra, in which m has the same value, are connected by the relation of dirhombohedral faces given in Art. 15; and their poles above the primitive lie in the diametral zones, [CV], [CV], [CV"] at equal distances from C. 34. The special case, where m = 0, gives a single plane coinciding with the equatorial plane, MM'M". For the apex coincides with the origin 0. Hence OR is Naumann's symbol for the pinakoid {HI}. Similarly, when m = oo , the vertex is infinitely distant, and the faces are those of the prism drawn through the lines MM', M'M", &c., all parallel to the triad axis. Hence ccR is Naumann's symbol for the hexagonal prism {21 1} described in Art. 27. 35. To find the Millerian symbols for the faces of mR. A face through M / M ti of Fig. 335 and an apex V m meets the axes of Y and Z at the same distance from the origin and on the same side of it, for OF and OZ a,re parallel to VM t and VM tl , respectively. Hence the second and third indices of the face are equal, for the parameters are equal. But the axis OX will be met at a different distance; and on the same side of the origin as OF and OZ, if the face is less steeply inclined to the equatorial plane than the corre- sponding face of the fundamental rhombohedron, i.e. if V m lies nearer to than V. The axis of X will be met on the opposite side of the origin, if V m is further away from than V. The face is, therefore, (hll) where h and I have the same, or opposite signs MILLERIAN SYMBOL FOR mR. 381 according as the pole of the face is nearer to C, or more remote from it, than ? (100). Let Fig. 340 represent part of a section of the rhombohedra mR and R in the plane 2 containing X and the polar edges V m M, V t M. The coign X is on the line drawn parallel to OM through T,, where 0r, = 0F w -r-3. Also, F m X is the polar face-diagonal and meets OX in L, OM in B and V t M in L t . Adopting F M as parameter, then OL= V t M+-h. But, from the similar triangles V,L t V m and OLV m , we have V,L, _V,V~ mc + c m+\ OL ~ OV m ~ me '' m ' From the similar triangles OLB and MBL t , we have L, M _ BM OL ~ OB since (Art. 31) OB=BM. Adding (28) and (29), we have = 1 (29); + 1 = 2m +1 .(30). h= V,M = OL ~ m m The face through V m \ of Fig. 340 meets the adjacent plane 2' in the polar edge F m \ which, since OV m >OV y must be prolonged to meet Y at K on the nega- tive side of the origin. Fig. 341 represents part of a section of R and mR in 2'. But, since VM t is parallel to 07, the triangles V m KO and V m M t V are similar : we therefore have VM t _ FP OK 0V = (31). m In (31) the lengths are all regarded as posi- tive ; but I is VM' -=- - OK. Hence, V,M' VM t _\-m OK = OK ~ m ' Fio. 341. The Millerian symbol (hll) is therefore given by the ratios: h I I 2m +1 1 m 1 m'" (32); and, if the Millerian symbol is known, the Naumannian symbol mR is found from *-' ...(33). 382 RHOMBOHEDRAL SYSTEM, CLASS III. The same equations are found when OV m is taken to be less than V. 36. Equations (32) and (33) can also be obtained from the A.R. of the four poles [Cgrm}\ where C, Fig. 342, is the pole (1 1 1 ), (hll), r (100) and m {21 T}. For, in Fig. 338, OBV m is the inclination to the pinakoid of the face of mR through V m \, and is therefore the angle Cg. Hence tan Cg = tan OBV m = OV m + OB Again, D = A Cr = A OB V ; and tanZ> = tan0F=c^6. Dividing the former tangent by the latter, we have tan Cg -r tan D = m (34). But, from the A.R. {Cgrm} of the four poles given above, we have sin Cg sin nig sin Cr sin mr = tan Cg H- tan Cr = 111 m 100 2TT m_ 2TT 100 h-l since mg = 90 - Cg, and mr = 90 - Cr. The angles on the left sides of (34) and (35) being the same, the terms on the right must be equal. h-l Hence, and 2m +1 1 m These are the results given in (32) and (33). The special forms given in Art. 34 are immediately deduced from the above equations ; for, if m = 0, h = I, and the face is (III) or (111). If m = oo, then h +21 = 0; and the face is (21 1) or (211), and the form is the hexagonal prism {2 II}. 1 A different type has here been used to represent the pole (2ll) to avoid confusing it with the Naumannian index m. MILLERIAN SYMBOL FOR mR. 37. The Millerian symbol for the inverse rhombohedron -mR is found from equations (32) by changing the sign of m : it can also be obtained from the geometry of the figure as follows. Let Fig. 343 be part of a section in the plane 2 of the rhornbohedra R and mR; and let the polar face- diagonal V m y meet OX in G and the lower polar edge r M of R in G t . Then from the similar triangles V m OG and V m VG t , we have V t G. VV m mc-c m-l ~OG = ~6T~ '' '" me =: ~~mT ' '' Again, from the similar triangles OBG and MBG t , we have GM BM = 1 _ ~OG~~OB :. adding (36) and (37), V M m-l .(37). i 2m -I (38). OG m m Fig. 344 represents a section of the same rhombohedron in the plane 2', where V m M t is the polar edge of the face through V m y of Fig. 343 and meets OY at F on the negative side of the origin. From the similar triangles V m FO, V m MV, we have VM_, _ VV m _ mc + c OF ~ OV~~~~ me .(39). Hence, the intercepts on the axes are OG, OF, -OF; or 2m - 1 ' l+m' l+m ' If now the upper face parallel to that taken is denoted by (hi 1), then the face through V m y is intercepts it makes on the axes are in the ratios: VM VM Hence, Also f l-2m~ l+m and the ...(40). It is clear that these expressions differ from those for mR given 384 RHOMBOHEDRAL SYSTEM, CLASS III. in (32) and (33) only in the sign of m. Hence, expressions (32) and (33) may be taken to apply to both cases, provided care is taken to give the correct sign to m. Expression (33) serves also as a ready test for determining whether a rhombohedron {hll} is a direct or an inverse one. Thus, when the indices of {011} are introduced in (33), we have for m the value \\ similarly, {111} gives m = - 2 ; and {31 T} gives m = 4. It follows therefore that the simplest way of drawing any rhombohedron, the Millerian symbol of which is given, is to find m from equation (33). The apices V m and V m at distances me from the origin can then be marked off on the vertical axis ; and the drawing can be made in the manner described in Art. 31. 38. The symbols of the faces of the rhombohedron {100} are given in table d of class II ; those of {Ml}, which may be either a direct or inverse rhombohedron, are : Ml IM llh III IM Uh (j). These rhombohedra are geometrically common to classes II, III and IV. 39. Since the median edges of rhombohedra and scalenohedra (Art. 40) are inclined to the equatorial plane and cross it in zigzag fashion, such forms can be quickly drawn in the following simple manner. The eye is supposed to be situated in the equatorial plane, and the triad axis to be in the paper : any horizontal plane is then reduced to a straight line. Two straight lines VOV, and FOF t , Fig. 345, are first drawn at FIG. 346. right angles to one another. On FOF t any six equal lengths OS=ST= TF^ OS=ST = TF t , are marked off. Lengths T and V t , DRAWINGS OF RHOMBOHEDRAL FORMS. 385 equal to the distances of the rhombohedral apices from the origin are determined on the vertical line, and the length intercepted between the apices is then trisected in the points t and t r Through the points of trisection lines are drawn parallel to FOF t to inter- sect the verticals through T, F, &c., in the coigns /*, p. iit - From the similar triangles OH'V n , r\V H , we have T X T V n mnc me -=-3 3n 1 mnc OM 3n-l .(47). FIG. 349. The trace of the face in the equatorial plane may be taken to coincide with the line HK I L I of Fig. 332, p. 372. The distance OL t on OM." is then given by the fact that the triangle K t OL f is made up of K t OH and HOL t , of which the angles at are 120 and 60. Hence, OK t . OL f sin 120 - OK r OH sin 60 + Off. OL t sin 60 ; OM OM qi/_3n+l 3n-l 1 " OL~OII oK~~toT' ~%ar~n ...... *'' since OK t - OH'. 43. To find h, k, I in terms of m and n. The Millerian indices A, I, k, are given by h = V / M + OS, RHOMBOHEDRAL SYSTEM, CLASS III. 1= V t M' + OT it k= V,M"+OU; if, as before, VM is the para- meter, and OS, OT t and OU are the intercepts on the axes, due attention being paid to the directions in which they are measured. By a semi-revolution about OA /f the polar edge V n \T^ Fig. 348, is brought to the position V n \H' which meets OX in S'. The intercept OT t , measured in the nega- tive direction along OY t is equal to OS' measured on OX. Hence Since OX is parallel to V / M, we can, by pairs of similar triangles, find V t M+OS and V f M '-f- OS', and therefore the indices h and I Let, in Fig. 349, / be the point of intersection of V t M and V n \, and/' that of V t M and V n \. From the similar triangles OHS, MHf, we have FIG. 349. fM OS HM OM-OH OH~ \JJJO. \JJCL , j e \\ 7m = ( from ( 45 )) 3n+l n + 1 ...(49). And from the similar triangles OSV n , V t fV n , we have V ,f V, V n ~OS = OV n Adding (49) and (50), we find V M n+\ mn + 1 mn + mn ,(50). 3mn + m + 2 OS 2n mn 2mn Again, from the similar triangles H'f'M, H'OS', we have 3n - 1 n-l (51). fM H'M OM-OH' -08' = OH' = -OW- 2n and from the similar triangles S'O V n , f V t V n , VJ' _ V,Vn _ mnc-c _ mn-l OS' ~ W n ~ mnc '' ~' mn .-. adding (52) and (53), we have (53). VM n-l mn 1 mn Liz (54) . 2mn RELATIONS BETWEEN EQUIVALENT SYMBOLS. 391 Again, if, in Fig. 350, V*L, is the trace of (hlk) in the plane 2,,, and if it meets OZ and V t M" produced in U and G, then from the similar triangles UOL jt GM"L,, M"G _ M"L i _ OM" OU ~ OL t ~ OL t + ^. .<, x, = (from (48)) i + 1 (55). And from the similar triangles V n OU, V n V G, we have Fl - 35 - V.O VV r^+l (56) Subtracting (56) from (55), we have 0-L + l-=!i -- ............. (57). OU n mn 2mn The above equation gives the numerical value of the ratio V,M" + OU. Since, however, OU is measured in the negative direction, the index k = -M"V > +OU=2(l-m) + 2mn ............ (58). Hence from (51), (54) and (58), 44. Equations (59) suffice for finding both m and n when h, k and I are given. For, if we add or subtract the numerators, and also the denominators, of any of the ratios, we obtain a ratio equal to any one of them. Thus, taking the first and second terms, we have by addition - 9\ = 9/i -- \ = (doubling the terms of the latter and sub- tracting from the former) = (by addition) . h + l-2k (60) ' Again, subtracting the numerator and denominator of the second member of (59) from those of the first, we have h-l k .h+k+l e^ = 2(r^)^ fromabove >-T-' h-l 1 h-l .'. n = i i . = i ,-- - _ . h + k + lm h + l-2k 392 EHOMBOHEDRAL SYSTEM, CLASS III. Knowing then the indices h, k, I of one face of the scalenohedron, we can find m and n ; and can then draw the form by the method given in Art. 40. 45. To find the symbols of all the faces of {hkl}. The pair of faces passing through X\ of Fig. 351 change places after a semi-revolution about 08 t/ . This axis, being perpendicular to OZ, interchanges positive with equal negative lengths on it ; and positive lengths on OX with equal negative ones on OY, and vice versl Hence, the face F n A\ being (hlk), the face V n XX, is (Ihk). This is also obvious from the discussion in the two last article's and Fig. 348. The above two faces are associated with two parallel faces drawn through the median edge parallel to AA,. The symbols of the latter faces are, therefore, (hlk) and (Ihk). The four faces are necessarily tautozonal with the four rhombohedral faces meeting in the same median edges, and with the two prism-faces (HO) and (110) truncating these edges. Again, by rotations of 120 about the triad axis the above four faces are brought into the positions of two other sets of four similar faces. We have already seen that the triads of interchangeable faces have their symbols in the same cyclical order; hence the faces of the scalenohedrou {hkl}, Fig. 351, taken in order from V n \\ and F n XA , have the fol- lowing symbols : hlk hkl khl Ihk Ikh klh\ Ihk Ikh klh hlk hkl khl) (k). The pairs of faces in the columns \ are interchangeable by semi-revo- lutions about the dyad axes 8 //} 8 f and 8, respectively. It is easy to see that the above faces are symmetrically placed with respect to the planes of symmetry. Thus the pairs of faces meeting in the edges V n X, V n \ meet OX in S and S' respectively, where POLES OF THE AUXILIARY RHOMBOHEDRON. 393 OS ' t = V,M+l, for I is a negative number (Art. 43). But the plane 2 bisects the angle between OY and OZ, and the intercepts on these axes are reciprocal reflexions. The face V n \\ being (hlk), the symbol of V n \\ t is (hid) ; and the face 7 n \\ f being (Ihk) the symbol of F n AA /x is (Ikh). Similar proofs can be applied to the pairs which meet in polar edges V n \, &c. ; and also to pairs of faces which do not meet in edges, such as F B \A" (klh) and K*A'A w (kid). 46. The relations of the poles P of the scalenohedron (hkl\ = inRn and of g those of the inscribed auxiliary rhombohedron rnJR, are shown in Fig. 352. The two faces of each form which meet in the edge XX / must be in a zone with 0^(110), the prism-face per- pendicular to the dyad axis 08 f/ ; and this face would, if developed, truncate the edge. It is clear, however, that the two faces of the scalenohedron make a smaller angle with (HO) than the two faces of the rhombohedron g. Hence the pole P' lies between a ti and g. If P' is (hlk) and g in [a,/"] is (hi 1) ; then, by Weiss's zone -law, hk + Ik - l t (h + l) = 0. This equation is satisfied by making h f h k + 1, and l t = k. The inscribed rhombohedron mR has therefore the symbols {h k + 1, k, k}. But, in Art. 36, it was shown that, if mR is identical with {hl t l t \, then m = (h t I) -=- (h t + 2l f ). Introducing into this expression the values of h t and l t just found, we have __*-*_+_*-* _h-2k + l _0-3k '~h^k + T^2k ~ h + k + l ~ ' the same result as is given in (60). The number n can now be found from equations (59). 47. Although, in the method of derivation employed in Art. 40 and in all the succeeding Articles, the number m has been supposed to be positive, the process is perfectly general ; and all the relations hold true if m is negative and the inscribed rhombohedron is the inverse form, mR. All that is necessary is to make m negative in equations (59) and in all equations into which m enters. 394 EHOMBOHEDRAL SYSTEM, CLASS III. Such a scalenohedron may, when it is desired to indicate its precise position with respect to the axes of reference, be called an inverse scalenohedron, mRn. It is clear therefore that, for the same numerical values of m and n, we have two tauto- morphous scalenohedra, the posi- tions of which in space are deter- mined by the sign of m; and the poles of which are shown in Fig. 353. It is required to find the relation between the Millerian indices of the direct scaleno- hedron {hkl} = mRn and the cor- relative inverse scalenohedron {pqr} = In Arts. 43 and 44 we have seen that for the face (hlk), h I k _h+k+l 3mn~2l-m~ 6 Changing the sign of m, we have, for the opposite face (2^q) ...(63). m + 2 3mn m + 2 + 3mn 2(l + m) If from double the numerator and denominator of the last term of equations (62) we subtract three times the numerator and de- nominator of each of the preceding terms in turn, we have l)-3h 3 (2 - m - 3mn) ~~ 3 (2 - m + 3mn) - (64). U I X T '/ft/ I The denominators in the last three terms of (64) are in the same ratio as the denominators of equation (63) : hence the numerators must be in the same ratios. ... . P ? _r_ (65) 2 (h + k + 1) 3h 2 (h + k + 1) 3k 2(h + k + l) 3l These are the same relations as were found in Art. 15 to connect the symbols of two dirhombohedral faces. As in that Article, it is easy to show that 2 (p + q + r) - 3p FORMULA OF COMPUTATION. 395 48. As a general rule the forms will fall into zones in which the angles can be measured, and in which the symbols of several faces are known or can be easily found. Hence, the anharmonic ratio of four tautozonal faces will usually give the Millerian symbols of the unknown faces. When, however, the symbols cannot be thus obtained, we can use the elegant relations between the angles of the scalenohedron and the face-indices which we pro- ceed to determine in this and the two next articles. Let the angles PF, P'P", P'P ttt , over the dissimilar edges F W X, V*\ and A\ of Fig. 351 be denoted ,< by 2, 2rj and 2, respectively. Then, in Fig. 354, = w7 y =90 -a'P' = 90 -a,P; ^ =a> ll P' = a'P. Fm. 354. Hence, from the right-angled spherical triangles dPN, a'PN, a it PN we have and ............ (67). cos = cos a'P = cos a' NCOS NP Adding the first and second equations, we have sin + sin >/ = cos NP (cos dN + cos a ti N) = cos NP {cos (60 + a'N) + cos (60 - a'N}} = 2 cos NP cos 60 cos a'N = cos NP cos a'N ; since 2 cos 60 = 1. .*. sin + sin 17 = cos ..................... (68). This equation can be readily thrown into a form suitable for logarithmic computation so that, when two of the angles have been found, the third can be calculated. Thus cos = 2 sin |(| + 77) cos ( - r,),\ sin = 2 sin $ (90 - ij + Q sin | (90 -? - ),[ ...... (69). sin r, - 2 sin (90 - + ) sin | (90 - - ) j 49. In Art. 15 the indices e, f, g of N expressed in terms of those of P(hkl), are shown to be 3/t - 0, Zk-0, 31-0; and in 396 RHOMBOHEDRAL SYSTEM, CLASS III. Art. 27 an expression (20) for tan a'N was found in terms of the indices of N. Hence, introducing the values of e and f in terms of h, k and I, into equation (20), we have But from (67), sin cos dN cos(QO+a'N) cos cos a'N cos a N = cos 60 - sin 60 tan a'N = (1 - ^3 tan a'N). Hence, introducing the value of 3 tan a'N', we have sin_|_ cos~ Similarly, sin t\ _ cos a t N _ cos (60 - a'N) cos ~ cos a'N ~ cos a'N k- (71). h-k And, since A (7P = 90 - in (67) sin CP 2(h-l) h-l- we have from the last equation cos cos a'N = V( 1 + tan 2 oUV) = (by transformation from (70)) \/2 {(A - k) 2 + (k- O 2 + (I - A) 2 Hence from (71) (73), sin _ sin t) __ cos _ h^l~ (A - 1) V3 ^3 sin <7P ...(73). (74). h-k - ^2{(A-A:) 2 + (A;-Z) 2 + (l-h)*} When therefore the indices and one of the above angles are known, equations (74) enable us to compute each of the remaining angles. 50. To connect the angles involved in (74) with the angular element of the crystal we shall employ equations (12) of Art. 14. Thus, cos YP = cos <7Fcos CP (1 + tan C Y tan CP cos m'N) = cos CX cos CP(\ tan CX tan CP sin a'N) ; since CY= CX, and m'N = 90 + a'N. FORMULAE OF COMPUTATION. 39- Introducing this value of COR YP into equations (13), we have 1 - tan CX tan CP sin a'N = -^ . v But from (3), tan CX = 2 cot D, And from (70), it can be shown that e-3k smaN = - . Introducing into equation (75), we have (76). If now the value of sin CP, deduced from (76), is introduced into equations (74), the latter can be given in the following form: sin _ sing _ cos _ sin 60 sin D k~^l ~h-k~ h^l ~ J(h + k + I? - 3 (Kk + kl + ^) sin 2 D " ' The last term of (77) is not however in a form suitable for log- arithmic computation : it is better therefore to compute CP from (76), and then to substitute the value in equations (74). Another easy way of finding CP is first to calculate Cp, p being the pole of the face truncating the obtuse edge F"A. of Fig. 351. Hence, the arc Cp of Fig. 354 is the angle V n HO of Fig. 349 ; /. tan Cp = tan V n HO = OV + OH; and OV n _ OV n OM _ mnc 3n+l _ 3mn + m OH ~ OM* OH~ 2acos30 X ~~2rT 4 since c -=- a cos 30 = tan D. But from (62), 3wm + m + 2 = 2h (1 - m) -=- k = 6h 4- ; 3mn + m 3A-0 2h ~k-l .-f***--^-***- **i> (78). Expression (76) for tan CP is easily deduced from (78) by the aid of the right-angled spherical triangle pCP, and the known value of a'N ; for tan CP = tan Cp + cos (mN = 30 - a'N). 51. If the scalenohedron P {hkl} is the only form present in a crystal of which the element D is known, the equations given in Arts. 48 50 enable us to determine the indices, when two of the 398 RHOMBOHEDRAL SYSTEM, CLASS III. angles 2, 2rj and 2 have been measured ; or ; assuming the indices to be any numbers consistent with (74), to find the angular element D which corresponds to the measured angles and the assumed indices. Thus two of the angles being measured, the third is found from equa- tions (69). The angle PaN can then be computed by the formula for finding the angle of a triangle of which three sides are known : thus in the triangle a Pa, Fig. 354, the sides are: aP=90-, a'P= and a'a=60. Hence, Gp = 90 - pm = 90 - PaN is known; and equation (78) gives a simple equation between the indices. A second simple equation is given by (74). The solution of these two equations gives the ratios h -=- 1, k + l, and the symbol of the face. Example. Miller gives the element of calcite as 4436'6', and for the scalenohedron v, Fig. 355, common on Derbyshire crystals he gives 2=35 36', 2^=75 22'. Hence, from (69) cos f= 2 sin 27 44-5' cos 9 56-5' ; and f= 23 30-5'. The sides of the triangle a' Pa are: aP=90- = 7212', a'P=2331', aa'=60. tania'aP-tanim - / sin 5 39 ' 5 ' sip 17 51 ' 5 ' and, by computation, a'P=mi> = 22 4-8'. Hence, by equation (78) h 2 .^~^~* =cot224-8'cot 44 36-6' = (by computation) 5 -=-2; IO ' 85 ' 2 (n + K + 1) Again, by equations (74) 1=0 (80). Subtracting (80) from (79), we have 5fc=0, .-. k=0: and therefore h + 2l=Q. The face is therefore (20l) and the form {201}. If it be desired to determine the element from the measured angles, it is necessary to assume definite indices, (201) say, for the face P. Then after computation of mp, we have cot D = 5 tan 22 4-8' -^-2. But it is clear that equations (74) will not be satisfied if any three numbers are assumed for indices : an arbitrary choice can only be made of two of them, and the third is then deduced from (74). 52. Each scalenohedron {hkl} = mSn has five associated rhom- bohedra, the symbols of which can be readily obtained by the aid of the stereogram, Fig. 356, and Weiss's zone-law, or from the geo- metrical relations given in preceding Articles. A knowledge of THE FIVE ASSOCIATED RHOMBOHEDRA. 309 these rhorabohedra frequently simplifies the drawing of a com- bination such as that in Fig. 374, p. 408. They are : i. The inscribed rhombohedron g = mR, having the same median edges as the scalenohedron. ii. The rhombohedron u, the faces of which truncate the acute polar edges F, t X, &c., Fig. 351, and have their polar diagonals parallel to these edges : the apex V u is formed by drawing through B in Fig. 349 (OB = OM-r 2) a line parallel to V n \. Hence OV U OB The figure is then readily drawn; and its symbol in terms of h, k and I can be found from the relations already established. iii. The rhombohedron p with faces truncating the obtuse polar edges V n \. The apex V p is found by drawing through B a line parallel to V n \ of Fig. 349. Hence, iv and v. The two remaining rhombohedra (iv) y, and (v) q have for polar edges the acute and obtuse polar edges of the scalenohedron, respectively. Their apices are at double the dis- tance of those of u and p, being found by drawing lines through M parallel to the edges V n \ and V n \ of Fig. 349. From the stereogram the symbols of the poles are readily obtained by Weiss's zone-law. For the zone-circle [cr] is [Oil], and [cr'] is [10T]. The symbols of the zone-circles [a'P 1 ], [aP], [a lt P], and [a,,/* 7 ] are readily found by Chap, v, table 10. They are: [a'P'] = [7, h + k, I], [aP] = [k + l, h, h], Hence, by Chap, v, table (23), i. g the intersection of [cr] and [a"P] is(h-k + l, k, k) ; ii. u' [cr'] [a'P 1 ] (h + k, 21, h + k); iii. p [cr] [aP] (2h, k + l, k + l); iv. y [cr] [a,P] (h + k-l, 1,1); v. q [cr'] [aP] (h, k + l-h, h). 400 RHOMBOHEDRAL SYSTEM, CLASS III. The poles having the above symbols are on the upper hemisphere, for the sum of the indices is in all cases either 6 or 20, a known positive number. By the aid of equations (32) and (59) the student can prove that V p has the value already given. Thus from (32) 2h-k-l = [from (59)] 3m + 3mn 3n + 1 ~12 C= ^T" The distances from the origin of the apices V u , V y , V q can be obtained in a similar manner. As illustrations of the above we may take the scalenohedra {20T} and {410} of calcite. For the former, the five rhombohedra are : (\){\W} = R; (ii) {lTl} = -2#; (iii) {4TT} = f#; (iv) {3lT} = 47?; (v) {232} = -5R. For the latter, the rhombohedra are : (i) {311}=%R; (ii) {101} = -iJZ; (iii) {811} = T Vtf; (iv) {100} = *; (v) The hexagonal bipyramid. 53. When the pole P of a face (hkl) lies in the zone [CaJ\ = [121], Fig. 354, the form {hkl} undergoes an important modification, al- though the indices h, k, I are unequal and their sum differs from zero. By Weiss's zone-law we then have h-2k + l = Q ........................... (81); i.e. one of the indices is the arithmetic mean of the other two. But the zone [a'P~\ contains the pole g of the auxiliary rhombohedron mR. The three upper poles g must, when [a'P] coincides with ['(?], coalesce in C (HI), and the rhombohedron be- comes the pinakoid 07?. The form {hkl} is then a bipyramid having its median edges all horizontal and its polar edges all equal; and the angles 2 and 2r) over these latter edges are therefore also equal. The particular instance of a bipyramid {3 IT}, not unfrequent on crystals of sapphire, is shown in Fig. 357. The relation (81) between the face-indices of a bipyramid can also be obtained from (60), which gives the value of m for the inscribed rhombohedron mR. When the median edges of this rhombohedron Fi THE HEXAGONAL BIPYRAMID. 401 become horizontal m = 0, and h - 2k + I = 0. It is clear that the distance of the apex cannot be obtained in the manner employed in the case of a scalenohedron. The form may, however, be con- structed as follows. Lines are drawn from the six points M, M t , M", &c., to meet the triad axis at apices V*, V n , where OV n - OV n = nc. We proceed to determine the relation between the pyramid- index n and the indices A, k and I of one of the faces. Let Fig. 358 be part of a section of the bipyramid by the plane of symmetry 2 containing the polar edges V n M, V n M, the axis OX and the parallel edge V M of the fundamental rhombohedron ; and let the polar edges meet OX in S and S'. The faces meeting in V n M are hkl and hlk those meeting in V n M are Ikh, Ihk. Then, from the similar triangles V t MV n , OSV n , we have VM VV* c + c = n+l from the similar triangles OS'V n , V f MV n , we have V t M = V\V n = nc-c = n-l As was the case with the scalenohedral face F n XX / having the symbol (hlk) (Art. 43), the inter- cept OT t on OF, of the face through MM t is equal to OS' but measured in the negative direction. Since the indices h, k, I, are whole numbers given in their lowest terms, and V t M is a definite FIG. 358. length selected arbitrarily as parameter, the plane given by V f M+h, V f M+l, V t M + k does not necessarily pass through MM t : it may be only parallel to the line. An arbitrary factor/ is therefore introduced when the ratios in (82) and (83) are expressed in terms of h and I. Hence, and from (81), Hence, n is inversely as that index which is the arithmetic mean of the other two. When the symbol (hlk) is given, equations (84) suffice to deter- mine n and/. Thus, for the face (201) of the pyramid {210} we have L. c. 26 402 RHOMBOHEDRAL SYSTEM, CLASS III. from the second equation n = 1 ; and then from the first equation / is also 1 : this result is also consistent with the third equation. Similarly, for {3Tl}, Fig. 357, we have from the third equation nf= I ; and from the first we then have 3 = n + 1. .'. n = 2. Equations (84) can be put in the following form, n+l l-n 1 2 nr :: ~ = * = (byaddltlon) ro = (by subtraction) - (85). fl 6 Hence, from the third and fourth terms, h- 2k + 1 = 0, the result given in (81); and from the third and fifth terms, n = k ~ k l (86). The index n, giving the length nc on the principal axis, is therefore known in terms of the Millerian indices ; and the pyramid is drawn by joining the apices V n and V n (0 V n = V n = nc) to the points M, M n &c., projected in the horizontal plane D'CA' of Fig. 51, or YOX t of Fig. 60. The edge MM t is bisected at right angles by 08 ti at a distance 08 /x (say) = OM cos 30 = 2a cos 2 30 = 3a + 2. It also meets the adjacent dyad axes 08 and 08' at distances OA = OA' = 08 /7 - cos 60 = 208" = 20M cos 30 = 3a. Naumann therefore denoted the form by the symbol nP2 ; indi- cating that the apex is at distance nc from the origin, and that each median edge meets the dyad axes inclined to it at 30 at double the distance at which the edge meets the axis perpendicular to it. His value for n is however two-thirds that adopted in this work; for he drew the faces through the points A, A', &c., on the dyad axes each at distance a from the origin. 54. We can now find the relation of the angle 2 over a median edge MM /t to 2 over a polar edge V n M, and that of these to the pyramid-index n and the indices h,k,l. For, 8' being the point in which the edge MM tl meets the dyad axis, then 08' F" is one-half the face-angle over MM it = arc CP, if P is the pole (hkl). Hence, cot = tan CP = tan OXV*=OV*+ 08' = 2nc 4- 3a. But, from (5), c = a cos 30 tan D ; .'. tan CP= 2n cos 30 tan D -f- 3 = n tan D -5- ^/3 AzltanZ)....(87). EXAMPLES. 403 If the angle measured is PP' = 2 of Art. 48, then, since = r), we have from equation (68) cos =2 sin ......................... (88). Since = 90 - CP, a relation can from (87) and (88) be found between and the indices. It is, however, simpler to compute from (88), and afterwards to introduce the value of CP in (87). 55. The hexagonal bipyramid is the special form in which the direct scalenohedron coalesces with the inverse form. For, if = 2k, then 6 = 3k, and 2k-h = l, &c. Hence jt? = 20-3h = (89). The opposite face (pqr) is therefore (Ikh), which is one of the faces of the pyramid {hkl}. 56. Crystals of the following substances are placed in this class : Substance. Composition. D. Common forms. Arsenic As 58 17' r. Antimony Sb 56 48 rcu{211}. Bismuth Bi 56 24 r,cf{IU},rcf. Tellurium Te 56 55 mr. Selenium Se 56 53 (nearly) mr. Graphite C 58(1) Ice H 2 58 18? (Nordenskiold) 35 10? (Kenngott) 1 Corundum A1A 57 34 ca, cr, era, n {311}, &c. Hematite Fe 2 3 57 37 cr, era, rnu {211}, &c. Sodium nitrate NaNO, 43 42 r. /-Calcite CaC0 3 44 36-6 e, cm, v {210}, and nu- merous other forms and combinations. Siderite (Chalybite) FeC0 3 43 23 r, e{011}, crvs {322} (Fig. 369). Rhodochrosite MnC0 3 43 23 r. Calamine ZnC0 3 42 57-3 r. The letters denoting forms in the above table will also be used to denote the same forms in other crystals of the system. Corundum includes the gems, ruby and sapphire, and the semi-trans- 1 Kenngott's (111) may possibly be Nordenskiold's (100) ; but on this as- sumption the latter's element D should be 54 38'. 262 404 RHOMBOHEDRAL SYSTEM, CLASS III. lucent dull-coloured variety of alumina more especially known as corundum. Ruby consists of the red crystals: in them the basal pinakoid c{lll} is largely developed ; and the common habit is shown in Figs. 359 62. In the blue and the colourless crystals of sapphire, pyramids n {3lT}, z {715}, FIG. 359. FIG. 360. a> {15, 11, 3}, &c., are the predominant forms. Simple pyramids, like Fig. 357, are sometimes found; but more commonly several pyramids occur together in association with the rhombohedron {100}, the basal FIG. SGI. FIG. 362. pinakoid and sometimes the prism {101}. Fig. 363 shows a combination of c, r, n, z and o>. Fig. 364 represents a crystal of sapphire from Emerald FIG. 363. FIG. 364. Bar, Montana, described by Dr. Pratt (Am. Jour, of Sci. [iv], iv, p. 417, 1897): the habit closely resembles that of hematite crystals from Elba. The rhombohedron x shown in Fig. 360 is {811}, with a computed angle e.#=67 2'5'. Figs. 361 and 362 are plans of the base showing the natural corrosion-figures of the face c: these are symmetrical to a triad axis and three planes of symmetry intersecting in it. EXAMPLES (CALCITE). 405 Hematite. Fig. 365 represents the habit of crystals from Elba. The faces u are generally much rounded, and accurate measurements of the angles between them and other faces are seldom possible: for u {211}, the angle ?i' = 372', and Ar=366'. The crystals from St Gotthard are often in rosettes consisting of plates in nearly parallel positions : the forms being c{lll}, a {101} and m{2ll}. Pio. 365. FIG. 366. 57. Calcite affords numerous instances of simple forms of different kinds, and an almost endless number of combinations in which a greater or less number of forms of different types enter. Thus we have e{110}= -\R, Fig. 366, in obtuse rhombohedra; /{Tll}= -2fl, Fig. 370, in acute rhombohedra. The fundamental rhombohedron r{lOO}=R is rare as a separate form, but is common in the allied minerals siderite, rhodochrosite, &c. ; and it is fairly common in combinations : it is con- spicuous by the perfect cleavage parallel to its faces. The scalenohedron v{2lO} = R3 is abundant in Derbyshire either as simple, or as twinned, crystals. It is one of the commonest forms in combinations: instances of its association with e{110}, and with e and m{211}, are shown in Figs. 367 and 368. Pyramids are rare as simple forms ; and they are far from common in combinations. The prism m{2n} and pinakoid enter fairly frequently into combinations. The prism a{10T} and dihexagonal prisms are in- frequent. A combination of m and c frequently found in the Harz Mts. is shown in Fig. 328 : it is geometrically the same whatever may be the substance. A combination of m and e is shown in Figs. 371 and 372. The above-mentioned figures have been drawn as follows. The clinographic cubic axes having been projected with the numbers n = s = 3 (Chap, vi, Art. 22); the points M, M t , M", &c., are determined in the way described in Chap, vi, Art. 19, placing 05,, in the axis OX of Fig. 60. Lengths OF=OF, = -8543 x OA" (see Art. 31) are then cut off on the vertical axis; the points of bisection and trisection of OF and OF,, and also the points at distances 2OF and 3OF are then marked. The fundamental rhombohedron is drawn in the manner de- scribed in Art. 31, and the median coigns /t are found by trisecting F,A/, &c. The rhombohedra e and / can be drawn in a similar manner, using for apices points at distances OF-7-2, and 20 F, respectively. 406 EHOMBOHEDRAL SYSTEM, CLASS III. The scalenohedron {201} is now drawn by joining the apices F 3 and F 3 at distances 30V to the coigns n in the way already described in Art. 40. To introduce the faces of {110} in Fig. 367, points c and c, are taken on the triad axis equally distant from the origin, and lines (not shown in the figure) are drawn through them parallel to the polar edges F/i", Vp', &c. of {100} : they meet the acute polar edges FV'> V 3 jj.', &c., in the points a", a' &c. These lines of construction are parallel to the diagonals of the faces (110), (101), &c.; for each face e truncates an edge of the rhombohedron r. From the same points c and c, edges are now drawn parallel to the lines joining the points of bisection of OF and OF, to the points M, M', &c. : they give the edges en, c,n,, &c., which meet the obtuse polar edges of the scalenohedron in n, n t , &c. The adjacent points a and n are then joined and the figure is completed. FIG. 368. FIG. 369. The faces of m {211} shown in Fig. 368 can be inserted on the median coigns of the preceding figure as follows. The lines 55,,, 5 t/ d', &c., joining the points of bisection of the median edges of {201} are bisected in points B at distances Oi/-f-2 from the origin. Through each point B a vertical line is drawn to meet the polar edges in the same plane 2 at points ft and y, respectively. The FIG. 371. EXAMPLES (CALCITE). 407 lines joining /9 and y to the adjacent points 5 give the edges in which the faces ?n and v intersect. The crystals are not often developed with the edges [mu] meeting at the points 5. Sometimes the faces in are much smaller, and often they are much larger. Having obtained the directions of the edges [mv], there is no difficulty in either diminishing or increasing the relative size of the faces m. Fig. 366 can be drawn, in the way described in Art. 31, by joining apices at the distance OF-r-2 to the points H, M', &c.; or by the following method. The polar edges of U={100} are all prolonged to double their length, and each of the points so obtained is joined to the apex opposite to that through which the respective lengthened line passes. That this construction is correct follows from the fact that each face of {110} truncates the edge of the fundamental rhombohedron ; the polar edges of the latter must therefore coincide in direction with the polar face-diagonals of the former, and the coigns of the two forms on the same face-diagonal lie on the horizontal lines which pass through t and t, , the points of trisection of FF,. A similar relation connects the rhombohedron /{111} and {100}, and is illustrated by Fig. 370. Each face of {100} truncates a polar edge of {111}. Hence each polar edge of {111} is one-half the corresponding polar face-diagonal of { 100 } . It is simpler however to remove the lo wer apex of { II 1 } to F 2 , doubling the size of the rhombohedron ._ The face-diagonals F/*, Fyu,, F^ of {100} give the upper polar edges of {111} ; the figure is then easily completed. All rhombohedra connected together by the relation that the one truncates the polar edges of the other, or has its polar edges truncated, can be drawn in a manner similar to that described for e and /, when one of the series is known. Thus the diagonals FV, F\", Ac. of {111}, Fig. 370, are the polar edges of {311}; and so on. In Fig. 371, e {110} is first drawn; the faces m are then introduced in the way described for obtaining the points /3 and y in Fig. 368. Thus in Fig. 371 the point g is on the vertical line bisecting the line S'5,,, and the lines gS', gS lt are the edges [me"], [me']. The edge [me] is horizontal and parallel to &'d, t . Fig. 372 gives the same combination of m and e as Fig. 371, but the triad axis has been displaced from the vertical to show the upper faces e more distinctly ; the figure representing one of the crystals on a specimen in the Cambridge Museum. Three grooves (represented by the short, black lines) filled with a dark earthy matter lie in the face-diagonals of {110} of each crystal. The triad axis in Fig. 372 was taken to coincide in direction with the cube-diagonal Op of Fig. 226, and the dyad axes of calcite coincide with 05"', OS 4 and 05 5 of that figure. But the lines Op and OS'" of Fig. 226 are to one another as ^/3 : >/2. If then 05'" is taken as the unit Fl - 372 - length, that on Op is Op^/2-^-^/3; and this length must be multiplied by cos 30 tan D, or -8543, to give c for calcite. A length = Op x 8543-j- N /6 has therefore been taken to give the apex of e {110} at distance c-r-2 from the origin. The points 5'", 5 4 , 5 5 are then the same points as those marked 5, &c., in Fig. 371. The figure is now completed in a manner similar to that employed for drawing Fig. 371. 408 KHOMBOHEDRAL SYSTEM, CLASS IV. The combination shown in Fig. 374 is common in crystals from Iceland. The forms are: r {100}, v {201}, w {410}. From the stereogram, Fig. 373, it is seen that {311} is the auxiliary rhombohedron of {410}, and that {100} is the auxiliary rhombohedron of {20l}. We therefore first construct {100}, its FIG. 373. median edges are also those of {201}, and the polar edges of this latter form are got by joining the coigns /j. to V s and V 3 at distances 30V from the origin. Again, it was shown in Art. 52 that {100} is the rhombohedron iy associated with {410} ; i.e.. the polar edges of {100} are parallel to the acute polar edges of {410}, and we may draw the faces of the latter through the edges V^, Vfi", &c. By (60) and (61), the apex of {410} is at a, where Oa=40F-=-5; and the coign X is in the vertical through /u at a distance c-=-5 nearer to the equatorial plane. For, by (33), the apex of {311} is at distance 2c-^5, whilst that of { 100} is at distance c: also \w = rt = Ot - Or. But (Arts. 31 and 33) O = c-=-3, and Or = 2c-f-15. .'. Vt = c-j-3-2c-:-15 = c-7-5. We therefore draw through V an edge [401, 410], parallel to a\, to meet the obtuse polar edge [210, 201] in the point r), the positions of which and of its homologue ??, are indicated by crosses. The lines joining /*,, /x" to i\ are the edges [401, 210], [410, 201]. The re- maining edges of the two forms are obtained in a similar manner. The rhombic faeces of {100} are now introduced; the coigns nearest to the apices V and V, being found by cutting off equal lengths on the polar edges F??, F,ij,, &c. IV. Trapezohedral class; a {hkl}. 58. In this class the triad axis Op is associated with three like and interchangeable dyad axes making 90 with it and 120 with one another; and there is no other element of symmetry. The above axes resemble those associated together in the preceding class, but the opposite ends of each dyad axis are no longer similar ; for the crystals of this class cannot be centro-symmetrical without intro- ducing a plane of symmetry perpendicular to each dyad axis. These AXES OF REFERENCE AND PARAMETERS. 409 axes are therefore uniterminal ; and they are pyro-electric axes : the opposite ends are in Figs. 377 380 indicated by letters 8 and d. 59. Any face inclined to the triad axis at a finite angle, other than 90, must be repeated in two other similar faces which meet at an apex on the axis. If, moreover, the face is parallel to one of the dyad axes, a parallel face must also be present; for a semi-revolution about a dyad axis brings the face into a position parallel to its original one. The same must be true of each face of the triad meeting at an apex on the principal axis. The form must therefore be a rhombohedron similar geometrically to those described in the two preceding classes. The method of construction has been fully given in Arts. 31 and 39. Again, the plane containing the three dyad axes is parallel to a pair of possible faces; for the dyad axes are possible zone-axes. Hence we obtain a second special form a pinakoid similar geometrically to that of the two preceding classes. We can now select as axes of reference lines parallel to the three polar edges of any rhombohedron possible on the crystal, and for parametral plane (111) a face of the pinakoid. The parameters are therefore equal ; for the axes are three similar lines equally inclined to the triad axis and interchangeable by rotations of 120 about this axis, whilst the parametral plane retains the same direction after each rotation. Hence all the analytical relations established in the preceding sections hold also for crystals of this class. The linear element c and the angular element Z) = 111A100 are deter- mined from the rhombohedron selected to give the axes of reference. 60. Crystals of this class have the following special forms : 1. The pinakoid {111}, having the faces 111, ITT. 2. Rhombohedra mR = {Ml\, and - mR = (hll^ : these are geometrically identical with those discussed in Arts. 31 39, and their geometrical relations are given in those Articles. 3. A trigonal prism a {Oil}, Fig. 375, each face of which is perpendicular to one of the dyad axes, and parallel therefore to one of the axes of reference. It is geometrically the same as r{OTl} of class I, Art. 9 ; and consists of the faces: a (Oil), a' (10T), a" (TlO). The complementary prism a {Oil} Fio. 375. has its faces parallel to those of a {Oil}. 410 RHOMBOHEDRAL SYSTEM, CLASS IV. Jf. A hexagonal prism {211}, the faces of which are parallel in pairs to one or other of the dyad axes : it includes the faces given in (g) of Art. 27. The face-symbols can be found in any of the ways given in Arts. 11 and 27. 5. The ditrigonal prism, a {hkl}, where h + k + I = 0. The prism a {hkl} consists of the faces : hkl, Ikh, Ihk, hlk, klh, khl (m). The alternate faces have their symbols in the same cyclical order, and are at 1 20 to one another. There is therefore but one independent angle, which may be taken to be hkl A Ikh : it is inde- pendent of the crystal-element, and can be computed from expression (20) of Art. 27, SiL^-:^..-^^"-!?!^ for it is double a'N of that equation. Fig. 376 represents the particular instance a {321}, in which 32T A 231 = 38 13'. The complementary ditrigonal prism p IG gyg a {hlk} has its faces parallel to those of a {hkl} ; and the two are geometrically tautomorphous, for the faces of the former can be brought into the position of those of the latter by a rotation of 180 about the triad axis : dissimilar ends of the dyad axes are however interchanged by such a rotation. 6. The trigonal bipyramid, a {hkl}, where h - 2k + I = 0, consists of six equal and interchangeable faces, each of which is an isosceles triangle. The faces of this form, of the ditrigonal prism and of the general form (Art. 61) have the general symbols given in tables m and n : in the special forms the indices of each of the faces are subject to the conditions given with the form-symbol. Geometri- cally the trigonal bipyramid may be supposed to consist of the alternate pairs of faces of the hexagonal bipyramid (Arts. 53 55) which are interchanged by rotations of 120 about the triad axis. The faces of the latter form were drawn through apices V n f V n and horizontal lines MM /} &c., where M, M t , &c., are the auxiliary points in the equatorial plane at distances 2a cos 30 from the origin. In the trigonal bipyramid we shall suppose three of these same horizontal edges to be retained : hence each of them is bisected by a dyad axis at a point d, where Od = OMcos 30 = 3a -=- 2 ; and has its extremities on the adjacent dyad axes at points 8, where 08= Od+cos 60 = 3a. The distance nc of the apex V n is given by equation (86) ; viz. n (h l)^- 2k. THE TRAPEZOHEDRON. 411 Particular instances, having the same symbols as trigonal bi- pyramids observed on crystals of quartz in association with a hexagonal prism and the rhombohedra {100}, {T22}, are represented by Figs. 377 and 378 the parameter c in these figures is not however that of quartz. The first is a {4 12} and consists of the faces : 412, 241, 124, 214, 421, 142. By (86) the value of n is 3. The figure is therefore constructed by joining apices V 3 , V 3 at distances 3c to points 8 on the similar extremities of the dyad axes at distances 3a from the origin. Fig. 378 represents the comple- mentary form a {421}, in which the median coigns are on the opposite side of the origin from those of the first. Geometrically the two forms are tautomorphous, but by a semi-revolution about the principal axis, opposite and dissimilar ends of the dyad axes are interchanged. 61. The trapezofiedron, a. {hkl}. When each of the faces meets the axes of reference at finite distances, and all the dyad axes at unequal distances from the origin, we have a six-faced form known as the trapezohedron a {hkl}, Fig. 379 ; for each face is bounded by four edges, which do not form a parallelogram. The face (hkl) must, owing to the triad axis, be accompanied by the faces (Ihk), (klh) having their symbols in cyclical order, and all meeting at an apex V n on the triad axis. The face (hkt) meets the adjacent dyad axes at points 8' and d if unequally distant from the origin and lying on parts of these axes which are not interchangeable. Since OB' is perpendicular to the axis of reference YY f and is equally inclined to XX , and ZZ^ a semi-revolution about 412 RHOMBOHEDRAL SYSTEM, CLASS IV. OB' interchanges positive with equal negative lengths on OF; and exchanges a positive length on OX with a negative one on OZ, and vice versa : hence the face is brought into a position in which it meets the triad axis at V n and has the symbol (Ikh). Similarly, Od tt is perpendicular to ZZ t ; and a semi-revolution about Od it brings (hkl) into a position given by (khl). The third face at V n is (hlk) ; the three faces meeting at V n being necessarily in cyclical_order. No other faces are introduced by a semi-revolution about dS; for this axis is perpendicular to XX t , and interchanges equal positive and negative lengths on it : hence the face (khl) is brought into a position given by (klh), and (Ikh) into that given by (Ihk). But these two faces are two of those meeting at V n . Hence, the tra- pezohedron a {hkl}, Fig. 379, consists of the six faces : hkl, Ihk, klh, Ikh, hlk, khl (n). The rule for the deduction of the symbols of the several faces from that of any one of them is that the three faces meeting at one of the apices have their indices in cyclical order ; the sign of each index, whether it be positive or negative, remaining the same to whichever axis of reference it may be transferred : at the oppo- site apex the cyclical order and also the signs of all the indices are changed. The positions of the poles of particular trapezohedra a. {421} and a {412} are shown in Figs. 384 and 385. The geometry of this form is most easily understood by con- structing it from one-half of the faces of the scalenohedron de- scribed in Art. 40 ; pairs of faces being drawn through each of the alternate median edges of the inscribed rhombohedron mR. But since there are no planes of symmetry in crystals of this class, the similar faces through adjacent median edges do not occur together. DRAWING THE TRAPEZOHEDRON. 413 We can therefore obtain two complementary trapezohedra a {hkl}, Fig. 379, and a {hlk}, Fig. 380, from the same scalenohedron. The two complementary forms are enantiomorphous, for the one is the reflexion of the other in any of the vertical planes containing an axis of reference. No crystal showing only a general form has yet been observed, but forms of the kind are common on crystals of quartz in combination with the rhombohedra {100}, {122} and the prism {211}. The method of drawing the form has therefore no great interest ; but when it is needed, the rhombohedron mR, where >n = (h-2k + l) + (h+k + l) (Art. 44) should be first constructed. The median edges of this rhombohedron are also those of the scalenohedron {hkl}. To obtain a {AH}, the apices V n and V n and the median edges through 8', 8" and 8 are retained : these edges have now to be extended to meet new polar edges in which the alternate scalenohedral faces at each apex meet. Thus the edge AX,,, in which the faces (hkl) and (Tkh) of the scalenohedron meet, is extended to meet the faces (Ihk) and (khl) at the points y lt and y, respectively. The extensions in the two directions are clearly equal ; for the edge is perpendicular to the dyad axis 08', and the two extremities are interchanged by a semi- revolution about the axis. The coign y is found by taking (see Chap, xix) The first expression for the factor of S'X is similar to that given in (61) for l-r, but the indices k and I have changed places ; the others give for the extension Xy = X,,y, ; = 2^8'X = 2^d'A... ....(91). 1 " Y " h-k sin 17 In these expressions the indices h, k and I are taken in descending order of magnitude, 2 is the angle over the obtuse polar edge and 2rj that over the acute polar edge of the scalenohedron {hkl}. Thus for a {712}, to which Fig. 379 corresponds, m = l-r-2, n = 3 and 8y = 28'X. The points y having been marked by proportional compasses on the alternate median edges, each of them is joined to the nearest apex V n or F n ; and the adjacent pairs of coigns y are joined to form the second set of median edges intersecting the dyad axes at the points d. The polar edges can also be found by the rule for finding a zone-axis (Chap, v, Art. 4). Thus V*y u is parallel to [hkl, lhk] = [k*-lh, P-hk, h*-kl\; and y,, is the intersection of a line through V n , parallel to this zone-axis, with the median edge XX . 62. Quartz, Si0 2 . Occasionally the crystals appear as fairly regular bipyramids, which may be represented by Fig. 381, consisting of the 414 RHOMBOHEDRAL SYSTEM, CLASS IV. two forms r{100j and 2 {122}. The faces (100) and (122) are connected by the relations given in Art. 15 and belong to dirhombohedral forms. Very rarely the faces r predominate to such an extent that those of z appear only as slight modifications on six coigns of the rhombohedron. Since ?r" = 8546', such a crystal looks much like a cube. Knowing the angle rr" we find./) by equa- tion (1) to be 51 47', when c = 1-0999. More commonly the crystals are hexagonal prisms {2TI} terminated by the dirhombohedral forms r{lQO} and z {122} : the faces of the prism are usually striated in a direction perpen- dicular to the triad axis ; and those of the rhombohedron r are generally more largely developed than those of z. The faces of the two rhombohedra and of the prism are, however, often very unequally developed ; and it is occasionally hard to make out the symmetry. Faces of a trigonal bi- 12 21 EXAMPLES (QUARTZ). 415 pyramid s= a {4 12} or *, = a{421}, and of a trapezohedron #=a{412} or #, = a{421} often occur as small modifications of the coigns of crystals of the bipyramidal habit ; but, owing to the remarkable tendency to twinning shown by quartz-crystals (see Chap, xvni), the homologous coigns are rarely all similarly modified. By the examination of a considerable number of crystals, it has been established that they are of two kinds. In the one, represented by Fig. 382 and the corresponding stereogram Fig. 384, the tautozonal faces /, * x, and m, and their poles, follow one another in this order along the spiral of a left-handed screw : such crystals will be called laevogyral or left-handed, and rotate the plane of polarization of light transmitted along the triad axis to the left ; i.e. to an observer receiving the light, the rotation is counter-clockwise. In Figs. 383 and 385 a similar combination of forms is represented, in which the faces z", s, x, m follow one another along the spiral of a right-handed screw : these crystals are dextrogyral or right-handed, and rotate the plane of polarization to the right ; i.e. to an observer receiving the light, the rota- tion is clockwise. The rule can also be given as follows. The prism being vertical, and a face r of the direct rhombohedron {100} facing the observer, then the faces s and x in the zone [z"sxm\ are to the right of the observer in a dextrogyral crystal ; and the faces * x, in the similar zone are to the left in a laevogyral crystal. The faces s of the trigonal bipyramid are parallelograms, and in a dextrogyral crystal lie each in two slanting zones ; for instance, the face s (412) lies in the zones [w,,r] = [021] and [z"m] = [102], and the form is a {4 12}: the faces are sometimes striated parallel to the edge [*r]. In a laevogyral crystal the faces s, lie each in two zones, such as [rro / ] = [012] and [Ym] = [120]: the face s, is (421), and the form is a {421}. The faces are occasionally striated parallel to [rs y ]. Measurement of the angles in the zone \mxsz"] gives : mx = 12 1', xs = 2557', tz" = 28 54', z"r' = 46 16' and r's (241) = 28 54'. The angles in the corresponding zone [m:r ; *,z'] of Fig. 382 are the- same. Knowing the symbols of m, z" and r', those of x and * can be found by the A.R. of four poles. Thus, taking .x to be (hkl), the A.B. {r'z"xm} gives sin r 1 ! sin mx sin r'z" ~ sin mz" h + 2k~l-k' according as the first and second columns are combined, or the second and third. L sin (r'z = 101 7') = 9-99177 L sin (rV = 46 16') = 9 85888 L sin (u"= 66 52') = 9-96360 L sin (mx = 12 l')=9-31847 19-95537 19-17735 19-17735 77802= log 5-998. .-. 3ft = 3/ =6. Hence, h= -4k, and l=2k. h + 2/c I k 416 RHOMBOHEDRAL SYSTEM, CLASS IV. The face x is therefore (4l2), and the form is a {412}. The symbols of the faces and poles to the front of the paper are inscribed in the diagrams. The dyad axes, being uniterminal, are pyro-electric axes ; and in crystals, such as those represented in Figs. 382 and 383, the analogous poles are on the alternate prism-edges where the faces s and x appear, the antilogous poles on those prism-edges where no such faces occur. Optical characters. With a few exceptions mentioned in Chap, xvm, plates of all quartz crystals, cut perpendicularly to the principal axis, rotate the plane of polarization of a beam of plane-polarised light trans- mitted along the axis. The amount of rotation of the plane of polarization depends on the thickness of the plate and on the colour of the light, and is the same for light of one colour in plates of equal thickness but opposite directions of rotation. Biot found that the angle of rotation (f> is given by the approximate expression, = + kt 4- X 2 ; where t is the thickness of the plate, X the wave-length of the light, and $ is taken to be positive or negative according as the plate is dextro- or Isevo-gyral. This expression for < is the first term of a series in which higher powers of 1 ~\ z enter. For plates 1 mm. thick the principal rotations are given in the following table (Stephan, Pogg. Ann. cxxii, p. 631, 1864). Fraunhofer's line. X. , the apex V n is infinitely distant ; and the pyramid becomes the ditrigonal prism p. {hkl}, where, by equation (60), h + k + I = 0. When OH=OK t , = oo: if, however, OV n is to remain finite, then m = 0. The two equations are satisfied by making in (60) and (Ql)h-2k + l- the expression already obtained for the hexagonal pyramids. 66. Crystals of sodium lithium sulphate, NaLiS0 4 ; of spangolite, (AlCl)S0 4 .6Cu(OH) 2 + 3H 5! 0; of greenockite, CdS ; of proustite, Ag 3 AsS 3 ; of pyrargyrite, Ag 3 SbS 3 ; and of tourmaline belong to this class. Tourmaline. Jannasch and Calb (Ber. Ch. Ges. xxn, p. 216, 1889) gave the formula (Mg,Fe,Li 2 ,H 2 ...) 3 A10. BO(Si0 4 ) 2 or R 9 I B0 2 (Si0 4 ) 2 , R 1 including the elements given in the first formula. Messrs. Penfield and Foote (Am. Jour, of Set. [iv], vn, p. 97, 1899) have shown that all tour- malines may be derived from 11206281402!, and have proposed as a special formula H 9 A1 3 (B. OH) 2 Si 4 19 , in which the nine hydrogen-atoms are replaced by Fe, Ca, Mg, Li, &c. The crystals often manifest the development of forms of this class in a striking manner. The trigonal prism /*{2ll} is sometimes the predominant form, but it is generally associated with the hexagonal EXAMPLES (TOURMALINE). 421 prism {Oil}. A section perpendicular to the triad axis of such a com- bination will either be a triangle having its angles bevelled, or else a hexagon having alternate angles truncated, according as the faces of /i{2ll} are large or small. Occasionally as shown in Figs. 389 (a) and (6) both the complementary prisms /i{2ll) and ^{211} represented by the letters m are present. The crystals are usually terminated by acleistous trigonal pyramids, which may or may not be complementary. Occasionally one end shows a pedion. Figs. 389 (a) and (b) represent the opposite ends of a small, brown crystal from Ceylon in the Cambridge Museum. At one (a) FIGS. 389. (b) end the trigonal pyramids r=/i{100}, =^{T22} (a rare form), * = /x{Tll}, K=/*{322} are associated with the pedion c= / *{lll}, one face of e = /x{110} and a single face of a ditrigonal pyramid u (302) not shown in the figure : this end is the antilogous pole. At the other end, the faces of the com- plementary pyramid r=/i{100} are very large, and those of e=^{IIO} are very narrow : this end is the analogous pole. The prisms are very short, a {Oil} being predominant and the two complementary prisms p {211} and /i{21l} being unequal. The crystal is markedly dichroic ; being more translucent across the crystal from a, to a' than from one end to the opposite ; although the thickness in the former direction is much the greater. Fig. 390 represents a crystal of common habit ; the trigonal prism m=n{%ll} is largely developed, and its edges are bevelled by the hexagonal prism a {Oil}. At one end the pyramid p {100} is shown ; and at the other end the complementary pyramid /*{TOO}. But though these forms, if they oc- curred alone, would compose a rhombohedron similar to that of classes II IV, the appearance of the two ends is different. At the upper end the faces r(100) and m(2ll) meet in a hori- zontal edge, and the same is true of the two other pairs of faces r and m : at the other end each face, such as r,, meets two faces m and m" at equal angles and the edges are not horizontal. FIG. 390. 422 RHOMBOHEDRAL SYSTEM, CLASS VI. In the former case the face r may be said to stand on the prism-face m ; in the latter the face r, may be said to stand on the prism-edge [mm"]. The rule connecting the electrification may generally be given thus : the analogous pole is at that end at which the faces r of /n{100} stand on the faces of the trigonal prism; the antilogous pole is at that end where the faces of p {100} stand on the edges of the trigonal prism. The crystals are optically negative ; the principal indices of refraction for sodium-light vary with the colour of the crystal, and sometimes even in its successive layers. Dark coloured varieties are strongly pleochroic, absorbing the ordinary beam to a far greater extent than the extraordinary. Hence the ordinary beam is practically extinguished by a thin plate of a dark coloured crystal cut parallel to the optic axis : and such plates of one or two mm. thickness are often employed for producing beams of plane- polarised light. In the zone [TCZSK] of Fig. 389 the following angles were measured : re = 27 30', 02=27 35', cs=46 6', c/c=68 45', cm =89 58J, mr=52 30'. Adopting r as (100) and c as (111), then m is (211) and m(2ll) ; and z, the dirhombohedral face to (100), is (122). Taking P to be any pole (hll) in [cm], we have from the A.K. {cPzm} tan cP -r- tan cz = ^-, r Hence we have, for s, l-h=2(2l + h) ; .-. l + h=0, and is (111). For K, l-h=5(2l+h), and3i + 2/t = 0; .-. K is (322), and the form is fj. {322}. Again u was found to lie in [r"ra'] t and the angle ru to be 41 58' : hence by the A.K. {e'rua'}, we have h-l=5(h + l). :. 7i=3, and 1= -2; and u is (302). VI. Trigonal bipyramidal class ; T {hkl, ])qr}. 67. The triad axis may be associated with a plane of symmetry, II, perpendicular to it, provided the crystal has no centre of symmetry. Were a centre of symmetry present, the principal axis would become one of even degree of symmetry, and would then be a hexad axis (Chap, ix, Prop. 4). No crystal has yet been discovered showing the symmetry of the trigonal bipyramidal class. The general form is a trigonal bipyramid similar geometrically to those represented in Figs. 377 and 378 ; but, whereas in class IV such forms have their faces limited to zones each containing the pinakoid and one of the faces of the trigonal prism a {Oil}, in this class every form is a bipyramid, save when the faces are parallel or perpendicular to the triad axis. TRIGONAL BIPYRAMIDAL CLASS. 423 The faces parallel to the plane of symmetry will form a pinakoid similar to that found in classes II IV ; and the triad axis, being the normal to a plane of symmetry, is a possible zone-axis (Chap, ix, Prop. 1). 68. We may take as axes of reference three lines parallel to the co-polar edges of any bipyramid, and a face of the pinakoid as parametral plane (111). Since the axes of reference are inter- changeable by rotations of 120 about the triad axis, whilst the parametral face retains the same direction, the parameters are equal. We therefore take as angular element D the inclination of the face of the axial pyramid to the plane of symmetry, and as linear element c a length on the triad axis connected with unit length in the equatorial plane by the equation c = cos 30 tan D (see Art. 6). The analytical expressions given in preceding sections apply therefore to crystals of this class. 69. The co-polar triad of faces meeting at an apex are represented by symbols in which the same indices are taken in cyclical order; for the three faces are geometrically similar to the triad forming the trigonal pyramid of class I. The pair of faces situated on opposite sides of the plane of symmetry generally need for their representation symbols having different indices : but a simple relation between their symbols can be found from the fact that the angle between them is bisected by the plane of symmetry parallel to (111), and their edge is truncated by a prism-face, so that the four faces form a harmonic ratio. Hence the face P above the plane of symmetry being denoted by (hkl), the homologous face below II is parallel to that which in Art. 15 was called the dirhombohedral face Q (pqr). The homo- logous face has therefore the symbol (pqr), where $ = S*-20, (92); r = 31 - 20 ; J being h + k + 1. We shall therefore denote the form by the symbol T {hkl, pq?}, in which the symbols of the two faces are connected by equations (92). It is immaterial which of the symbols is derived from the other; for by addition, p + q + r = - 3 (h + k + I) - - 36. 424 RHOMBOHEDRAL SYSTEM, CLASS VII. Hence, 9A = 3p + 60 = 3p - 2 (p + q + r), =3q-2(p + q + r), The form T {hkl, pqr} consists of the six faces : hkl, Ihk, klh, pqr, rpq, qrp .................. (q). 70. The special forms are : 1. Thepinakoid {111}. 2. Trigonal prisms T {hkl}, geometrically identical with those of class I. The prisms r{011} and r{211} are particular instances, the faces of which have simple relations to the particular edges selected to give the axes : but by a change of the bipyramid selected to give the axial planes general symbols, such as T {321}, may be assigned to either. The bipyramids represented by Figs. 377 and 378 need only one set of indices and may be represented by TV {41 2}, TV {421}; but they are not special forms. The faces are tautozonal, each with the pinakoid and a face of the prisms 7-{10l} and r{lTO}. They cannot, however, be represented by the symbols r{412}, r{421}; for these are the symbols of trigonal pyramids of class I. The a- subscript may be used to indicate that the faces are symmetrical to the equatorial plane ; or the forms can be given by the general symbol T {412, 214}, T {421, 24T}. The same is true of any other bipyramids T ff {h, (h + l) + 2, 1} having their faces in the same vertical zones. VII. Ditrigonal bipyramidal class; K{hkl, pqr}. 71. The only other arrangement of elements of symmetry consistent with a single triad axis is one in which the elements of symmetry of class V are associated with a plane of symmetry II perpendicular to the triad axis and to each of the like planes of symmetry 2. The lines of intersection of II with each of the 2 planes are dyad axes (Chap, ix, Prop. 8). The elements of symmetry are therefore : p, n, 32, 3A. The dyad axes are uniterminal, and should be pyro-electric axes. No crystal has however been discovered showing the symmetry of this class. DITRIGONAL BIPYRAMIDAL CLASS. 425 72. The forms possible in this class can be readily derived from those of class V in a manner similar to that by which forms of class VI can be obtained from class I. For to obtain any form of class VII, all that is needed is to unite with the corresponding form of class V a similar one, such that the two are reciprocal reflections in the plane of symmetry II. But retaining the axial arrangement adopted for crystals of class V, the pair of homologous faces symmetrical with respect to n are connected by the relations (92) given in Art. 69. Hence the general form n{hkl, pqr], Fig. 391, consists of the faces : Ml, khl, Ihk, Ikh, klh, hlk, \ , pqr, qpf, rpq, fqp, qfp, prq. } ' ' The two apices are interchangeable by rotations of 180 about one or other of the dyad axes. The drawing can be made in the manner described for p. {hkl}. 73. The special forms are : 1. Thepinakoid{lll}. 2. A trigonal prism K {2TT} geometrically the same as that of class V. 3. A hexagonal prism {Oil}, having its faces parallel in pairs to the three like planes of sym- metry 2. This prism is common to classes II, III, V and VII. 4- A ditrigonal prism K {hkl}, geometrically the same as p. {hkl} of class V, and in which h + k 4- 1 - 0. 5. A hexagonal bipyramid {hkl}, where h - 2k + 1 = 0. This is geometrically similar to the corresponding form of class III, and can be constructed in the way described in Art. 53. 6. A trigonal bipyramid K {hll, prf}, geometrically the same as the corresponding bipyramid of class VI. In this class the trigonal bipyramids are limited to the zones in which the faces are perpen- dicular to one or other of the like planes 2; whilst in class VI every form having faces inclined to the axis and equatorial plane at finite angles is a trigonal bipyramid. The particular bipyramid K {100, 122} has its upper polar edges parallel to the axes of reference. CHAPTER XVII. THE HEXAGONAL SYSTEM. 1. THIS system is characterised by having a single hexad axis. It was shown in Chap, ix, Art. 11, that no crystal can have an axis of symmetry of degree higher than six; and in Chap, ix, Art. 21, i. that no crystal can have more than one such axis. The system comprises five classes, which differ in the elements of symmetry associated with the hexad axis. Since there can be only one hexad axis, it follows that a centre of symmetry, planes of symmetry, parallel and perpendicular to the axis, and dyad axes perpendicular to it are alone admissible. The five classes are : I. The acleistous hexagonal (hexagonal-pyramidal, hemimorphic- hemihedral) class, in which the hexad axis is the only element of symmetry. II. The diplohedral hexagonal (hexagonal-bipyramidal, pyra- midal-hemihedral) class, in which the hexad axis is associated with a centre of symmetry and a plane of symmetry, EC, perpendicular to it. III. The acleistous dihexagonal (dihexagonal-pyramidal, hexa- gonal-hemimorphic) class, in which the hexad axis is associated with six planes of symmetry intersecting in it at angles of 30. Owing to the hexad axis, these planes consist of two triads of like and interchangeable planes S and 2, arranged alternately so that the S planes bisect, respectively, the angles between the 2 planes ; and vice versa. IV. The diplohedral dihexagonal (dihexagonal-bipyramidal, holohedral-hexagonaV) class, in which the elements of symmetry of class III are associated with a centre of symmetry. The crystals, consequently, have a plane of symmetry, II, and six dyad axes CLASSES OF THE HEXAGONAL SYSTEM. 427 perpendicular to the hexad axis. The lines of intersection of the planes $ and II are three like dyad axes 8 ; and the lines of inter- section of the planes 2 and II are three like dyad axes A. The dyad axes 8 are interchangeable by rotations of 60 about the principal axis ; and so are the A axes ; but an axis of one kind is not interchangeable with one of the other ; nor is a 2 plane inter- changeable with an S plane. The elements may be given by : H, 3S, 32, C, n, 38, 3A. V. The trapezohedral (Itexagonal-trapezohedral, trapezohedral- hemihedral) class, in which the hexad axis is associated only with six dyad axes perpendicular to it. The dyad axes consist of a triad of like and interchangeable axes 8 inclined to one another at angles of 60, and of three like and interchangeable axes A also inclined to one another at angles of 60. Adjacent 8 and A axes are inclined to one another at angles of 30. 2. The hexad axis is perpendicular to a possible face (Chap, ix, Prop. 3), which belongs to a pedion in classes I and III, and to a pinakoid in the other classes ; the plane through the origin parallel to this face will be called the equatorial plane. When the faces of a form are inclined to the hexad axis at finite angles, other than 90, the homologous faces and edges meeting at the same apex will be six or twelve in number, and will be said to be co-polar. In classes I, II and V the number of co-polar faces and edges is in all cases six ; in classes III and IV the number is six only when the faces are perpendicular each to one of the six tauto- zonal planes of symmetry S and 5. When treating of the several classes we shall see that certain of these forms, or all of them, are either acleistous or diplohedral hexagonal pyramids, similar in shape to those represented in Figs. 392 and 393. For purposes of analysis, axes of reference may be selected in two different ways, which we may denote as (i) rhombohedral (or Mil- lerian) axes, and (ii) hexagonal axes. i. Rhombohedral axes. 3. The axes are taken parallel to the three possible edges in which alternate co-polar faces of a hexagonal pyramid would, if produced, intersect; and the possible face perpendicular to the hexad axis is taken as parametral plane (111). The pyramid selected to give the directions of the axes we shall call the 428 HEXAGONAL SYSTEM. fundamental pyramid. Thus, in Figs. 392 and 393, the axes of X, Y and Z are parallel to the lines joining the apex V to the points M, M /} M ti which lie on the lines OB, OB t , OB" at distances 2 x OB from the origin and are interchanged by rotations of 120 about the hexad axis. FIG. 392. FIG. 393. We shall, for the sake of easy comparison with the relations of rhombohedral crystals, represent the three co-polar faces meeting in VM, VM t , VM tl by the letters r, r', r" ; and we shall take OX to be parallel to VM in which r and r" intersect, OY to VM, in which r" and r intersect, and OZ to VM it the line of intersection of r and r'. The three faces have therefore the symbols: r (100), r (010), r' (001). We shall take for unit length the equal semi-diagonals OA, OA', OA" of the hexagon formed by the fundamental pyramid in the equatorial plane, and OV= c for the linear parameter of the crystal. But from Figs. 392 and 393 it is clear that OB = OA' cos 30, and tan VBO = V -=- OB = V + OA' cos 30. But A VBO is equal to the angle between the principal axis and the normal to the face r (100) : this angle we shall denote as the angular element D of a hexagonal crystal. Hence, when OA' = 1, c = cos30tan V0 = cos 30tanZ> (1). The parameter c is the only crystal-element which in the hexagonal system varies with the substance. From the fact that the three axes of reference are interchanged by rotations of 120 = 2 x 60 about the hexad axis, whilst the para- metral plane retains the same direction, it follows that the para- meters on the Millerian axes are all equal, and may be taken to be any three equal lengths parallel to the axes, such as VM, VM t , VM ti . RHOMBOHEDRAL AXES. 429 4. To find the Millerian symbols of the faces of a hexagonal pyramid. Rotations of 120, double the least rotation characteristic of a hexad axis, being always possible, it is clear that any face (hkl) must be associated with two others (Ihk) and (klh) which meet the principal axis at an apex V n , these faces having their symbols in cyclical order. Further, if the crystal is centre-symmetrical, the parallel faces will also be present; viz. (hkl), (Ihk), (klh). And, again, if planes of symmetry pass through the principal axis and each of the axes of reference, the above triads must be associated with like triads having the same indices taken in the reverse cyclical order. For purposes of analysis a selection of one half of the faces of a hexagonal form may be made, which can all be represented in exactly the same manner as if the crystals belonged to one or other of classes I V of the rhombohedral system. But the faces homologous to (hkl) obtained by rotations of 60 or 180 about the hexad axis cannot be represent- ed by the same indices. The face connected with (hkl) by a rotation of 180 is that denoted by (pqr) in Chap, xvi, Art. 15. Hence, the second half of the faces of a hexagonal form may be regarded as consist- ing of the faces of the dirhombohedral form. The face of this form opposite to (hkl) being (pqr), the symbols are connected by the equations (see Chap, xvi, Art. 15), p = 2 (h + k + I) - 3h = 20 - 3h, \ FIG. 394. Hence the six co-polar faces meeting at V in Fig. 394 which are connected by rotations of 60 about the hexad axis are : hkl, qrp, Ihk, pqr, klh, rpq ... (a). The poles of six homologous faces may also be represented by the stereogram, Fig. 395, in which the poles of one triad are denoted by letters P and those of the homologous co-polar triad by Q. The axes meet the spheje at the points X, Y, Z indicated by crosses. Fio. 395. 430 HEXAGONAL SYSTEM. In the particular instance of the fundamental pyramid, the co-polar triads are represented by letters r and z, and the symbols are : 100, 22T, 010, T22, 001, 2l2. We shall therefore adopt for the forms of hexagonal crystals symbols in which triads hkl and pqr are both included. Such symbols have been already employed to represent the forms in classes VI and VII of the rhombohedral system. It follows from what precedes that the analytical formulae given in Chap, xvi hold for crystals of this system. Thus, the equations of the pole P (hkl) are cos XP cos YP cos ZP 3 cos CX cos CP ~hT ~T~ ~~T~ C being the pole (111) at the centre of the primitive. ii. Hexagonal axes. 5. Many crystallographers adopt a set of four axes of reference, OA, OA', OA it and OF of Figs. 392 and 393, which enable them to represent all the faces of a hexagonal form by a symbol {hkil}. The positions of the several co-polar faces of a hexagonal pyramid are given by taking the three first indices (referring to the like axes in the equatorial plane) in a single cyclical order, and also changing all their signs together. The index 1, referring to the hexad axis which will be denoted by OZ, always occupies the last place, and has the same sign for the co-polar faces of a form. To obtain the symbols of parallel faces, the signs of all four indices are changed. In the case of a dihexagonal pyramid the three first indices must be taken in the two possible cyclical orders, and their signs must also be changed together. Such a set of axes we shall call hexagonal axes, and the three horizontal ones equatorial axes; and we shall use different type to distinguish them and the corresponding symbols from the Millerian axes and symbols. We proceed to give the relation between the hexagonal and Millerian axes, and the equi- valent symbols for the faces of the fundamental pyramid, which correspond to the positions of these axes adopted in this book. We shall take OX to coincide in direction with OA of Figs. 392 and 393, and OA (=a or 1) to be the parameter on it, and A to be on the positive side of the origin. OX is therefore, from the Mil- lerian symbols assigned to the faces of the fundamental pyramid, perpendicular to Miller's axis OX. Similarly, OY is taken along HEXAGONAL AXES. 431 OA', and the positive direction is taken to the right : the parameter OA' is equal to OA, and the axis is perpendicular to Miller's OY. We shall denote by Oil the third equatorial axis having the direction OA /t , and we shall take OA jt = OA to be the parameter on it measured on the negative side of the origin : this axis is perpen- dicular to Miller's OZ. The three first indices will be written in the order in which they refer to the axes OX, OY and OU, respectively. Since by rotations of 120 about the hexad axis, the positive directions of OX, OY and OU are interchanged, and also equal positive lengths or equal negative lengths on them, it is convenient to take equal parameters on the three equatorial axes ; and we shall denote by a t a length a when measured on OY, and by a ti when measured on OU. We can now take OF of Figs. 392 and 393 to be the parameter c on OZ, when the face (100) of the fundamental pyramid becomes (0111); for VA' A ti , passing through A'A it , is parallel to OX, and OA tl = - OA'. The face therefore meets the axes at distances given by the ratios a -f- : a, : a 7/ -=- T : c ; and its symbol is (OlTl ). Simi- larly, (010) is (T011), and (001) is (1T01). Again, the face VAA tl Miller's (2T2) meets the axes at distances given by the ratios a : a, -4- : a n -f- T : c ; and has the symbol (lOTl). The two remaining co-polar faces of the pyramids have the equivalent symbols (T22) = (OTll) and (221) = (Tl01). There are cases however in which Miller has not adopted the same fundamental pyramid as other crystal lographers ; and OX, OY, OZ may accordingly coincide with OA, OA', OA" , or with OB, OB' and OB". In the former case, Miller's (100) becomes (Ohhl) ; and the two values of c given by equation (1) are in the ratio of h : 1, and depend on the change in D only. In the second case, Miller's (100) becomes (2h, h, h, 1) ; and the two values of c are not commen- surable when the faces of the fundamental pyramid pass through points on OB, &c., at unit distance from the origin ; for OB - a cos 30. We shall however assume that the same fundamental pyramid is adopted in both notations, unless the contrary is stated. The student will discover whether this is the case in any particular substance by comparing the symbols and the corresponding angles given by the authors consulted. Hence, for the same value of D or c, we have the following equivalent symbols 100 22T 010 T22 001 2T2; ) OlTl T101 T011 OT11 1T01 10T1.I ( '' 432 HEXAGONAL SYSTEM. "v :F i 1 j___J\ 15 ill I 211 1 * ~.--*-X'k i* T m, . TO : FIG. 3< )6. * 121 n )0 1010.J ..(C). 6. The faces perpendicular to the hexad axis have the symbols 0001 and 0001, according as they meet it on the positive or negative side of the origin: they correspond, respectively, to Miller's (111) and (ITT); and they form pedions or a pinakoid according to the elements of symmetry associated with the hexad axis. Prism-faces parallel to OZ and to one of the equatorial axes of reference OX (say) meet the other pair of axes at equal distances OA', OA //t &c., from the origin ; but the one intercept is positive whilst the other is negative. The faces have therefore the symbols (OlTO) and (OTlO) : they must occur together, for they are both parallel to a hexad axis. They are repeated in two other pairs ; and the three pairs form a hexagonal prism, Fig. 396, represented by the following equivalent symbols : {2TT}, 2TT 112 T2T 211 ] {OlTO}, OlTO TlOO T010 OTlO ] The reader will, on reference to Chap, xvi, Art. 27, see that the prism-face through A A ti , coinciding with 8'8 tl , is (2TT). The other hexagonal prism {110} described in Chap, xvi, Art. 27, belongs also to this system : its faces are perpendicular each to one of the axes OA, OA', &c., of Fig. 392, and parallel therefore to the corresponding Millerian axis. We may now suppose one of the faces to pass through A and A', for the line A A' is perpendicular to and bisects OA /t . The intercepts on the axis are a:a / : a ;/ -=-2 : c -=-0 ; and the symbol is (1120). But' this face is Miller's (1TO). Hence this hexagonal prism, Fig. 397, has the equivalent symbols : {1TO}, 1TO 10T OlT TlO 101 OTln {1120}, 1120 12TO 2110 TT20 1210 2TTO.J" The above prisms are common to all classes of the hexagonal system, and are also geometrically the same in classes II and III of the rhombohedral system. 7. When the intercepts made by a face (hkil) on two of the equatorial axes are known, the points in which it meets any other FIG. 397. RELATION BETWEEN THREE INDICES OF A FACE. 433 known lines in the equatorial plane can be found ; the intercept on the third axis can therefore be determined. Let, in Fig. 398, ELK be the trace of (hkil) in the equatorial plane. Then OE = a + \i, 08' = a, -=- k, Od it = a lt -=- i. Now the whole triangle EOd it is made up of the two tri- angles ^08', o'0d it . = OE . 08' sin 60 + 08' . Od tl sin 60 ; and sin 120 = sin 60. Dividing by OE. O8'.0d /t sin 60, we have L-- J_ J_ 08' ~ Od + OE' Fl - 398. In obtaining this equation all the sides of the triangles have been treated as positive lengths. When, however, their values in terms of the indices are introduced, attention must be paid to the signs of the latter. But in the particular case represented by the figure, h and i are both negative, for the trace meets OX and OU on the negative sides of the origin. Consequently OE must be replaced by a H- h, Od a by a H- i and 08' by a H- k. k i h Hence - ; a a a (4). A similar relation can be readily established for any other position of the face, when the order or signs of h, k and i have to be changed to correspond with the intercepts on the axis. Hence the algebraic sum of the three indices referring to t/ie equatorial axes is always zero. It may be noticed that in the face actually taken the positive index is the greatest, and the corresponding intercept OS' is the least ; also that 08' divides the larger triangle into the two smaller ones. Thus the face may be (T321) or (2531). For a face like that through the trace EdL^ the shortest intercept is now negative and lies on OU r The signs of the three first indices must all be changed; and the symbol is given in the general case by (hikl), and in the particular cases by (1231) and (2351). 8. To find the symbols of the six co-polar faces connected by rotations of 60 about the hexad axis. Let, in Fig. 399, a face (hkil) meet the equatorial plane in the trace Kd H H$E (E on OX not being shown) and let the intercepts L.C. 28 434 HEXAGONAL SYSTEM. on the axes be OE=a + \\, O8' = a t ^-k, Od i/ = a /t + i. A rotation clockwise of 120 about OZ (perpendicular to the paper) brings OX to OU, OU to OY, and OY to OX ; and also the trace KHV into the position H"^K t . The intercept OE is now measured on OU and must be written a^n-h, OS' has become 08 on OX and must be written -=-k, and Od tl is measured on OY ', and must be written a t H- i. The face through the trace H"8K I has therefore the symbol (kihl). A second rotation of 120 in the same direction again changes the same axes and brings the trace to ff'o", the face through which has the symbol (ihkl). The first three indices are therefore changed in cyclical order; and the three faces have the symbols (hkil), (kihl), (ihkl). If the first face has the Millerian symbol (hkl), the three correspond to (hkl), (Mh), (Ihk) respectively. When however the crystal is turned about OZ through 180 a rotation interchanging homologous faces which may be said to be opposite equal positive and negative lengths are interchanged on each equatorial axis. Thus, the face (kihl) through H"^K t is brought into a position in which it passes through H ti K" : the symbol of the opposite face is therefore (kihl). The triad of faces opposite to the first triad are (hkil,) (kihl), (ihkl). But a semi- revolution about the hexad axis interchanges the Millerian triad (hkl), (Mh), (Ihk) with the dirhombohedral triad (pqr), (qrp), (rpq). Hence the six faces connected by the hexad axis are, when taken in order counter-clockwise, given by the following equivalent symbols: hkl qrp Ihk pqr klh rpq hkil klhi ihkl hkii kihi ihki SYMBOLS OF HOMOLOGOUS CO-POLAR FACES. 435 The traces of these faces in the equatorial plane are shown in Figs. 399 and 400 ; in the former some of the symbols have been omitted for the sake of greater clearness. 9. To find the symbols of the six co-polar faces symmetrical to those in table e, when planes of symmetry intersecting in the hexad axis are present. If planes of symmetry parallel to the hexad axis are present, there must be six. Three of the planes, denoted by letters S, pass through opposite co-polar edges of the fundamental pyramid and meet the equatorial plane in the three equatorial axes : the three others, de- noted by letters 2, are perpendicular each to a pair of opposite faces of the pyramid and meet the equatorial plane in the lines OB, OS', OB" . The planes forming each triad are like planes of symmetry, for they are interchangeable by rotations of 60 about the hexad axis. Now the Millerian axes OX, OT, OZ lie in the planes 2 ; and it was shown in Chap, xvi, Art. 45, that (hkl) being one face of the scalenohedron, the adjacent face meeting the first in the polar edge V n \ has the symbol (Idk) ; and a similar relation holds for each pair of faces symmetrically placed with respect to each of the planes 2. Similar relations hold for pairs of faces of the inverse scalenohedron {pqr}, and also for ditrigonal pyramids p. {hkl} and fi {pqr} of class V of the rhombohedral system. But each of the planes 2 is perpendicular to one of the equatorial axes of reference and bisects the angle between the other two. Thus 2 passing through BOM of Fig. 399 is perpendicular to OX and bisects the angle between the axes OY and Oil ; . Hence the face (hkil) through the trace HS'L is repeated in a similar face through KHL^ The intercepts on OX are equal but opposite in sign, whilst those on OK and OU are interchanged. But e^n-k measured on OK is positive, and is therefore changed to + when measured along OU t . Again, /( H-i on OU is negative, and must be changed to , -^-i when the intercept is measured along OY. The signs of the three first indices are therefore all changed, and also the cyclical order ; and the symbol of the face through the trace KHL t is (hikl). This face is by successive rotations, each of 60, brought into five other positions, in which the cyclical order of the first three indices remains the same. The six faces have therefore the following equivalent symbols : hlk rqp khl prq Ikh qp r \ ,*. hikl ikhl khil hikl ikhl khilj'"^ '' 282 436 HEXAGONAL SYSTEM. The two sets of six faces given in (e) and (f) occur together or alone, according as the crystal has, or has not, the planes of symmetry S and 2j. They may be also associated with similar sets of six when the crystals are centro-symmetrical ; or (e) may be associated with the set of faces parallel to (f) in the trapezohedral class, or vice versa. The sets of faces parallel to those given in (e) and ( f ) are found by changing the signs of all the indices together. 10. We now proceed to determine the relation between the symbols (hkl) and (hkil) representing the same face of a dihex- agonal pyramid or scalenohedron. In Chap, xvi, Art. 42, the scalenohedral face (hlk) is drawn through a point 8 /t or A tt at distance a on the axis OU ; and the points H and K, in which it meets OM and OM t , Fig. 348, are given by equations (45) and (47) of that Article ; viz. 2n 2naJ3 and OK t = - - - OM= ^ ^ ; ' 3n 1 3n - 1 where n has the value given in (61) of Art. 44. Similarly, the face (hkl) is drawn through A' to meet OM" at L, where OL = OK, = ^^ . By shifting the face parallel to itself O?i 1 until it passes through 8' in Pigs. 399 and 400, the ratios of the intercepts on OM, t , and OV H on the axis OZ are changed in the ratio 08 '+OA' = (3 (say). Let the new apex be at V p , where OV p = (3.0V n , and let, in Fig. 401, the trace of (hkl) on the equatorial plane be given by ELS'Hd^, in which the intercepts OH and OL given by the above equations have to be multiplied by ft. Then, since the triangles B'OL and LOE make up the whole triangle 8'OE, , _ (08' . OL + OL . OE) sin 30= 08' . O^sin 60; 1 1 tan603rc-l 1 n-\ (5) ' Fl0 ' 40L EQUIVALENT SYMBOLS FOR A FACE. 437 Similarly, the triangle 8'0d lt is divided into two by Oil, and we have (08' . OH+ OH. OdJ sin 30 = OX . Od /t sin 60 ; 1 1 tan 60 3n + " Od~ /f *0>8' = ~OBT = Znafi 1 3w + 1 1 n + 1 ( '' Hence fth = l-n _ OC l+n "~2n 00 2 ..(7). O^ 1 and Od it are both treated as positive lengths in (5) and (6), but for the particular face taken they are measured in the negative directions on the axes OX. and OU : hence the introduction of the minus signs in (7). We therefore have h __ ll_ i J_ \-n~2n -(l+)~2^m " These equations give the hexagonal symbols corresponding to a face of the scalenohedron inRn. In Chap, xvi, Art. 44, the values of m and n in terms of the Millerian indices /*, k, I are given by equations (60) and (61); viz. 1_ h + k + l m and Hence, i_k~ h-r k-h If k and I are interchanged, i.e. if we take the face (hlk), it is clear from (9) that the signs of h, k, i are changed, and also their 438 HEXAGONAL SYSTEM. cyclical order ; but not their numerical values. Thus the denomi- nator of the last term is unchanged, that of the first becomes k - 1, the second h k, and the third I h. But these four numbers are proportional to 1, h, i, k, respectively. Hence, (hikl) is the equiva- lent symbol of (hlk). This can also be seen from the geometry of Fig. 401. For the trace of (hlk) is given by E'S^d'; where OE'=-OE, Od' = -0d tl , and 08,, = - OS'. Hence, as in (7), the hexagonal indices are given by : P.OA +OE' = (n-l) + 2n = /3h, ft. OA' -r Od' = (1 + n) + 2n = 1, p.OA"+0* a = -l =ftk, OC + OV* = \ + nvn = (31. Equations (9) enable us therefore to find the hexagonal symbols of all the faces of {hkl}. It is also easy to show that by a change in the signs of h, k, i, but not in their cyclical order, equations (9) give the equivalent symbol for the opposite face (pqr); and that therefore {hkil} is the equivalent of {hkl, pqr}. For /. r-q = 26-3l -(26-3k) = p-r = 20-3h-(26-3l) = q-p = 20-3k-(26-3h) = p +q + r = Hence from (9), r q p r q -p Therefore (hkil) is the equivalent symbol for (pqr) ; and similarly, (hikl) for (prq). It is therefore immaterial which of the faces of {hkl} or of {pqr} is employed to determine the equivalent hexagonal symbol, for {hkil} includes both {hkl} and {pqr}. 11. Equations (9) and (10) can, by addition and subtraction of the numerators and denominators, be transformed so as to give (hkl) and (pqr) when either of the equivalent symbols (hkil) or (hkil) is known. For each term of (9) is equal to 1 + k-i _l + k-i_l + i-h_l + h-k 3h 3k 3 1 Hence, = = - ^. : r = i i - r ............... (!!) 1+h-k EQUIVALENT SYMBOLS OF DIUHOMBOHEDRAL FACES. 439 Again, each term of (9) is equal to 1-k+i _ 1-k+i 1-i+h h+k+l-h+l+k-h ~ ^h + lk^l)^3h = 2(JTnfeTJp 3k 1-h + k But the denominators of these equations are p, q and r respectively. Hence the opposite face (pqr) is given by These latter equations accord with the values derived in a similar manner from equations (10) ; and they can be derived from (11) by changing the signs of h, k, i. As, however, both (hid) and (pqr) are included in every hexagonal pyramid, equations (11) and (12) can be given in the following form : _*-_ =_^_ * (13). l(k-i) l(i-h) l(h-k)- where the plus signs give the symbol of one face and the minus signs that of the dirhombohedral or opposite face. In obtaining the above equations the face (hkl) was supposed to make the least intercept on OA' (OY) ', but the equations are perfectly general, and it is immaterial which face of a form is taken in order to obtain the equivalent symbol. 12. In computations such as the determination of zone-indices, the indices of a face which is common to two zones, or the anharmonic ratio of four tautozonal faces, we omit one of the equatorial axes. We then have three axes OX, OY, OZ (say) which may be regarded as a set of axes of an oblique crystal in which XOY=12Q, and ZOX = /OK = 90. When the indices referring to the three axes have been obtained from the general relations mentioned, the index referring to the fourth axis Oil is obtained from the relation h + k + i = 0. 13. The equations of the normal OP, or of the pole P, are obtained in a manner similar to that given in Chap, iv, Art. 15. Taking OA to be unity, they are : cosXP cosYP cos UP ccosZP OP = h + k + 1 p (14). 440 HEXAGONAL SYSTEM. We may take P to be the pole (hkl) of a rhombohedral scaleno- hedron, Fig. 402 (see also Figs. 412 and 413, p. 454). Using the notation of Chap, xvi, Arts. 48 50, we have : UP = a"P = 90 + 77, and IP = CP. Hence, cos XP = sin , cos YP = cos , cos UP - - sin 77. But by equation (68) of that Chapter cos - sin 77 - sin = ; hence, the numerator of the last term of (14) is zero, and also the denominator. the result already given in (4). But introducing into equations (74) of Chap, xvi the values of |, 77, , we have cosXP = cosKP^cosf// J = J3 sin CP Dividing each term of these equations by the corresponding one of (14), we have _h k _i_ l^/StanCP l^k~ ~h^t~ k-h~ c^2{(k- /*) + (h - If + (l- hf] " By adding the squares of the numerators of the first three terms of (16) and those of the denominators, and then extracting the square root of the ratio, a term is obtained equal to each of the ratios ; viz. Hence, If in (16), the value of h : 1 given in (9) is introduced, we have -- Again, N being the pole in which the zone [CP] intersects the zone [wiw'J, we have from (70) of Chap, xvi, I A o z, i~ ~ tan YN - tan a'N= -= ,-~= (fro TRANSFORMATION OF SYMBOLS. 441 From the right-angled triangle YPN, Fig. 413, p. 454, we have cos YP= cos PN cos YN= sin IP cos YN. Therefore from the second and fourth terms of (14), cos YN = cos YP + sin IP = y c cot ZP= ^ cos 30 tan D cot ZP = (from (9))- cos 30 tan Z> cot CT ............ (19). The equations given above together with the relations holding between tautozonal poles suffice for the solutions of most of the problems which the student is likely to meet with. 14. In the preceding Articles, and also in the discussion of the forms characteristic of classes I to V, it is assumed that the face (100) is the same as (OlTl) ; but as pointed out in Art. 5, the same pyramid is not always selected by different crystallographers to give the crystal element D, or c. Thus it frequently happens that, as in the case of apatite, Miller's (210) is Dana's (OlTl). If it is desired to find the equivalent symbols of the forms corresponding to the different pyramid adopted, the transformation may be carried out (i) in two steps, or (ii) in a single step. i. We first transform the Millerian symbol (hkl) to its equiva- lent (h'k'l") in the same notation when three co-polar faces of a form {efg} interchangeable by rotations of 120 about the principal axis become (100), (010) and (001), respectively; the parametral plane remaining unaltered. But (efg) being known to be the face represented by (OlTl) in hexagonal notation, the values of h', k', V are now introduced into equations (9), and give the equivalent symbol (hkil). In the transformation from hexagonal to Millerian symbols, we should first find the symbol (h'k'l 1 ) corresponding to (hkil) from equations (11) ; and we should then transform from one set of Millerian axes to another. Assume (efg) to become (100), then (gef) becomes (010), and (fge) (001); the parametral plane (111) remaining (111). Then etc., of Chap, vin, Art. 20, are: [*-/& f*-ge, g'-ef] = new [100]; [w^MVl = [f - ef, *-fg,f*- ge] = [010] ; [u 3 Y 3 w 3 ] = \f'-ge,f- ef, e* - fg] = [001]. 442 HEXAGONAL SYSTEM. Hence by equations (31) of Chap, vm, (20). The values of h', k', I' obtained from equations (20) are now used in equations (9) instead of h, k, I; and the equivalent symbol (hkil) is determined. Thus, taking the case of apatite in which Miller's (210) is Dana's (0111), we first change the Millerian axes, (210) becoming (100). Hence in equations (20), 6 = 2, f=I, g = 0; and we have: \ (21). The equivalent symbols for the faces given in the first and fourth columns of the following table are now determined and are those occupy- ing the third and sixth columns. Miller's. New. Miller's. New. X 210 100 x, 012 T22 r 321 411 r, 123 Oil y 311 5TT y t 113 Til ' V 411 5T2 v Oil T52 8, 100 421 s 122 241 V 322 20T u, 104 425 n 423 8T4 n, 052 212 m 21 T 1TO a OlT 112. The symbols of the faces being now known when the Millerian axes are parallel to the edges of intersection of alternate co-polar faces of the pyramid adopted by Professor Dana to be {OlTl}, we can apply equations (9) to find the hexagonal symbols. They are : New. Dana's. New. Dana's. X 100 OlTl x / T22 OTll r 411 01T2 r , Oil OT12 y 5TT 0221 y, Til 0221 V 512 1122 V 152 TT22 s/ 421 1121 s 241 TT21 u 20T T321 w , 425 1321 n 8T4 T431 n , 052 1431 m 110 1120 a 112 1100. TRANSFORMATION OF SYMBOLS. 443 ii. In practice, time is saved by introducing the values of /*', k', I' given by such equations as (20) and (21) into equations (9); and afterwards introducing the particular values of h, k, I corre- sponding to each form. The general formula), though cumbrous, are very symmetrical. Thus, since h' y k', I' occur in the first degree in each denominator of (9), the denominators of the values given in (20) may be cancelled. Taking now the first term in (9), we have h The second and third terms of (9) are, from the symmetry, clearly similar; and the last term is 2x1 The final equations are therefore h l (e - f)-h(e-f) + k(f-g) + l(g- e) In the particular instance of apatite, in which e = 2, f = 1 and g - 0, we have h-'2k + l~ h + k 2l~ -2h + k + l~ h + k + I Equations (23) serve generally to transform from Miller's symbols to Dana's when Miller's (210) is the latter's (OlTl). I. Acleiatous hexagonal class ; T {hkl, pqr] - T {hkil}. 15. In this class the hexad axis is the only element of symmetry present; and every form, with the exception of the pedions, has six like and interchangeable faces. When the faces of the form are parallel to the hexad axis, they form a hexagonal prism ; and when they are inclined to the axis, they form an acleistous hexagonal pyramid, Fig. 392. As stated in the preceding 444 HEXAGONAL SYSTEM, CLASS I. Articles the alternate faces of one of the possible pyramids are selected to give the Millerian axes, and the faces have the symbols (100), (010), (001) ; the three remaining faces being (T22), (2T2), (22T): the symbol of the fundamental pyramid is r{100, T22}. The pedion, having its face perpendicular to the hexad axis, is (111) or (111), according as it meets the hexad axis on the same side of the origin as the apex V of the pyramid r{100, T22}, or on the opposite side. In hexagonal notation the fundamental pyramid has the symbol rjOlTl} ; and the face (OlTl) meets the axis OY at distance a from the origin, and the axis OZ at distance c ; where c a cos 30 tan D. The equivalent symbols of the six faces of the pyramid in both notations are given in table b. The pedions are (0001) and (OOOT) respectively. 16. The hexagonal prisms have the symbols: {2TI} = {OlTO} (table C), {110} = {1120} (table d) ; and r {hkl} = r {hkiO}, in which h + k + I = 0. In the first two prisms the Greek prefix is omitted, for they are common to all classes of the system. The prism T{M}=-r{hkiO} has the following faces: hkl klh Ihk hkl klh Ilk } hkiO kihO ihkO hkiO kihO ihkOJ ' There is, however, no essential difference between the three prisms, for the symbols depend on the pyramid selected as the fundamental one. The ratios of the indices of the prisrn given by (g) can be determined by equation (18). 17. The acleistous hexagonal pyramid T {hkl, pqr} = r{hkil} has six faces, the symbols of which are given in table e. When however the pyramid makes equal intercepts on two of the equatorial axes, we have two cases, according as one of the faces supposed to be shifted parallel to itself so as to pass through A' meets (i) the adjacent axis OU at A ti or (ii) OX at A. i. In this case the trace in the equatorial plane is parallel to OX, and the first index is zero : the corresponding face is therefore (Ohhl), and is tautozonal with pairs of opposite faces of the funda- mental pyramid ; and its Millerian symbol is (hll). The pyramid has therefore the equivalent symbols T {Ml, prr} = T {Ohhl} ; and includes the faces : hll rrp Ihl prr llh rpr | ,,. ohhi iihoi iiohi ohhi hhoi hoiii j ( >' ACLEISTOUS HEXAGONAL PYRAMIDS. 445 ii. In the second case the equatorial trace is perpendicular to OU and bisects OA it : the corresponding face has the symbol (h, h, 2h, 1), and is tautozonal with (1 1 20) and (0001). Since (1 120) is (1TO) and (0001) is (111), the face has the symbol (hlk), where h + 1 - 2k - 0. Here (Ihk) is the opposite face (prq) of the general case ; and if the indices h, k, I are taken in the two cyclical orders, the form may be represented by a symbol containing only the in- dices of a single face. For the sake of uniformity, we shall however denote the form by r {hkl, Ikh} ; for the general rule of the class does not admit of opposite cyclical orders. The pyramid r {hkl, lkh\ = T {h, 2h, h, 1} has the faces : hkl kM Ihk Ikh klh hlk h,2h,h,l 2h,h,h,l h,h,2h,l h,2h,h,l 2h,h,h,l h,h,2 The pyramids of the two series r{0hhl} and T {h, 2h,h,l} are not in this class special forms; the exceptional relation between the indices is due to their horizontal traces being respectively parallel and perpendicular to the edges which have been selected arbitrarily to give the directions of the equatorial axes. Further, the angles of a pyramid of one series can never be equal to those of a pyramid of any other of the possible series. If, for instance, the face (h,h, 2h,l) were inclined to the pedion (0001) at the same angle as a face (OhjU,), it would be the same as if the latter face were turned through 30 about the hexad axis. Assuming the two faces to pass through a common apex and A', the face (OhJ^l,) would after rotation meet OB of Fig. 392 at a distance equal to OA, and OB H- OA would, by the law of rational indices, be commensurable. But OB = OA' cos 30, and OB -=- OA is incommensurable. The two faces cannot therefore have the same inclination to the equatorial plane or pedion. Since the inclinations to the pedion are different, the angles over the polar edges of a pyramid of one series must be different from those of any pyramid of the other series. A similar proof can be given in the case of a face (hkil), only in this case the angle of supposed rotation would be a'N of Fig. 402. 18. The hexad axis is one of uni terminal symmetry and should be a pyro-electric axis. The crystals may also be regarded as enantiomorphous, i.e. correlative forms are possible in which the faces are inclined to one another at the same angles ; and the two crystals may be placed in positions in which they are reciprocal reflexions in a mirror parallel to the common direction of the hexad axis. This being the 446 HEXAGONAL SYSTEM, CLASS I. case, the crystals may be expected to rotate the plane of polarization of light transmitted along the hexad axis. 19. The crystals of lithium potassium sulphate, LiKS0 4 , of Li(NH 4 )S0 4 , of LiRbS0 4 and of LiKSeO 4 are placed in this class; for the above sulphates rotate the plane of polarization, and the seleniate shows on the pyramid-faces the same kind of unsymmetrical corrosion- figures as the sulphates. Lithium potassium sulphate. .0 = 62 39'; c= 1-6743. The crystals are easily obtained by the evaporation of a solution containing the two sulphates, and occur as apparently simple hexagonal prisms terminated by both pedions and by similarly developed hexagonal pyramids at both ends. The habit of the crystals is therefore similar to that of the crystals of apatite and mimetite shown in Fig. 405. But the crystals are twins, occasionally of a simple character, but frequently of great complexity (Dr H. Traube. N. Jahrb. f. Min. n, 1892, p. 58; I, 1894, p. 171). The opposite ends of the crystals are often positively electrified with^ falling temperature, whilst a broad central zone is negatively electrified. Such crystals are twins joined along the face (TIT), which is at the analogous pole of the two components. The latter moreover rotate the plane of polarization in opposite directions ; and occasionally Airy's spirals have been seen in a plate cut from such a crystal. Other ap- parently simple crystals, showing opposite electrifications at the two ends, have been shown to be complex twins. The amount of rotation for a plate 1 mm. thick is given by Traube as 3-44, but this may be too low; for it is possible that no plate has been obtained free from twin-lamellae of opposite rotations. The crystals of nepheline, Na 8 Al 8 Si 9 O 34 , of strontium- and lead- antimonyl dextro-tartrates are also placed in this class ; for the cor- rosion-figures on the prism-faces are unsymmetrical trapezia, but those on adjacent faces are congruent when the crystals are turned through 60 about the principal axis. The tartrates are also pyro-electric ; but neither their crystals nor those of nepheline show any rotation of the plane of polarization. Strontium-antimonyl dextro-tartrate, Sr 2 (SbO) 2 (C 4 H 4 O 6 ) 2 . A crystal after Traube (N. Jahrb. f. Min. Beil. Bd. vm, 1893, p. 270) is shown in Fig. 403. = 44 16-5', c= -8442. The forms shown are : m {2ll } = {OlTO}, O=T {100, 122} = T {OlTl}, x = r {111, 511} = {022T}. The measured angle ;o7=lTT A 151 = 52 50' was used to determine the element. If we denote by x' the pole 5TT, by x" the pole (111) and by C the pole (111) = (0001), we have, from the isosceles spherical triangle Cx'x" and the A. R. {Coxfm}, the following equations : tan =tan (0^ = 62 51')-j-2; .-. Z> = 4416'5'; c = cos 30 tan (Z> = 44 1 6'5') = -8442. EXAMPLES (DEXTRO-TARTRATES). 447 The obtusely terminated end at which o is developed is the antilogous pole, that at which x appears is the analogous pole. The cor- rosion-figures on two of the adjacent prism-faces are shown in Fig. 403. They are four-sided pits, having two sides parallel to the edges [mo], but otherwise unsymmetrical ; and those on adjacent faces are congruent when the crystals are turned through 60. No evidence of twinning similar to that in lithium potassium sulphate was observed in the pyro- electric characters or in the arrangement of the corrosion- figures on the prism-faces. A crystal of lead-antimonyl dextro-tartrate after Traube (loc. cit.} is shown in Fig. 404. D=U 9', c = -84064. The forms are : m = {2TO} = {0110}, o=r{100, T22} = r{OlTl}, #=r{l!T, 511}=r{0221}. FIG. 403. Traube gives as measured angles those in column (2) of the following table of angles, and computes the element and the angles in column (3) from the measured angle xx = 111 A 151 = 52 47'. Assuming ox= 100 Mil = 72 51' the sum of the last two measured angles of column (2) to be as trustworthy as xx, we may find the angles in column (4) by equation (21) of Chap, vin, Art. 18. For in the A.K. {Comx} we know the angles Cm = 90 and ox ; and the transformation needed to find om or Co falls under case (e) of that Chapter. Hence equation (21) of p. 102 becomes mcos(90 + oa;-2om) = (m - 1) cos (90 - ox) + cos (90 + ox), Also l-Hm = A.R. {Comx} = 3. Fio. 404. .-. sin (2om - ox) = sin 72 51'-=-3, . 2om- 72 51' =18 34-4', and om = 45 42-7'. J 1 ) (2) (3) (4) 111 A 151 *5247' 52 60-4' 100 A 212 4049 40 46' 4052 100 A 211 4541 4551 4542-7 2llAlIl 27 10 27 15 27 8 -3 111 A 100 ( = Z>) 44 9 44 17'3 The value of c corresponding to D of column (4) is -8448. The method of computation by equation (21) of Chap, vin can also be applied to the strontium salt, and gives results in close accordance with the measured angles. The pyro-electric and optical characters, aud also the corrosion-figures, are similar to those of crystals of the isomorphous strontium salt. 448 HEXAGONAL SYSTEM, CLASS II. ..{k}. II. Diplohedral hexagonal class; ir{hkl, pqr} = 7r{hkil}. 20. If the hexad axis is associated with a centre of symmetry, there will also be a plane of symmetry II perpendicular to the axis (Chap, ix, Prop. 4). The hexagonal pyramid of the last class will then be associated with a like one, having each of its faces parallel to a face of the first pyramid. The bipyramid TT \hkl, pqr} = TT (hkil} consists therefore of the following faces : hid qrp Ihk pqr klh rpq hkil kihl ihkl hkil kihl ihkl pqr klh rpq hkl qrp Ihk hkil kihl ihkl hkil kihl Ihkl In the above table the two faces in each column are symmetrically placed with respect to the equatorial plane II ; and the parallel faces are derived the one from the other by changing the signs of all the indices. The particular case {100, T22} = {OlTl}, which may be repre- sented by Fig. 393, differs in no essential respect from TT {hkil}. The particular values of the indices in the two notations result from the arbitrary selection of this pyramid for the fundamental one. The element is obtained from this pyramid in the manner already given. Similarly, the two series of hexagonal bipyramids {Ohhl} and {h, 2h, h, 1}, which can be derived by the addition of parallel faces to those given in tables h and j, are not special forms. The symbols of the new faces, being obtained by changing the signs of all the indices of each face in these tables, need not be given here. The Greek prefix is omitted in these cases, for the pyramids are geometrically identical with similar pyramids of class IV. 21. The hexagonal prisms of the last class belong also to this, so that there is no geometrical distinction between tr {hkiO} and r {hkiO}. The pinakoid {111} = {0001} has its faces perpendicular to the axis and parallel to the plane of symmetry : it consists of the two faces, 0001 and OOOl. 22. Apatite, 3Ca 3 P 2 8 . Ca (Cl, F) 2 , affords a good illustration of forms of this class. In the crystal shown in Fig_._405 the forms are: c {0001} = {111}, m{0ll0} = {2ll}, and x {Olll} = {100, 122}. The relative magnitudes of the faces of the several forms vary greatly; the habit being sometimes prismatic FIG. 405. EXAMPLES (APATITE). 449 owing to the predominance of the prism-faces, and sometimes tabular through the predominance of the pinakoid. A crystal having several additional forms is represented by Figs. 406 8. It can be completely determined by the observation of zones and by measurement of the following angles in the three zones [m"am t ], [crxym], [mnusx,]: 30 0' 30 |~cr 22 59' ex 40 18-3 r* Ml 59 29 90 mn 22 41' mu 30 20 44 17 mar, 71 8 i. The angles in the zone [ma] are fixed, as was shown in Chap, xvi. Art. 27. Take m to be (2ll) = (0ll0), and a; to be (100) = (Olll),then, by (1) of Art. 3, the parameter c = cos 30 tan 40 18-3' = -7346. ii. Again, let p be any face (Ohhl) in the zone [carm]; then, from A.R. {cxpm}, OlIO Ohhl 0110 0111 0001 Ohhl 0001 Olll Hence, when p coincides with r, h 1 Fio. 406. , and .-.h = l, 1 = 2. The face r is (01 12). and the form is {0112} equivalent, from (11), to {411,011}. Similarly, when p coincides with y, cp = cy = 5$ 29', and h-=-l = 2; therefore y belongs to the form {0221} = (from (11)) {511,111}. iii. In the remaining measured zone we know m to be (Oll0) = (2ll), the face x, on its right to be (Il01) = (22l). and m,, in [ex,] to be (1100) = (112). The arcs joining the poles m, x, , m lt of these faces, Fig. 407, form a right- angled triangle. Hence, by Napier's mnemonic, cos ar,roi,, = tan 60 cot 71 8'. Let us now determine the arc-distance from m of the pole a in which the zones [ca,] = [11 1,101] and [nur / ] = [2ll,22l] intersect. From Weiss's law we find the symbol of = 7627', c = 3'5936. The crystals are hexagonal prisms {2TT} = {OlTl} terminated by the pinakoid {111} = {0001}, and by narrow faces r{100, 122} = {OlTl} modifying the edges [cm]. The physical characters of the two salts are much the same. The crystals show no pyro-electric poles, nor do they rotate the plane of polarization. The determination of the class rests solely on the character of the corrosion-figures observed on the faces of the pinakoid and prism. When plates parallel to the pinakoid are observed in parallel light between crossed Nicols, they show segments similar to those characteristic of triplets of biaxal crystals such as witherite, &c. (see Chap, xviu) ; and the characteristic figure of a biaxal crystal can occasionally be seen in places, when convergent light is used. But the segmentation is irregular, and the internal structure manifested is very complex. Again, plates cut from near the surface sometimes give a central portion which is uniaxal. Plates parallel to the prism-faces examined between crossed Nicols are also seen to have a lamellar structure ; and occasionally, when convergent light is used, the brushes characteristic of a biaxal crystal may be seen. The structure is sometimes that of plates bounded by parallel planes, and sometimes such as is found in bodies having a fibrous structure. The determination of the class, and even of the system, is therefore doubtful. CHAPTER XVIII. TWIN-CRYSTALS AND OTHER COMPOSITE CRYSTALS. General Introduction. 1. COMPOSITE crystals often occur, in which the several portions have different orientations governed by regular and definite laws. When the crystallization of a substance held in solution is hurried by rapid evaporation of the solvent, the crystals usually grow together in groups, in which the arrangement of the several members is purely accidental. But it was observed at a very early date that crystals of certain minerals, in particular those of cassiterite and spinel, are joined together in a regular and constant manner to form a well-defined individual. Such regularly formed composite crystals will be the subject of this Chapter. Rome de 1'Isle was the first to attempt an explanation of the composite character of the crystals of spinel and cassiterite ; and he introduced the word made to denote the kind of composite crystal which we now call a twin. Werner employed the word zunlling (= twin) at present used by German crystallographers ; and later on Haiiy introduced the word hemitrope (from 17/11- = half, and TpoVos = a turn) ; for he perceived that the orientation of the two portions of every well-defined twin known to him is given by the following law. A complete crystal, bounded by the forms observable on the twin, is divided along a central plane which is parallel to a possible face ; and the half on one side of the plane is then turned through 180 about its normal, the two halves remaining in contact to form the twin. This law gives in very many cases the relative orienta- tions of the two portions united together in a twin-crystal ; it offers no suggestion as to the cause of twinning, and supplies no explanation of the growth of the twin. A crystal or form of normal character and uniform orientation will be said to be simple, when it is necessary to distinguish it from a composite crystal or a twinned form. 462 TWIN-CRYSTALS. 2. With very few exceptions, which however are important, twins consist of portions, the relative orientations of which are such that a semi-revolution of one portion about a line having a definite direction brings the rotated part into the same orientation as the fixed part. The line of rotation we shall call the twin-axis : its direction may be that of (i) the normal to a possible face, (ii) a zone-axis, or (iii) a line lying in a crystal face perpendicular to a zone-axis (this last is very doubtful). The twin-axis has clearly the same relation to both portions. Such twins we may designate hemitropic twins, when we require to distinguish them from those rare composite crystals in which the orientation of the portions can only be given by regarding the one as the reflexion of the other in a definite plane ; these latter being then called symmetric twins. Taking the hemitropic twins, in which the orientation of the portions can be connected by an axis of rotation, we can divide them into two main classes according to the manner in which the portions are united : 1. Juxtaposed twins, in which the portions are united along a plane surface, and lie on opposite sides of it. 2. Interpenetrant twins, in which the portions are intimately com- mingled without any regular surface of separation between their matter. 1. Juxtaposed twins. The plane surface along which the two portions of a juxtaposed twin are united will be called the com- bination-plane. It may be (i) parallel to a possible face which is perpendicular to the twin-axis, or (ii) it may be parallel to the twin-axis. i. This group of twins includes those called by Haiiy hemi- tropes, in which the twin-orientation is fully expressed by the single statement that the twin-face is that particular face parallel to which the portions are united after the one has been turned through 180 about the face-normal. For example, in spinel the twins have a face of the octahedron for twin-face. Provided the two portions belong to a centro-symmetrical crystal, they are situated symmetrically to the combination-plane ; and, if they are also equal, the coigns will lie at equal distances on straight lines perpendicular to this plane. Instances of equably and symmetrically developed twins are shown in the drawings of spinel, cassiterite and calcite. ii. When the combination-plane is parallel to the twin-axis, the directions of both must be expressly specified. This arrange- INTERPENETRANT AND SYMMETRIC TWINS. 463 ment is common when the twin-axis is a zone-axis which is not perpendicular to a possible face ; e.g. the so-called Carlsbad twin of orthoclase, and several of the twins of anorthite and the other plagioclastic felspars. In these twins the two portions do not, after a semi-revolution of one of them about the twin-axis, form a com- plete crystal. They ai-e two like halves of two separate crystals placed in parallel orientation. The hemitropes placed under (i) may, as we shall show, in many cases be referred to a twin-axis parallel to the combination-plane, but the statement of the twin-orientation is more precise when they are referred to a twin-face. In many cases the twins of classes i and ii cross one another in the middle, so that portions at the opposite ends and on opposite sides of the combination-plane are in like orientation, those at the same end are in twin-orientation. They are generally called intercrossing twins. The surface at which the individuals seem to cross is often fairly well-defined. In drawings it is often taken to be a plane perpendicular to the combination-plane. 2. Interpenetrant twins. However intimate the intergrowth of interpenetrant twins may seem to be, it has been found by examining the cleavages and other physical characters, and more especially by examining in plane-polarised light plates cut across the twins, that the matter of the different portions remains distinct and separate. Just at the boundary of the individuals the optical phenomena are often indistinct in consequence of the interlocking and overlapping of the matter of different portions. Fluor gives a good instance of interpenetrating twins. Symmetric twins. A very large number of the hemitropic twins falling under the preceding subdivisions are symmetrical to certain definite planes. Thus, in most of the juxtaposed twins of sub- class (i), the portions on opposite sides of the combination-plane are reciprocal reflexions in this plane. But a few composite crystals are known in which the physical structure of the two portions can only be connected by regarding the one as the reflexion of the other in certain definite planes. Such composite crystals, of which instances occur in sodium chlorate, sodium periodate and quartz will be distinguished as symmetric twins. The composite crystals known as complementary and mimetic twins fall under one or other of the preceding divisions. By com- plementary twin is meant one composed of two individuals which belong to a class of inferior symmetry in the system ; the two 464 TWIN-CRYSTALS. individuals being combined in such a way that the homologous faces of the two individuals taken together produce a form identical with that of greatest symmetry in the same system. Thus the two interpenetrating pentagonal dodecahedra of pyrites, Fig. 431, p. 476, make a complementary twin, in which the faces taken together would compose a tetrakis-hexahedron of class II of the cubic system. Similarly, the juxtaposed twin of pyrargyrite, Fig. 499, p. 527, is another complementary twin. These complementary twins serve as an introduction to the complex twins of harmotome and phillipsite, in which an apparently simple prismatic crystal consists of intercrossing twinned individuals of the oblique or anorthic system. These prismatic crystals are again twinned so as to approximate in external appearance to a tetragonal, or even to a cubic, crystal. These complex twins and other similar twins, which we shall discuss further on, give an insight into those curious cases known as mimetic twins in which apparently simple crystals, having the external form characteristic of a class with complex symmetry, are composed of a number of portions of crystals of inferior symmetry. The statement which defines the twin-orientation in a given case is called the twin-law for instance, that of spinel is : twin-face a face of the octahedron. Multiple twins. In certain substances, calcite, labradorite, fec., the twinning is sometimes repeated several times parallel to the same twin-face, and the crystal consists of a series of thin plates twinned according to the same law. The lamellae and twins are said to be poly synthetic. The physical characters of homologous faces and edges being the same, twinning may occur at the same time with respect to each of the homologous faces of a form to which a twin-face belongs, and with respect to all the homologous lines of which the twin-axis is one. Such multiple twins occur among the crystals of some substances, e.g. rutile and cassiterite ; but they are, as a rule, far less common than the twins which are composed of only two portions. A twin of two individuals we shall call a doublet, when we wish to emphasize the fact that it consists of only two portions ; one consisting of portions of three crystals a triplet ; one of four different portions a quartet; and so on for five, six and eight. 3. Tests of twinning. Many twins are distinguished by the TESTS OF TWINNING. 465 presence of re-entrant angles', which may be made by like or unlike faces. The angles between the faces meeting in such edges will be indicated by affixing a minus sign ; thus, in Fig. 420, p. 466, the angle 111 A (111) = 38 56'. Re-entrant angles may, however, occur on untwinned crystals, and be due either to an irregularity in the deposition of the matter or to the presence of some obstacle. It is therefore necessary to determine whether the portions forming the re-entrant angle have a parallel or a twin- orientation. Inspec- tion generally suffices to show whether all the like faces in the neighbourhood of the re-entrant angle are parallel or not. Optical test. When the crystals are translucent, the identity or difference of orientation of the several parts can in most cases be decided by examination in plane-polarised light ; this method fails, however, in the case of isotropic (cubic) crystals, and also in twins of the rhombohedral system which have the triad axis for twin-axis, and in a few cases in other systems. The optical investigation in polarised light is the most delicate test that can be applied. By its means, for instance, a twinned structure has been revealed in mimetite which had not been previously suspected, and of which there is no external evidence. S trice and corrosion. The presence of twinning is sometimes shown by barbed striae, meeting in a line on a face of an apparently .simple crystal. It is also sometimes revealed by a difference of orientation of the corrosion -figures observed on the same face of a crystal ; thus, etched crystals of succin-iodimide have been observed showing at one end of a prism-face triangular pits orientated as shown in Fig. 215, p. 269, whilst at the other end of the same face, when composite, the apices are all turned the other way. Electrical test. The difference of orientation is sometimes ren- dered manifest by the electrification resulting from change of temperature; thus, the opposite ends of the twinned crystals of succin-iodimide just mentioned are both analogous poles, and are negatively electrified with falling temperature. The apparently simple hexagonal crystals of lithium potassium sulphate (Chap, xvn, Art. 19) often reveal a twinned structure by the fact that, as the temperature falls, the two similarly developed ends become positively electrified whilst a central zone is negatively electrified. 1 The edges at which faces meet in re-entrant angles will be indicated by lines of interrupted strokes, except when the figure is taken without modification from a memoir on the particular twin, or from Dana's Mineralogy. 466 TWINS OF THE CUBIC SYSTEM. 4. Before a group, composed of differently orientated portions, can be accepted as a twin, the association must be definite and regular, and occur with sufficient frequency to prove that it is not merely accidental. Possibly a single case of a well-formed twin of simple law may be accepted, provided the law itself is not un- common in the system to which the crystal belongs. i. TWINS OF THE CUBIC SYSTEM. 5. In this system the twin-axis is usually one of the triad axes, i.e. the normal to a face of the octahedron or tetrahedron ; and the juxtaposed twins are generally combined along a plane perpen- dicular to the twin-axis. Such twins are hemitropes having a face of {111} for twin-face. A few instances, some of which are given in Art. 11, have been described in which combination takes place along one of those faces of the form {211} which are parallel to the twin-axis. Some twins of crystals of class V require a normal (211) for twin-axis; for instance, diamond. The above twins will form the main subject of sub-section A. Complementary twins, having for twin-axis the normal to a dodecahedral face and an angle of rotation of 180 about the axis, will be discussed in sub-section B. The only instance of a hemitropic twin, which does not fall under subdivisions A and B, is a twin of galena having a face of {441} for twin face. This law gives rise to a lamellar structure clue to repetition of the twinning parallel to the same face. Repeated twinning also occurs in the same crystal parallel to two or more faces of the form. A. Twin-axis a triad axis. 6. Spinel. Let us suppose an octahedron of spinel or magnetite to be divided into two equal parts by a plane parallel to the face Fio. 420. Fio. 421. TWINS OF SPINEL AND GALENA. 407 ^11 1), and the front half to be turned through 180 about the normal to the plane of section; the two portions being then joined together along this plane, the twinned crystal, Fig. 420, is con- structed. The edges in the plane of section form a regular hexagon, the sides of which are parallel to the dyad axes of the crystal : these edges are the intersections of portions of opposite faces which are parallel in the simple crystal; and the angles over opposite edges are equal salient and re-entrant angles. Thus the salient angle over the edge d tl d' is 111 A (III) = 180 - 2 x 70 32' = 38 56' ; the re-en- trant angle over the opposite edge is equal to that over dd' and is - 38 56'. The twin often acquires a more or less strongly marked tabular habit by the disproportionate development of the faces parallel to the combination-plane. Twins of this habit also occur in crystals of gold and diamond. Galena. Fig. 421 represents a similar twin of galena in the Cambridge Museum, in which the forms are: o{lll}, d{101} and a {100}. In Chap, xv, Art. 17, it was shown that six faces of the dodecahedron are parallel to each triad axis; opposite pairs of these faces being parallel, and adjacent faces making angles of 60 with one another. Hence the twin-axis being [111], the pairs of parallel faces 101, (TOI); HO, (110); Oil, (Oil) are co-planar after a semi- revolution of one portion about the twin-axis. This is well seen in the specimen, and is indicated in the drawing by the omission of a dividing line between the co-planar portions of faces of the two individuals, such as 101 and (101). In these figures, and in those of other twins, we enclose in parentheses the symbols of the faces on the portion which is supposed to have been rotated. When the twin is a triplet, quartet, &c., the face-symbols of the other portions are enclosed in brackets, braces, &c. When letters are used to indicate the faces, those of the rotated portion are usually underlined, and the additional portions of multiple twins are indicated either by different type or by affixing to the letter an index-number. The different portions are also sometimes indicated by capital Roman numbers. 7. To determine the twin-law of the spinel and diamond doublets. The determination of the twin-law in simple cases, such as those represented by Figs. 420 and 421, is easy; for two octahedral faces on different portions of the twin are parallel, as can be proved by 302 468 TWINS OF THE CUBIC SYSTEM. measurement of the angles in the zone of which dd' is the direction of the zone-axis. The twin-axis must therefore be either parallel or perpendicular to these faces. By trial with a cardboard model of an octahedron bisected by a plane parallel to (111), the student can convince himself that a semi-revolution of one half about the triad axis perpendicular to the plane of section transforms the simple octahedron into the twinned form. A semi-revolution about the line lying in the plane of section and bisecting at right angles the edge dd' in Fig. 420 brings the rotated portion into the position of the fixed half. By such a rotation the twin is represented as consisting of two like halves of separate crystals. Semi-revolutions about each of the two ho- mologous lines in the plane d t dd' bisecting, respectively, dd" and d'd it also bring the rotated half into the position of the fixed one. These positions of the twin-axis satisfy the geometry of the twin, and in the particular instances of spinel, magnetite, and galena, also the physical relations ; and there is nothing to discriminate between them and the triad axis as the twin-axis. The latter is adopted for the sake of greater simplicity in the expression of the twin law. Diamond. The twin of diamond, Fig. 422, is best interpreted as having for twin-axis the line which lies in the combination-plane and joins an obtuse to an opposite acute coign. This line is the normal to one of the faces 112, 121, or 211, which are parallel to the triad axis [111]. The two portions are similar portions of separate crystals. A semi-revolution of the front half about the triad axis [111] brings similar faces on the two portions into parallelism ; but opposite faces in a crystal of class V of the cubic system are dissimilar. A FlG> 422 - simple crystal is therefore not produced by such a rotation. The twin may also be described as a symmetric twin, in which the two portions are symmetrical to a combination-plane parallel to (111). The same interpretation should be applied to the spinel-like twins of diamond, in which faces 111 alone occur. 8. To draw the spinel-doublet. We begin with a drawing of the cube, Fig. 423, such as was described in Chap, vi; and we inscribe the octahedron by joining the middle points A, A', &c., of the cubic faces. Selecting for DRAWING THE SPINEL-DOUBLET. 469 FIG. 423. Produce A'R to twin-axis the diagonal p"p,,> we determine the position of the cubic axes of the rotated half, and the points of bisection of the edges A A", AA t , A t A llt &c. The edges dd\ &c., of Fig. 420, forming the hexagonal section in the combination-plane, can then be drawn. To determine the position of the cubic axes of the rotated portion we proceed as follows. The plane AA'A it meets the twin-axis at R, where OR= Op tl +3. On the axis cut off a length Rto equal p . to OR. The point O is the origin of the rotated cubic axes, which are to A, IA', SlA tl in direction and magnitude ; these being the reflexions of the origi- nal axes in the plane AA'A tt . To prove this : let Fig. 424 be part of a section of the cube and octa- hedron by the plane containing OA', Op tl and the dodecahedral axis 0S 4 ; and let d t be the point in which A'R meets 08 4 . a,, where a. f R = RA' ; and join Oa. t , toa. f and QA'. The four-sided figure, OA'to^ is a rhombus, for the diagonals Oto and .4 'a are bisected at right angles at R. Hence QA' = fia ( = OA'. The angles OA 'a /} tiA'a, and fio/' are also equal; and so are the angles A'QR, a.fill. Hence a semi-revolution about Oft interchanges toA' with no,, which is equal and parallel to OA'. O A' is therefore the direction of the rotated axis of Y ; but it is the negative direction, for, if the original axes are shifted without rotation so as to pass through O, OA' and Qa, are measured in opposite directions. Similarly, it can be shown that Q,A is the rotated axis of A' and the parametral length a measured in the negative direction; and that toA tt is the rotated axis of Z and the length a measured on it in the positive direction. The coigns A,, A_', A^ t of the rotated portion of the doublet are now found by producing through li the axes O A, toA', flA lt in Fig. 423 to points at the same parametral distance from li. The point A it is joined to d and d', and similarly for the other edges, so that Fig. 420 is quickly completed. FIG. 424. 470 TWINS OF THE CUBIC SYSTEM. The coigns A', J.', A can however be most easily found by drawing through A", A t , &c., lines parallel and equal in length to 0O ; so that, if the spinel-twin is alone needed, the determination of the rotated axes is unnecessary. Twin-crystals of this habit, but in which a certain number of subordinate forms are introduced, can be drawn in a similar manner ; the directions of any rotated edges being found by Weiss's rule, using the rotated axes. The drawing of the twin of galena, Fig. 421, needs no separate description. 9. Blende. Crystals of blende occur, in which the comple- mentary tetrahedra /A {111} and //.{111} are nearly of equal size: they are distinguished from true octahedra by the difference in lustre, markings, and other physical characters of opposite faces. When such crystals are twinned in the same way as those of spinel and galena, the twin has geometrically the appearance given in Fig. 420 ; any variation depending on the presence of modifying faces, such as those of the cube, dodecahedron, &c. But the twin is no longer physically symmetrical to the combination-plane, for the faces which meet in edges in this plane are portions of opposite faces : the difference in physical character on opposite sides of the combination- plane is well seen in the specimens. The twin-orientation does not in this case permit of a double interpretation in the manner shown in Art. 7 to be possible in the twins of spinel and galena. For a semi-revolution of one of two similar halves of separate crystals would give a twin which is physically and geometrically symmetrical to the combination-plane. Such a symmetric twin is possible in blende. When the prominent faces on the crystal are those of the rhombic dodecahedron, the doublets with a triad axis for twin-axis acquire a very different aspect. Suppose a rhombic dodecahedron to be bisected by a plane parallel to (111), and the front half to be turned through 180 about the normal to the plane of section ; then a twin is constructed such as is shown in Fig. 425. As already explained in describing the twin of galena, six faces of the rhombic FIG. 425. TWINS OK BLENDE AND COPPER. 471 dodecahedron, inclined to one another at angles of 60, are tauto- zonal and have the twin axis Op tl for zone-axis. A semi-revolution of the front half brings the face (Oil) into the same plane as Oil. Similarly, 101 and (TOT), 110 and (110), &c., are co-planar. The twin therefore resembles a hexagonal prism, which is terminated at each end of the twin-axis by three faces of the dodecahedron belonging to separate portions. The faces at opposite ends are no longer parallel, but are symmetrically placed with respect to the com bination-plane. The crystals of blende are never simple rhombic dodecahedra, but are usually combinations of this form with /u,{lll}, {100}, &c. Further, the twinning is not, as supposed in the ideal twin just described, limited to the production of a doublet, but is repeated several times. When the several portions are twinned parallel to the same face of the tetrahedron, the twin has six tautozonal dodecahedral faces, the portions of which belonging to the different individuals are co-planar and show no trace of twinning save at the edges in which they meet faces not belonging to their zone. When the faces of the cube and tetrahedron are largely developed, and the lamellae are numerous and thin, the faces of these forms intersect at salient and re-entrant angles in a manner which renders it often difficult to discriminate between portions of a cubic and a tetra- hedral face. Further, the twinning is not, as a rule, limited to one twin-face but is repeated parallel to different tetrahedral faces. This is clearly seen on breaking some of the crystals, when interruptions in the cleavages due to twin-lamellae having different orientations will be perceived. 10. Copper. Fig. 426 represents a rare twin of copper which resembles a doubly terminated hexagonal pyramid of the rhom- bohedral or hexagonal systems. This is due to the fact that the faces present are those of the te- trakis-hexahedron {210}, in which the angles between adjacent pairs of faces are all equal (p. 290). Hence a semi-revolution about Op tl of the six faces meeting at the ditrigonal coign p lt in Fig. 231 FIG. 426. 472 TWINS OF THE CUBIC SYSTEM. brings them into a position congruent with their original position. But, as shown in Fig. 426, those edges of the simple form which diverge from the opposite ditrigonal coigns p" and p ti will, if pro- duced, meet in the central plane perpendicular to p"p lt . The two sets of faces meet in a hexagon in this plane ; and the middle points of the sides of the hexagon lie in the dyad axes which are at right angles to the triad axis. The bi-pyramid so formed is similar to that of the rhombohedral system discussed in Chap, xvi, Arts. 53 55. A semi-revolution of the six faces meeting at either apex brings the matter into twin orientation, but leaves the geometrical aspect of the bipyramid unaltered. It is clearly immaterial which half of the twin is the rotated portion : the symbols inscribed on the faces correspond with a semi-revolution of the six faces meeting at p". The twins often reveal their com- posite character by the presence of grooves modifying the edges and coigns which lie in the combination-plane. In the figure several of the edges of the tetrakis-hexahedron are shown by lines of short strokes. 11. Sadebeck (Zeitsch. d. Deutsch. geol Ges, xxiv, p. 427, 1872, and xxvi, p. 617, 1874) has shown that twins of galena and fahlerz sometimes occur in which the twin-axis is a triad axis, and the combination-plane is one of those faces of the form {211} which are parallel to the twin- axis. Galena. Fig. 427 is a copy of Sadebeck's diagram to illustrate the position of the two portions in the twin of galena : it is a plan of two cubo-octahedra placed in the twin-orientation and projected on a plane perpendicular to the twin- axis. In each crystal the edges parallel to the paper are at 60 to one another; each of them being the p IG 427. intersection of a cubic and an octa- hedral face, and the direction of a dyad axis. Those faces of {21 1} which are parallel to the twin-axis would, if developed, replace these edges, and would in each crystal compose a hexagonal prism. The two crystals having been placed side by side in similar orientations, that to the right has been turned through 180 about the twin-axis : the two crystals touch in a common face of the hexagonal prisms. The ideal twin is now obtained by bisecting both crystals by a plane TWINS OF GALENA AND FAHLEHZ. 473 parallel to the common face, and uniting the more remote halves in the plane of section. The twin has two parallel octahedral faces ; the boundaries of which on one side, i.e. above the paper in Fig. 427, consist of four octahedral and two cubic faces, and on the other side of four cubic and two octahedral faces. The twin growth is usually manifested by deep furrows running irregularly across a largely developed octahedral face on thin tabular crystals. If the furrow starts from an edge of the plate, it either traverses the plate to an opposite edge; or it bends round and returns to the same, or to a neighbouring edge. In the latter case it encloses a portion of one individual interpolated at one side of the other. Occasionally the furrow completely encloses a definite portion in the midst of the plate, the furrow forming a closed curve. If the twin is broken across a furrow, the twin-orientation is re- vealed by the different positions of the cubic cleavages on its two sides. Fahlerz. An ideal twin of fahlerz, in which a triad axis [111] is the twin-axis, and a face (112) parallel to it is the combination- plane, is shown in Fig. 428 (after Sadebeck). Owing to the tetra- liedral habit of fahlerz, two modifications of the twin are found. In one, the two like tetrahedra /-(.{111} are united in a plane (112), which, as in Fig. 428, modifies a like edge of both crystals. Two of the tetra- hedral faces Til and (111) are co-planar: they are perpendicular to the twin-axis and are represented by the clotted lines. The other tetrahedral faces meet in pairs in the combination-plane ; one pair, having the symbols 111 and (111), form a re-entrant angle of - 2 x 70 32' = - 141 4', and the two other pairs form equal salient angles, viz. TlT A (HI) = HI A (TlT) = 56 15'. In the second modification the positions of the two individuals in Fig. 428 are interchanged, their faces retaining the same relative directions, i.e. the individual with barred letters is transferred to the right hand of that having unbarred letters. They are united along a face 112 truncating the short parallel dodecahedral edges which, in the previous case, are to the extreme right and left. By copying the individual on tlie left on tracing pai>er, and transferring the copy to the right of the unbarred individual, FIG. 428. 474 TWINS OF THE CUBIC SYSTEM. the student can easily make a diagram which gives fairly well the relations of the twin. The co-planar faces 111 and (111) are united in the short edges [ofo], and the common face resembles the figure produced by joining two deltas at their apices. The two pairs of tetrahedral faces, which in the first modification, meet in salient angles now meet in equal re-entrant angles of 56 15' : the remaining pair of faces o do not meet. Interpenetrani twins. 12. Fluor. Fig. 429 represents an ideal interpeuetrant twin of cubes of fluor having p"p tl for twin-axis. Those cubic edges, which do not meet on p"p tt , bisect one another at the points S', 8 t , fee., of Figs. 423 and 429. The edges of the rotated cube meeting at p" and p /t are equal and parallel to the rotated axes in Fig. 423. The directions arid length of these edges being known, all that is needed is to complete the parallelograms which represent the faces of the rotated cube. The individuals are so related that a wedge- like portion of one cube, such as p^So", protrudes from a cubic face of the other. The re-entrant angles at which the faces intersect are of two kinds ; the angle 100 A (GOT) = 010 A (OOT) - - 48 11', and the angle 001 A (OOT) = - 70 32'. P' FIG. 429. The crystals are not intergrown as regularly as is represented in the figure. Usually a small portion of a second individual protrudes TWINS OF FLUOR AND SODALITE. 475 in twin-orientation from a face of the cube, and similar but ap- parently unconnected portions of different magnitudes often protrude from other faces. But however numerous and independent these several portions may seem to be, they are all twinned to the large individual about the same triad axis, and there is no repetition of the twinning about the other homologous axes. 13. Sodalite. A remarkable interpenetrant twin of sodalite, Na 4 (AlCl) Al 2 Si 3 O 12 , shown in Fig. 430, is sometimes observed in crystals from Vesuvius. This twin consists of two rhombic dodeca- hedra twinned about one of the triad axes. As in the twin of blende, the six faces of both individuals parallel to the twin-axis coalesce to form a hexagonal prism. In this case, however, each of these faces is di- vided into four parts, which, for want of precise knowledge as to the boundaries of the individuals, we may consider to be equal ; the boun- dary-lines join points like those marked xx and nn t on one of the FIG. 430 composite faces. Two portions of this face belong to 101 and two to (TOT), arranged cross-wise so that opposite segments belong to one individual and are in like orientation. The terminal faces meet in two sets of six like edges. The one set of edges, like AR ti , is shown by continuous lines: they are the edges of the rhombic dodecahedra, and the angle over each of them is 60. These edges also lie in pairs, interchangeable by a semi- revolution about the twin-axis, in planes parallel respectively to two faces of the hexagonal prism ; as, for instance, the pair AR t/ , AR lt , which are parallel to the face A'A^A'A,. The other set of edges, like R t x, is formed by faces of different individuals, and the angles over them are re-entrant, each being 38 57'. In the diagram the dimensions are such that an equably de- veloped rhombic dodecahedron can be formed by prolonging each edge of either individual whilst the cubic coigns remain unchanged. In the specimens at Cambridge the prism edges, such as A'A lt are much elongated as compared with the terminal edges, AR lt , elongs to one individual L. c. 31 482 TWINS OF THE TETRAGONAL SYSTEM. fills the space included between the axial planes, and each face belongs to a single individual. By drawing the truncating planes further away from the centre, so that only a portion of each of the projecting pyramids is cut off, a mimetic octahedron having grooved edges, like Fig. 436, is produced : the sides of the grooves are faces of the tetrahedron which is complementary to that to which the octahedral face belongs. The same explanation may be given of the crystal of diamond, Fig. 294. This can be derived from twinned hexakis-tetrahedra which would somewhat resemble the twin of eulytine, Fig. 432. Each ditrigonal coign is then modified by a tetrahedral face (111), which cuts off the greater portion of the protruding pyramid. ii. TWINS OF THE TETRAGONAL SYSTEM. A. Twin-axis the normal to a face of {/iQl} or {hhl}. 19. Cassiterite and rutile. These twins have a face of the pyramid {101} for twin-face. In the twin of cassiterite, Fig. 438, a face a of {100} of both individuals is co-planar: the twin-axis is therefore parallel or ^^^ perpendicular to this face. But the ~"~^ x > ** normal to a is a dyad axis ; and a semi- revolution about it gives an orientation identical with the first. The twin-axis is therefore parallel to the face, and, since the crystal is symmetrical to the face and to a centre, the twin-axis may be either perpendicular or parallel to the FIG. 438. combination-plane. By a semi-revolution about the normal (Oil), the faces on the lower half of Fig. 438 are brought into parallelism with those on the upper half. By a semi-revolution about the line in the combination-plane parallel to [as] = [100, 111], the lower half is brought into congruence with the upper half, and the twin is represented as if formed by the union of two like halves of separate crystals. The twin is also symmetrical with respect to the combination-plane. We shall, however, speak of it as a hemitrope with Oil for twin-face. In both minerals the faces a and m are usually well developed. TWIN- AXIS OF CASSITKUITE. 483 In cassiterite the faces in their zone are striated vertically : this character enables us to distinguish them from the faces s, which have a somewhat glazed aspect. In Fig. 438 the faces present make salient angles with one another; but when the prism-edges are short, faces * make a trough with two re-entrant angles in the neighbourhood where m it m and a t meet. The prism-faces m make two salient and two re-entrant angles of equal magnitude. 20. To determine the direction of the twin-axis in a doublet similar to Fig. 438. i. When the faces a of both portions are seen to be co-planar, measurement of the salient or re- entrant angle a A a, suffices. For C in Fig. 439 being 001 and 7* (Oil), we have a A a, = 2CT; and in cassi- terite CT=33 55', and in rutile 32 47-25'. Therefore a' A a, is 67 50' or 65 34-5'. Measurement of any one of the angles, m A m, m i A ', or s f\ s also suffices ; for m/ Am' = -rA=180 -2wi / 7 T , and s A = 180 - 2*7 t . For cassiterite, m t T= 66 45', m t A m' = 46 29' ; * A * = 38 29' ; rutile, =6729, =45 2; =4239. ii. When however the faces a are not present, measurement of two of the above angles is needed to establish the position of the twin-axis. Thus, in Fig. 439, measurement in the drawn zones [7W|, \in t Tm^ of the angles s A & c -> as shown in Fig. 444. The presence of other faces m and e {101} near these edges much facilitates the detection of the twin character. The trapezohedral faces 12 , &c. are generally very even and smooth, and show little or no trace of their composite character ; but one of the specimens in the Cambridge Museum shows a well-defined re-entrant angle. Unfortunately the faces of this specimen are too dull to admit of measurement, and it is thus impossible to ascertain whether the faces meeting in the edge make the angle 12' between 8 and ttj, which regular twinning from 1 to 8 requires. Nest-like octets of the black variety (nigrine) found in Arkansas, U.S.A., also occur, in which the twin-axes have the same ar- rangement and lie in a zone [m? 7 ]. The difference in habit arises from the fact that the faces developed belong to the ditetragonal prism {210} the form denoted by the letters I in Fig. 441. The faces I are deeply striated parallel to the principal axis, and these axes zig-zag across the plane [mT] containing the twin-axes in the same way as the edges [ra^], &c., in Fig. 444. The prisms are usually slender, so that a depression is left in the middle of the twin. 25. Multiple twins of cassiterite in which individuals are twinned about homologous faces e of the pyramid {101} also occur; but the twin-axes of the successive components rarely remain in the same plane. A frequent habit is that in which a prism i{110} forms the larger portion. Suppose now a cup-like depression to be formed at one end of this prism by the four pyramid faces e parallel to those which in the simple crystal form the salient pyramid at the opposite end. Along each face of the tetragonal cup a small indi- vidual is united in twin-orientation with the larger prism. Four individuals are similarly united to the prism at the other end. We TWINS OF ZIRCON. 491 thus get a twin of nine individuals; the four small individuals at each end are not in twin-association with one another, nor with those at the other end ; but the pair of components situated at opposite coigns of the prism are in like orientation, and may, if they touch, be regarded as portions of one individual. Complete twins of this kind are very rare. 26. Zircon. Although the crystals of zircon are isomorphous with those of rutile and cassiterite, it was not known to twin until Herr O. Meyer discovered microscopic twins in rock -sections (Zeitsch. d. Deutsch. geol Ges. xxx, p. 10, 1878). Shortly after- wards macroscopic twins, similar in habit to the doublet repre- sented in Fig. 438, were discovered in Renfrew Co., Canada ; and were almost simultaneously described by Mr. L. Fletcher (Phil. Mag. [v], xn, p. 26, 1881) and Mr. W. E. Hidden (Ami Jour, of Sci. [iii], xxi, p. 507, 1881). The twin-face is, as in the twins of rutile and cassiterite, a face of the pyramid {101}, and the principal axes of the two portions are inclined at 65 16' to one another. A similar twin and five other twins of zircon from Henderson Co., N. Carolina, U.S.A., have recently been described by Messrs. Hidden and Pratt (Am. Jour, of Sci. [iv], vi, p. 323, 1898); to whose courtesy and that of the Editor of the Journal I am indebted for the follow- ing figures illustrating the twins. The five twins are intercrossing doublets in which the twin-face is parallel to a face of different pyramids of the series [IM\ : a face m of both individuals is co- planar in all of them. Fig. 446 represents an intercrossing doublet having (Oil) for Fio. 446. Fio. 447. twin-face: the forms are m {110} and p {111}. The twin from Renfrew Co. has the same twin-law, but the individuals do not 492 TWINS OF THE TETRAGONAL SYSTEM. cross : it can be drawn by omitting the prolongations beyond the combination-plane of the individuals in Fig. 446. FIG. 448. FIG. 449. FIG. 450. In Fig. 447 a pair of faces p, parallel to the combination-plane, and also a pair of faces m perpendicular to the former are co-planar. The normal to the prism-face m being a dyad axis, the twin-axis is the normal to the face p(lll), and the combination-plane is parallel to this pyramid face. Since Op = 90 mp - 42 10', the two principal axes, given by the prism-edges [mm] and [mm], are inclined to one another at angles of 84 20' and 95 40'. Fig. 448 represents a similar intercrossing doublet in which the angle between the edges [mm] and [wm] was found by measurement to be 112 23' and 67 37'. The authors take a face (553) for twin- face, which corresponds with an angle between the tetrad axes of 112 57'. In Fig. 449 the twin-face is (221), the calculated angle between the principal axes being 122 12' and the measured 122. In Fig. 450 the twin-face is (331) ; the calculated and measured angles between the principal axes are 139 35', and 139 10', The authors have not given a figure of the twin in which (774) is twin-face. The corresponding angle between the principal axes is 115 30'; and their measurements gave 115 49'. 27. Copper pyrites. The simple crystals of this substance were described in Chap, xiv, Art. 19. The twins are combined according to three different laws : (1) juxtaposed and interpene- trant twins having the normal (111) for twin-axis, and in the former case the face (HI) for combination- plane ; (#) complementary interpenetrant twins geometrically very like the twinned tetra- hedra, Fig. 435 ; (3) juxtaposed symmetric twins with a face of the pyramid {011} for combination-plane. TWINS OF COPPER PYRITES. 493 1. Twin-face (111). Since the angles of the sphenoids, o = K{lll} and to intersect in this plane, in the same way as occurs in the juxtaposed twins according to the first law. In the TWINS OF THE TETRAGONAL SYSTEM. composite figure the four faces which have \CA^\ for zone-axis are single, and the four other faces are each composed of portions of dissimilar faces inclined to one another at very small salient and re-entrant angles. When the angle too/ is 108 40', the crystal- element = A COT is 44 34-5' : the angle To is 89 18-25', and the salient angle ow and the re-entrant angle ow' are both 1 2 3 -5'. The salient and re-entrant angles between the corresponding portions of faces at the back of the figure have the same values. A composite crystal, similar to the ideal one shown in the figure, is an asymmetric twin with (Oil) for twin-face: one instance of it from the Junge HoheBirke Mine, Frei- berg, has been described by Sadebeck. The orientation of the two portions in the above composite figure is also given by a semi-revolution about the line MM, which is the zone- axis parallel to the edge [CA t ]. But in the twin represented by Fig. 452 the faces meeting in the edges [cow], [oo] are like faces; and the twin is physically as well as geometrically symmetrical with respect to the combination- plane through these edges. Such an orientation of the portions composing the doublet can be obtained by taking two equal octaids placed with the faces o and to of the one similarly situated respec- tively to the faces o and w of the other. One of the octaids is then turned through 90 about its principal axis, when its faces of Fig. 453, in which A,c<=45 is one-half the angle of rotation about the principal axis, and f\ce l (&=9Q a is half the angle of rotation about the twin- axis. Hence o> (ill) is the extremity of the equivalent axis of rotation ; and cos e,o>c = sin 45 cos (ce t = 44 34 - 5'). Therefore the internal angle e t is 59 45-3'. Twice the external angle is 240 29'4' = 360 - 1 1 9 30"6'. The rotated portion is in the same position as if it had been turned once about the normal ( 1 Tl ) through 1 1 9 30'6'. Fio. 453. 496 TWINS OF THE TETRAGONAL SYSTEM. The twin-orientation is completely denned by the statement that the two portions are physically and geometrically symmetrical with respect to the combination- plane. The twinning according to this law is often repeated, and figures somewhat resembling octrahedra can be obtained in which each face consists of portions of three like faces which are nearly, but not quite, co-planar. The portions of these composite faces are separated by grooves at which well-marked striae meet in a manner similar to that shown between o and o in the doublet, Fig. 452. B. Twin-axis the normal to a face of {100} or {110}. 28. Scheelite. This substance affords good examples of simple juxtaposed twins and of complementary interpenetrant twins having the same axis of rotation. In these twins, which have been ably described by Professor Max Bauer (Jahr. Ver. Wurtt. p. 129, 1871), the axis of rotation may be taken to be parallel to any one of the horizontal edges of one or other of the tetragonal pyramids e {101} and o {111} (see Fig. 209, p. 262). For the sake of simplicity, we shall suppose the horizontal edge of e{101}, which was selected as the axis of X, to be the twin-axis. Fig. 454 is a basal plan of two crystals having the faces given in p. 262 and placed side by side in twin-orientation. The two crystals are then reciprocal reflexions in a plane through the common edge parallel to the principal axes. The individual to the right can be brought into a like orientation with that on the left by a semi-revolution about the common edge or about any of the edges in which the faces e and o meet the plane Fio. 454. of the paper: it can also be brought into a like orientation by a quarter-revolution about its principal axis. As far as the twin- orientation is concerned, it is immaterial which of the axes is TWINS OF SCHEELITE. 497 selected, provided the rotation is 180 if about a horizontal axis, and 90 about the principal axis. The juxtaposed twins are sometimes combined in a plane which is approximately the plane through OX and the principal axis. The faces e meeting in the combination-surface are co-planar ; but the irregular line of separation traversing the face is easily recognised, being the axis of more or less strongly marked barbed striae parallel respectively to the edges [eV] of the two portions. Occasionally an indentation, formed by faces o and s of the two individuals, is seen where the line of separation of the portions e 3 and e 3 meets their horizontal edge. Fig. 455 is a plan on the common basal plane of an interpenetrant twin consisting of eight segments which are united together along vertical planes in the same way as the doublet. As in that twin, the faces e divided by interrupted lines are co-planar and frequently show fine barbed striae meeting in the line of separation. The portions meeting in the polar edges of the geometrically simple pyramid e are also in twin- orientation to one another ; and this is sometimes proved by the way in which the pyramid -coigns are modified by faces h, o, and belonging to the forms 7r{313}, {111} and 7r{13l}. The faces o at each coign are co-planar ; and the faces 8 form an indentation similar to that occasionally seen in the middle of the horizontal edge of a face e. The surfaces of separation are not, as a rule, true planes, and the lines of separation are somewhat irregular. Twins of this kind are not common ; they are principally found in Schlaggenwald, Bohemia. Professor Bauer describes twins of the same kind united along a surface more or less approximately parallel to the base. Further, the relative dimensions of the individuals are very unequal ; and it often happens that a mere corner is, as it were, cut out of a crystal and replaced by an equal amount of matter in twin-orientation. FIG. 455. iii. TWINS OP THE PRISMATIC SYSTEM. 29. In crystals of this system the twin-axis is commonly the normal either of a prism {hkO}, or of one of the domes {Qkl} or {hOl} ; L. c. 32 498 TWINS OF THE PRISMATIC SYSTEM. and twins of this kind are especially frequent when the faces of the form make angles of nearly 60 and 120 with one another. The twins are often juxtaposed with the plane perpendicular to the twin-axis for combination-plane : the twin-orientation may then be given by stating the twin-face. As instances of substances, having for twin-face a face m of a prism {110} with angles of nearly 60, we have: redruthite, Cu 2 S, mm, = 60 24'; mispickel, FeAsS, mm, = 68 13'; stephanite, Ag 5 SbS 4 , mm =64 21'; aragonite, CaCO 3 , alstonite, (Ca,Ba)CO 3 , witherite, BaCO 3 , strontianite, SrCO 3 , and cerussite, PbCO 3 , in which mm, varies between the limits 63 48' and 62 15'; potassium sulphate, K 2 SO 4 , and ammonium sulphate, (H 4 N) 2 SO 4 , ram, = 59 36' for both. As instances of domes with angles of nearly 60 between the faces, one of which is perpendicular to the twin-axis, we may mention : marcasite, FeS 2 , 101 A10T = 63 40'; mispickel, 101 A10T = 59 22'; manganite, Mn 2 O 3 .H 2 O, Oil A Oil = 57 10' ; chrysoberyl, BeAlO 4 , Oil A Oil = 60 14' and 031 A 03T = 59 46'. Kokscharow gave i(011) as the twin-face of chrysoberyl, Hessenberg and Cathrein make p (031) the twin-face. The angle P ,i = 03T A Oil =89 46' ; and it is not easy to distinguish between the two representations of the law of twinning, when the crystals are developed, as is frequently the case, in large triplets or sextets, which do not admit of accurate measurement, and are not sufficiently translucent to enable us to determine the optical orientation of the several portions. Another class of twins similar to the preceding is that in which the twin-axis is perpendicular to a face of a prism or dome, the faces of which are inclined to one another at angles of nearly 90. Thus, bournonite forms multiple twins, like Fig. 456, with m (110) for twin-face, mm, being 86 20'. Staurolite, H 4 (Fe,Mg) 6 (Al,Fe) M SinO^ (?), is another instance. Twins with the normal to a pyramid-face for twin-axis are rare. In Art. 41 we shall describe one of staurolite, in which a normal to (232) is the twin-axis. Aragonite group. 30. The determination of a doublet of aragonite, such as those shown in Figs. 457 and 458, is easy ; for the prism-edges are all parallel, and the basal planes of the two portions are, when present, TWINS OF ARAGONITE. 499 co-planar. The twin-axis is therefore parallel to the base ; for the normal to the base is a dyad axis, a semi-revolution about which leaves a crystal in an orientation identical with the first. Again, it is easy to see in the crystal, as is evident from the basal plan, Fig. 457, that two prism-faces m and m' are parallel : the axis of semi-revolution is therefore perpendicular to these faces. Further, since the crystals are, as a rule, translucent, it is often possible to see, by total reflexion from a plane surface traversing the doublet, that the indi- viduals are combined along a plane parallel to Flo< 457- m t . Hence the doublet has the face m t (HO) for twin-face. In the zone [ra^m] we can measure the following angles ; and confirm the accuracy of the determination of the twin-face made bv inspection. mtn i = mm' = 63 48', bm = bm i = bm' = bin = 58 6', all angles of the simple crystal. Again, we have the following angles between faces on different individuals : bb = 1 80 - 2 x 58 G' = 63 48' = - bb t , mm = 180 - 2 x 63 48' = 52 24', and mb, = mb = -5 42'. Provided the faces can be distinguished with certainty, measurement of any one of these angles suffices to give the positions of all the faces, and of the twin-axis. The faces k and k of the domes {011} make at each end of the prism a salient and re-entrant angle, both of equal magnitude. Since m t k' = 72 1', k'k, = - kk = 180 - 2 x 72 1' = 35 58'. 31. To draw the basal plan, Fig. 457, two lines intersecting in at right angles are taken in the paper, and convenient lengths OA, OA t , and OB, OB t are measured off on them in the ratio a : b. A series of lines parallel to OA are drawn at distances apart corresponding fairly with the width of the faces k, k' and c, when the crystal is viewed endwise. Lines parallel to AB and AB t give the traces m, m t ; and complete the simple crystal. This is then bisected by a vertical plane passing through parallel to m t : the lines of section are shown in the edges [A'A,], [kk]. To get the rotated axes, the twin-axis is drawn through the origin perpendicular to A it and produced to an equal distance beyond AB. The point Q so found is the origin of the rotated axes, 322 500 TWINS OF THE PRISMATIC SYSTEM. which are given in direction and magnitude by flA t , QB, and a length c on the vertical. From the points in the trace of the bisecting plane the edges [6&J, [^/]i *^ c -> are drawn parallel to O4 / to meet lines from the coigns bmk, &c., parallel to the twin- axis. Through the points so found, the traces m, m' are now drawn, completing the figure. To make the clinographic drawing of the doublet, Fig. 458, the cubic axes are projected in the way described in Chap, vi, Arts. 22 and 23, and lengths are measured off on them in the ratios a : b : c, as described in Chap, vi, Art. 14. The simple crystal represented by unbarred letters can then be drawn, and bisected by a central plane parallel to the twin-face m i (110). The directions of the rotated axes of X and Y of the doublet are found in the same way as the rotated axes of Z and Y were found in Art. 22. The points B and A t on the axes of Y and X are joined, and a point M in BA t determined, where BM : MA f = cot 2 F : 1 = cot 2 31 54' : 1. The line OM is produced to where QM= MO. The rotated axes of X and Y are O^ and SIB ; the axis OZ remains vertical. Lines are now drawn through the points in which the edges of the fixed portion meet the combination-plane parallel to dA t . They give the edges \b_1e\, [/&'], &c., of the rotated portion. The coigns on these edges, such as b t mk, are the points in which the edges meet lines parallel to the twin-axis drawn through the corresponding coigns of the fixed portion. The figure can then be completed by drawing the vertical FIG. 458. 32. Multiple twins often occur, in which the same normal is the twin-axis of all the individuals. Three indi- viduals thus twinned are shown in Fig. 459, in which individual n appears as a thin plate sand- wiched between the two outer individuals which necessarily have the same orientation. Very thin lamellae twinned in this way may be often perceived in apparently simple crystals by the linear interruptions which they produce on the faces k and b. They may also be discerned in sections by internal reflexions and by the con- fusion they produce in the optical phenomena seen in a polariscope. FIG. 459. TWINS OF ARAGONITE. 501 Fio. 460. If four individuals are thus twinned the two outer portions will be in twin-orientation, and the twin will only differ from a doublet in having two twin-plates interposed. The process may be further repeated in a similar manner, so that the outer individuals are in similar orientation when the number of individuals is odd ; and in twin-orientation when the number is even. 33. When the individuals in a doublet are continued beyond the twin-face, i.e. intercross, they are ar- ranged in the way shown in the plan, Fig. 460. Tn this diagram the combination- plane is projected in the line aa,; and there are two re-entrant angles kk and k'k f = - 35 58' over this plane. The vertical plane containing the twin-axis TT t is not parallel to a possible face ; and the re-entrant edges in which the faces kk f and kk' meet are not straight and regular. The plane approximates to the face (130), for b A 130 = 28 10' and m,A!30 = 93 44'. Were the intercrossing doublet formed of individuals having the forms {110} and {001} only, and were the prism-faces m so extended as to meet at the points TT t and aa, we should have a pseudo- hexagonal prism having four angles mm i - 63 48' and two angles at a and a, each equal to 52 24'. The bases would be co-planar for they are parallel to the twin-axis : they would each consist of four seg- ments divided by the vertical planes through TT t and aa t . Usually, however, such pseudo-hexagonal composite crystals are formed of three, or more, indi- viduals; and as shown in Fig. 461, the combination may occur very irregularly. In this figure the twin is regularly developed for five portions; but the sixth is replaced by two smaller individuals, which are in twin-orientation to the adjacent larger por- tions, but are not in regular association , ; , with one another. Fio. 461. 34. Witherite and alstonite. These crystals are all multiple twins conforming to the same laws of development as the pseudo- 502 TWINS OF THE PRISMATIC SYSTEM. hexagonal twins of aragonite. Witherite is usually in short pseudo- hexagonal prisms terminated by one or two pseudo-hexagonal pyramids. The section, Fig. 462 (after Des Cloizeaux), is cut perpendicularly to the common vertical axis. It shows that, in its main features, the crystal is an intercrossing triplet, in which the faces forming the hexagonal prism all belong to the pinakoid I {010}. The series of lines marked m indicate that each individual of the triplet is traversed by numerous lamellae, each series being FIG. 462. parallel to the same twin-face m of the form {110} in the manner illustrated by Fig. 459. The lines connecting two circlets show the directions of the plane of the optic axes of the main portion of the segment in which each of them lies. The crystals of alstonite appear as acute hexagonal pyramids with striated faces, on each of which an interruption, approximately in a plane corresponding to that marked TT \ in Fig. 460, shows the composite nature of the face. A section parallel to the base shows that the crystal is an intercrossing sextet consisting of twelve segments. Each segment is bounded by a well-defined line, which lies in the vertical plane containing opposite edges of the pyramid. But the dividing lines joining the middle points of opposite horizontal edges are irregular. The line of extinction between crossed Nicols is in each segment inclined at nearly 30 to the horizontal edge of the pyramid, and the plane of the optic axes is parallel to (100) and Bx a to OZ. The composite faces belong to a prismatic pyramid of the series {hhl}, and not, as in witherite, to a dome {Okl}. 35. Cerussite. Intercrossing triplets with m(110) for twin- face are common, and often have the habit shown in Figs. 463 and 464. The faces b {010} are often striated horizontally; and the shading in Fig. 463 has been employed to show the depth of the re-entrant angles. In Fig. 463 the unlettered portions to the right and left belong to a single crystal having its face 6(010) parallel to the paper : the portion carrying p and m is twinned to this crystal about the normal to the prism-face lying to the right ; and as shown by the absence of dividing lines between the pyramid-faces, the prism- and pyramid- faces to the right are co-planar. Similarly, the portion bearing p T\VI\S OF POTASSIUM SULPHATE. 503 and m is twinned to the unlettered crystal about the normal to the prism-face to the left ; and the prism- and pyramid-faces to the left FIG. 463. Fio. 464. are co-planar. But the portions bearing letters are not in twiu-orien- tatioii with one another ; for the prism-angle mm i - 62 46'. Hence mAm = -818'; and p f\p = - 6 44', Cp = 00l Alll being 54 14'. In Fig. 464 the faces k belong to {Oil}, and v to {031}. 36. Potassium sulphate. Twins of this substance having much the appearance of hexagonal prisms and pyramids, similar to those of aragonite and witherite, are obtained by evaporation from aqueous solution. Sections of them are easily made, so that the character of the twin-orientation can be determined by the examina- tion of plates in plane-polarised light. Fig. 465 represents an actual doublet which was fully determined in the Cambridge Museum. The forms observed were: o{lll}, a {100}, ?{110}, /{130J, and two or three faces irregularly developed and alternating, so as to form step-like grooves where they met the combination-plane: they were {011} with {021} and possibly {031}. The faces o, a, m and / were smooth and bright, and gave good images. The face /, / is composite, .and the two portions make a very small salient angle with one another. The principal angles are : am = 29 48', mf= 30 0', // = 89 36', om = 33 39' and oo" = 48 52'. Hence a : b : c = -5727 Fio. 465. 7494. The plane of the optic axes is (100) and Bx a is parallel to OZ. 504 TWINS OF THE PRISMATIC SYSTEM. Hence a plate, ground down on roughened glass perpendicularly to the edges [am/*], gives an optic figure in convergent polarised light, in which the line joining the 'eyes,' shown in the diagrams by a line joining two circlets, is perpendicular to the edge \bc\. 1. In the doublet the twin-axis is the normal to (iTO), and the two individuals are combined parallel to this face. From the angles given above for the simple crystal, we can compute the angles mm,^and oq: they are mm=180 2mm / =6048 / , ^=180 -2x8936' = 48', and oo = 180 - 2mo = 49 49-5'. The salient angle ff was found by measurement to give values varying between 35' and 46' ; the other measured angles agreed well with the theoretical values. A section of the crystal, Fig. 466, was carefully prepared, leaving as far as possible the prism-faces and portions of the pyramid-faces intact, so that the position of the plane of the optic axes could be accurately correlated with the external form. The directions of the planes of the optic axes are shown by the lines joining circlets, and prove the twin-face to be m y (110) : the trace of the combination-plane is shown by a line of alternating strokes and dots. The triplets are sometimes very regular, and closely resemble a hexagonal prism terminated by a hexagonal pyramid. When sections of such triplets are made, several different arrangements of the component individuals are found. FIG. 466. i. The hexagonal section may consist of three rhombic segments, Fig. 467 (), the combination-planes passing through alternate edges of the hexagonal prism and meeting in a central axis. The faces TWINS OF POTASSIUM SULPHATE. 505 of the pseudo-hexagonal prism are all prisra-faces m{110}, and the pyramid is formed by o faces. The plane of the optic axes is in each segment parallel to the longer diagonal of the rhombus. ii The hexagonal section may consist of six similar triangular segments, Fig. 467 (6), and the combination-planes pass across the central axis and through opposite edges of the prism. The plane of the optic axes in opposite segments is coincident, and is, as shown by the line of eyes, perpendicular to the face of the bounding prism. These prism-faces must therefore all belong to the pinakoid b {010}, which is at right angles to the plane of the optic axes. iii. Another common arrangement is shown by the section, Fig. 467 (c), in which two-thirds of the section belongs to a single crystal, to which two other small segments are twinned. Four of the external faces of the hexagonal prism belong to the pinakoid, and two to the prism m ; but the prisms do not as a rule give very regular hexagons, although, as shown in the diagram, the faces m and b are sometimes very nearly equal in size. As indicated by the irregular line the two inserted segments are not in twin-combina- tion with one another. The re-entrant angle mm t within the large individual is 180 - 59 36' = 120 24' : if then two wedges, bounded by m faces, are inserted, the joint angle is 2 x 59 36', and a thin wedge having an angle of 1 1 2', would be left vacant in the place of the irregular line. The tilling up of this gap takes place irregu- larly and produces a want of distinctness in the optic character in its neighbourhood. The above and other sections made by the students in the Cambridge Museum were all found to obey the same twin-law. 2. Scaccbi gives for potassium sulphate a second twin-law in which the twin-face is /(130) ; and he states that the twin occurs frequently, when a small amount of sodium sulphate is added to the solution of potas- sium sulphate. Many efforts were made by the author to obtain such twins, but without success. In sections of regular sextets or intercrossing triplets according to this law, we should have a hexagon bounded by faces m as in Fig. 467 (a) ; but the combination-planes bisect opposite sides of the hexagon forming trapezia, which each include one of its corners. The planes of the optic axes in these segments are perpendicular to the line joining the edge [mm,] to the centre. In a doublet of this kind faces m, and m', o' and o, become almost co-planar. On one side we should have a salient angle m, m' = 48', and on the opj>osite side a re-entrant angle m'm,= - 48 ; and the adjacent faces o make similar angles of 4(X. 506 TWINS OF THE PRISMATIC SYSTEM. Fm. 468. 37. Mimetite. The twins of witherite and potassium sulphate which have been described in the preceding Articles enable us to explain the mimetic twins of mimetite, which so closely resemble crystals of the hexagonal system that their true character was only discovered in 1881. The common habit of the crystals is shown in Fig. 468, and the crystals are so perfect that they were re- garded as an undoubted member of the isomorphous series of hexagonal crystals of class II formed by apatite, pyromorphite and vanadinite ; and this belief was supported by the fact that the chloro-phosphate, pyromor- phite, and the chloro-arsenate, mimetite can only be distinguished by analysis, and are often intermingled in one crystal. M. Bertrand (Bull. Soc. /rang, de Min. iv, p. 36, 1881) discovered that sections, cut parallel to the face c, are composed of six triangular segments similar to those of witherite and potassium sulphate shown in Figs. 462 and 467 (&), respectively. In the crystals from Johanngeorgenstadt, Saxony, the plane of the optic axes in each segment is parallel to its external side, i.e. to the face m of the pseudo-hexagonal prism, and the section resembles that of witherite without the twin lamellae ; the acute bisectrix is parallel to the prism-edges [mm], and the angle of the optic axes is about 64 in air. The crystals are therefore sextets of the prismatic system similar to those of witherite and potassium sulphate, having a face of the pinakoid {010} for external face, and twinned about a face (110) ; the angle 010 A 110 being 60, or very nearly 60. The crystals from Roughten Gill in Cumberland containing both phosphoric and arsenic acids were found to consist of a central prism of pyromorphite which is uniaxal, and of a surrounding zone of mimetite formed of six biaxal segments. The planes of the optic axes in these segments appear to be parallel to the diagonals of the hexagonal section, and not to the external sides of the segments. A plate cut perpendicularly to the prism-edges of a crystal from Wheal Alfred, Cornwall, was found by the author, on examination in the polariscope, to be irregularly divided into segments, in which the light is extinguished in directions making nearly 12 with the prism-edges. In convergent light the hyperbolic brushes of a biaxal plate having a small angle between the optic axes are seen. Further, TWINS OF CHRYSOBERYL AND MISPICKEL. 507 the main segments are traversed by narrow plates in a manner similar to that in the section of witherite shown in Fig. 462. 38. Chrysoberyl. The orientation of a prismatic crystal adopted in its representation is arbitrarily chosen, and it happens that chry- soberyl has been so placed that the twin-axis is perpendicular to a face of a dome {Okl}, which may equally well be placed with its edges vertical, when it would be called a prism. Consequently, in the in- tercrossing triplets and sextets of chrysoberyl, Figs. 469 and 470, FIG. 469. FIG. 470. the twin-face is (031), although they are similar to the twins of witherite, alstonite and potassium sulphate discussed in the preced- ing Articles. In Fig. 469 the faces shown are; a {100}, o{lll}, n { 1 21 }, b {010}. The faces n and b make in some cases indentations in the pyramid-edges as shown in the figure. Now 010 A 031 -29 53', 010 A Oil =59 53', aAo = 438', and 1 11 A 111 = 40 1' : hence 031 A Oil = 90 14' and 031 A HI = 90 9'. If then (031) is the twin-face, the portions o and o separated by lines of interrupted strokes are not co-planar, but include a re- entrant angle of 18'; and the pyramid-edges in the paper include a re-entrant angle of 28'. If the face (Oil) is the twin-face, then o and o are co-planar, and the pyramid-edges are co-linear. Fig. 470 represents an intercrossing sextet with faces a {100} and i{011} alone developed. Each segment of the pinakoid having barbed striae is a doublet with (031) for twin-face; and these doublets are united along faces of {Oil}. 39. Afispickel and the isomorphous glaucodote, (Co, Fe) AsS, afford good examples of substances twinned according to two different laws. In one a face m of {110} is the twin-face; in the other the twin-axis is perpendicular to a face e of {101}, the individuals 508 TWINS OF THE PRISMATIC SYSTEM. generally interpenetrating as shown in Fig. 472, and the twins being sometimes juxtaposed sextets. Fig. 471 represents a large sextet or intercrossing triplet of mispickel from Cornwall in the Cambridge Museum, in which m(110) is the twin-face. The individuals having symbols in parentheses are twinned to the fixed individual, i.e. to that having unmodified letters and symbols, with 110 for twin-face; the in- dividuals having their symbols in brackets are twinned to the fixed individual with 110 for twin-face. The basal pinakoid is common to all the individuals, and the lines of separation in the combi- nation-planes are shown by straight lines. The individuals having differently modified symbols are not in twin-association with one another ; and this is indicated in the figure by an irregular line of separation. The faces c of the form {001} and u of {014} are much striated and the edges [cw] are not so well defined as is implied in the figure. FIG. 471. FIG. 472. To show the appearance of the twin more clearly, the vertical axis of the twin has been put in the position 08' of the cube, Fig. 226, p. 284 : the axis of X of the fixed individual is in 08* and OY in OT'. Hence a length 08' x c 4- >/2 = 08' x -84 gave the parametral length OC, and 08* x a+j2 = 08* x -479 gave OA. The positions of the twin-axes and of the rotated axes lying in the plane T'OP were found by the construction given in Art. 22. Fig. 472 represents an interpenetrant twin of mispickel in which the normal to e(101) is the twin-axis; the fixed individual being in the conventional position. The twinned individuals are usually of different sizes, and the axial plane XOZ of the one is rarely coincident, as represented in the figure, with that of the other. TWINS OF REDRUTHITE AND STAUROLITE. 509 Fio. 473. 40. Redruthite (chalcocite). This mineral is twinned according to three different laws, (i) twin-face m(110), (ii) intercrossing twins with (032) for twin-face, and (iii) with twin-face v(112). Fig. 473 represents a twin from Bristol, Conn., U.S.A., described by J. D. Dana, in which two laws occur. The basal pinakoid is placed vertically, and is striated parallel to the edge [be]. In the larger crystal a wedge formed of two indi- viduals twinned to the first crystal about faces m is inserted. The resulting twin is similar to that represented by Fig. 467 (c). To the large crystal another, in which the faces b and c are also ver- tical, is twinned with twin-face (032) : the angle between the pinakoids c being 11 TO' and 69 0'. 41. Staurolite. Figs. 474 479 serve to elucidate the twins of staurolite. The simple crystal is usually a stout prism 7n{110}, associated with the pinakoids b {010} and c{001}, and occasionally with r {101}. The angles are mm, = 50 40', bm = 64 40', cr = 55 16'. 1. Twin-aids a; (032). Fig. 474 is an ideal representation of a common twin. It consists of two interpenetrating crystals, which cross one another nearly at 90, in such a way that the faces b and c of the two individuals are tautozonal. The possible face with low indices which most nearly truncates the edge [be], i.e. makes 45 with b and c, is x (032) ; the angle ex being 45 41'. Adopting, as has been done in Fig. 474, the normal to this face as the twin-axis with an angle of rotation of 180, the pair of faces b make at the edge through g a re- entrant angle 010 A(010)=8838', and the faces c a salient angle 001 A (001) = 91 22'. The prism-faces meet in two sets of re- entrant angles : the edges [da], [ae] are in the plane parallel to (032) ; the edges [go] and [ah] are not in a plane parallel to a possible face. Usually one individual is much smaller than the other ; and the prism-edges [mm,], [mm 1 ] do not inter- sect, as in the figure. 2. Twin-axis (232). An ideal drawing of this twin is given in Fig. 476 : in it the normal to a possible face (232) is taken as the twin-axis; this normal makes angles of nearly 60 with c(001) Fio. 474. 510 TWINS OF THE PRISMATIC SYSTEM. and b (010) (see table of angles, p. 511). The individuals are, as in the first twin, rarely equal ; and seldom cross one another in the very symmetrical manner shown in the diagram. FIG. 475. FIG. 476. 3. Twin-axis (130). Fig. 477 represents a rare twin de- scribed by Professor E. S. Dana. The twin-axis may be taken to be (130) or (230). The former requires an angle 66 = 70 18', the latter an angle 70 46', whilst measurement gave 70 30'. Professor Cesaro (Bull. Soc. fran$. de Min. x, p. 244, 1887) has shown that the crystals may, in accordance with views propounded by Mallard in the same journal (vm, p. 452, 1885) to explain the formation of twins, be regarded as pseudo-cubic ; i.e. that the particles are p IG 477 arranged in a manner approximating to one of the arrangements possible in cubic crystals. Thus the crystal of staurolite is in some respects compared to one of pyrites ; the axis OX of the former being parallel to a cubic ejjge of the latter. This axis is a pseudo-tetrad axis ; and the zone [be] is a pseudo-tetragonal zone, having c#=4541' and 6^=44 19' as approximations to 45. Again, the poles of the octahedron {111} lie in zones each of which contains a pole of the cube and one of the rhombic dodecahedron, the angle ao being 54 44'. Now, the possible faces (130) and x(l 2 )> Fig. 475, of staurolite would together make a figure differing but little from the octahedron ; for a = 5451', ax = 54 12' and x = 7019'. The faces b and c then occupy the positions of the pair of dodecahedral faces parallel to the pseudo- tetrad axis OX. The remaining faces of the rhombic dodecahedron must lie in the zones \ax\ ; and their poles in Fig. 475 must make angles of nearly 45 with a and 60 with b and c. Computations from these requirements and the angles already given prove to be the poles of the possible pyramid (232). The following tables give some of the TWINS OF STAUROLITE. 511 most important angles between the actual and possible poles of staurolite, and the angles between some of the corresponding cubic faces. -am 25 20' Staurolite. prAl34 25 30' Cubic crystal. 100 A 311 25 14' ay (230) 35 23 a 54 fil Tar 34 44' a x 54 12 *x' M 65 12 29 ,,211 ,,111 35 54 K; it ab 90 lac 90 f '.in 58 ,,011 90 90 14 Lr x/ 88 56 f' A 134 68 28 211 ,,111 90 ' -f A 162 M 53 fM84 L; A 134 31 31 29-5 31 29-5 af 44 47 45 13 101 ,,101 100 001 By changing the parametral plane of staurolite, we can show the affinity to a cubic crystal in another way. If m is taken to be (310) and x(011\ the faces have the following symbols : r=(201\ x = ( 101 \ = (110) and (=(211). The parameters are now given by c -r b = cot &#= 1*024, and a 4- b = 3 tan am = 1 '420 = ^2, nearly. Hence a : b : c= 1-420 : 1 : 1-024 = ^2 : 1 : 1, nearly. The latter ratios are the parameters of a cubic crystal, when the axes of reference are a tetrad axis and the pair of dyad axes perpendicular to it, and two of the octahedral faces become (110) and (Oil), respectively. Now in a cubic substance the interfacial relations are exactly similar in azimuths about a dyad axis which differ by 180, and in azimuths about a tetrad axis which differ by 90. The arrangement of the particles about any point must have the same symmetry, although like groups of particles may only be interchanged by screw- rotation about an axis of symmetry. We may argue that similar relations hold for the possible arrangements of the particles in a pseudo-cubic substance ; and that the crystals can grow when the particles are in positions or in orientations differing by rotations of 180 or 90 about the pseudo-axes of symmetry from those positions or orientations which hold for the growth*bf the simple crystal ; the difference of azimuth admissible being 90 when the line of rotation is a pseudo-tetrad axis, and 180 when it is a pseudo-dyad axis. This hypothesis offers an explanation of the three twin-laws. The rectangular twin, Fig. 474, resembles the interpenetrating twins of pyrites described in Art. 15. The twin-axis should be regarded as OX, a pseudo-tetrad axis, and the angle of rotation 90. The angle cc should be then 90, and not 91 22'. The inclined twin, Fig. 476, has for twin-axis a pseudo-dyad axis, the normal f, and an angle of rotation of 180. The third twin, Fig. 477. has for twin-axis a pseudo-triad axis, if (130) is taken as the twin-axis; and it then conforms with the law of twinning common in cubic crystals. The method of drawing Fig. 476 will be now explained. In Fig. 475 the poles of the fixed crystal are represented by dots, a (100) being placed 512 TWINS OF THE PRISMATIC SYSTEM. at the centre and c (001) at the top : the poles of the rotated crystal are indicated by crosses and barred letters. From the table of angles given it will be seen that a semi-revolution about brings b to b, which is nearly coincident with ', and c to c nearly coincident with ,, ; and that the poles a and x nearly change places. Assuming that the rotation produces exact coincidence, we can find indices of the rotated prism-faces m as if they were, in their new positions, faces of the fixed simple crystal. For the faces lie in the zone [aft], which is now the same as [#'] = [623]. This zone intersects [ac] in ^(102) and [ab] in , (130). The angles x-^ and x, can be calculated from the right- angled triangles cx% and >/,, : they are, f\xx = 55 29', A #/, = 54 1 1 ?'. In the A.R. {xmx'} the symbols of three poles and all the angles are known, for xm = am = 25 20'. Hence m is (134). Similarly, from the A. R. {x &'}, the pole m' is (162). The poles m and m' are not exactly in [#'], but the displacement of m from (134) is only about half-a-degree. When m and m' are taken to be (134) and (162), it is easy by the rule of Chap, v to find the directions in which any pairs of the faces intersect, and therefore to draw the twin. Fig. 478 shows the relation of the principal zones which has led to the ideal drawing, Fig. 476. The zone [x, wi fz] of Fig. 475 is placed in the primitive. Now x, i 8 at 90 from 6(010) and at 88 56' from r(101) : it is therefore nearly the pole of the zone-circle [6fr], which is in Fig. 478 projected in a diameter perpendicular to x,X- Further, the angles f&, f& are nearly 60; and the zones [x,?"]? [x/ m /&]> [x/'*/l are nearly at 60 to one another. The pairs of faces m and m, m, and 6, m' and &,, will therefore intersect in a regular hexagon in a plane per- pendicular to xx,' this plane is the base of Figs. 476 and 479. Further, the axial poles c and a, and the axes OZ and OX, of each crystal lie in a zone-circle through x, bisecting the angle ff (Fig. 475) = {b (Fig. 478), where x/ c / = 3548' and x,a=5412'. A hexagon, Fig. 479, is now projected in the way described in Chap, vi, Art. 19, having the back-and-fore cubic axis for the diagonal tin, . The right- and-left cubic axis is the twin-axis ff, and the vertical axis is the normal XX,- From Fig. 478 it is seen that OB / and OB are in the plane of the hexagon at 60 to ff. They are the bisectors of the other pairs of opposite sides of the hexagon. Again OA" and On are unit lengths on the vertical axis and on a horizontal axis. But the semi-diagonal of a hexagon is equal to each of its sides. The sides of the hexagons therefore give unit length along any line parallel to them. Further, since the angles at B t , B and f are right angles, OB t = OB = Of = On cos 30 : the projected lines are therefore the unit length in their directions multiplied by cos30, i.e. by TWINS OF STAUROLITE. 513 The axes of Z and X of each individual are in the vertical plane perpendicular to the axis of Y : hence those of one crystal lie in the plane LOd perpen- dicular to OB,, and those of the other in LOd, perpendicular to OB. In the plane LOd construct a parallelogram OSXe having for sides OS=OA" sin 35 48' ; and Oe = 0dcos3548'. The diagonal OX is the direction of, and unit length on, the axis of X. This construction is identical with that given for the axis OX of an oblique crystal (Chap, vi, Art. 15). Similarly, the parallelogram LOfZ, with sides OL = CU"cos3548' and O/=Od8in3548', FIG. 479. has for diagonal the unit-length on OZ. The unit-lengths on OX, and OZ, are found by constructing the parallelograms OSX.e, and OLZJ,, where the points e, and /, are got by drawing lines through e and / parallel to ft". OB and OB, being left unchanged, the lengths OX and OX, are multiplied by a cos 30 giving OA and OA,, the lengths OZ and OZ, by c cos 30. The axes of the two crystals have been now projected, and forms present on either crystal can be drawn. Lines through the corners of the hexagon parallel to OZ and OZ, give the edges [mm,] and [bin] of the two individuals. Their lengths are immaterial, but, in any particular case, they should be made to correspond approximately with the lengths of the edges. The edges [cm] and [be] of the crystals are easily found; for [cm,] is parallel to AB,, [cb] to OX, and so on. Again, the faces in Fig. 476 intersect in a second hexagon gnp.... From Fig. 478 it is seen that b and b, are adjacent faces, and since their poles are in a plane perpendicular to xx> ( tne vertical line in the drawing), their line of inter- section is parallel to xx>- The re-entrant edge ng is therefore vertical. Again, it is clear that m and the face 110 parallel to m must intersect in a horizontal line parallel to rff of Fig. 479. The remaining edge np in which m, and m' intersect is the zone-axis [fw,], and is inclined to the vertical at an angle of nearly 34 45'. iv. TWINS OP THE RHOMBOHEDRAL SYSTEM. 42. As in cubic crystals, the triad axis is commonly the twin-axis in rhombohedral crystals, and it is immaterial whether the angle of rotation is taken to be 60 or 180 ; for azimuths about the axis which differ by 120 are interchangeable by a rotation about it of 120. For the sake of uniformity in the statement of the twin-orientation, it is usual to take 180 to be the angle of rotation. The combination-plane is often the base (111), when the law may be given by stating that the base is the twin-face. The crystals are sometimes united along a face of a hexagonal prism; and they 33 514 TWINS OF THE RHOMBOHEDRAL SYSTEM. FIG. 480. sometimes interpenetrate one another, as is the case in quartz and chabazite. Twins with inclined triad axes occur, and in them the twin-axis is generally the normal to a face of a rhombohedron or of a trigonal pyramid ; and the twins are most frequently combined along a plane perpendicular to the twin-axis. We shall discuss the twins of calcite and quartz, and describe briefly some interesting twins of a few other substances. Calcite. 43. 1. Twin-axis [111]. Fig. 480 represents a twinned fundamental rhombohedron with (111) for twin-face. This is sometimes seen in fragments of calcite obtained by cleavage ; but in complete crystals, it is more common in dolomite, (Ca,Mg)CO 3 . The lower half has been rotated, and occupies the position of the inverse rhombohedron {122} : its polar edges are therefore the lines joining V t to M, M t , M ti of Fig. 335, p. 376 ; and the median coigns on these edges are the points of trisection nearest to the equatorial plane, which coincides with the combination-plane. The median edges are bisected in this plane at the points 8 on the dyad axes. The salient and re-entrant edges join adjacent pairs of these points ; and the angles rr are 90 47. Fig. 481 represents a scalenohedron {2lO} twinned according to the same law. This twin is readily drawn with the aid of the twin shown in Fig. 480. Each of the rhombohedra in the latter is completed in fine pencil, and the median coigns are then joined to apices thrice as distant from the combination-plane as the apices of the auxiliary rhombohedra. The obtuse polar edges join the apex to coigns on the further side of the combina- tion-plane, and these obtuse edges intersect in pairs in the points H. The median edges cross the combination-plane at the points S', &c. ; and these points united to the adjacent Flo TWINS OF CALCITE. 515 points 77 give the horizontal edges of the twin : these latter edges are in pairs which are alternately salient and re-entrant edges, the angles over them being in all cases 41 56'. The twinning is frequently repeated. If repeated only once, the portions at the opposite apices are in like orientation with a twin- lamella interposed ; and if this lamella is very thin, it appears only as a fine line running horizontally across each face of the crystal. Twinning according to this law is sometimes perceived in crystals in which the prism {2TI} is the conspicuous form. Since the faces are parallel to the twin-axis and pairs of them to one another, twinning does not cause any interruption on them ; for a portion of a rotated face is co-planar with one of the fixed crystal, as in the twinned rhombic dodecahedron of blende, Art. 9. The twin, if terminated by rhombohedral faces e {110} at both ends, will differ from the simple crystal in Fig. 371, inasmuch as alternate prism-faces will be met in a like manner at both ends by faces e t and the equatorial plane will seem to be a plane of symmetry. If terminated by the pinakoid, the twin will be indistinguishable geometrically from a simple crystal, Fig. 328, but the cleavages at the two ends will be symmetrical to the combination-plane and not parallel. A variety of the law is described by Haidinger as sometimes occurring in which a face of the prism {211} is the combination- plane ; and another variety occurs in which the individuals pene- trate one another to a greater or less extent : the latter is common in dolomite. 44. 2. Twin-face e(110). This is the most common twin-law in calcite, frequently giving rise to polysynthetic twin-lamellse, which are generally, though not always, parallel to one face of the form {110}. The lamellae can be recognised by the striae which they produce on some of the cleavage-faces, and by the internal reflexion caused at the combination-plane. In a cleavage-rhomb of Iceland spar traversed by such lamellae, the four cleavages tautozonal with the twin-face are smooth and even ; the remaining pair of cleavage- faces meet the twin-face at angles of 70 52' and 109 8'. These latter faces are therefore traversed by alternate salient and re- entrant edges parallel to the horizontal diagonal of the rhombus, the normal-angle over each edge being 38 16' = 180 - 2 x 70 52'. If a thin plate is prepared parallel to this rhombic diagonal, and if the 332 516 TWINS OF THE RHOMBOHEDRAL SYSTEM. plate is inserted in parallel light between crossed Nicols, the light is extinguished simultaneously in all the lamellae, for the principal planes are all perpendicular to the diagonal. If a plate is prepared parallel to one of the smooth faces and similarly tested, the light in alternate lamellae is extinguished simultaneously, but not that traversing adjacent lamellae ; the extinctions in the latter make a minimum angle of 11 55' with one another, for the principal planes perpendicular to the plate are inclined to one another at 101 55'. The twin-lamellae are often of secondary origin ; i.e. produced after the formation of the crystal. They can, for instance, be produced in the laboratory by squeezing a cleavage-rhomb between the jaws of a vice, parallel polar edges being placed in contact with the jaws ; or they can be produced by pressing a knife-edge placed across a polar edge into the substance. A doublet of rare occurrence with (Oil) for twin-face is shown in Fig. 482. The upright individual is drawn in one of the ways given in Chap, xvi, Art. 57. The edges of the section in the com- bination-plane, and the directions of the twin-axis and of the triad-axis of the rotated crystal are then determined. The prism- edges through the corners of the section and lines through the middle points of its alternate sides are now drawn parallel to the rotated triad-axis. Lines parallel to the twin-axis are also drawn through the coigns in the faces e of the fixed crystal : they meet the edges and lines of construction already drawn in corresponding coigns of the rotated crystal, which can be then completed. 45. 3. Twin-face r(100). Doublets of calcite according to this twin-law are fairly common in Cumberland. The twins vary much in aspect, the habit depending on the forms present and on the relative size of the faces either of the same or of different forms. Thus the twins sometimes consist of hexagonal prisms {2TT} termi- nated either by {111}, or by the rhombohedron {110}, Fig. 486, sometimes of scalenohedra {20T}, Fig. 485. The principal axes are inclined to one another at angles of 89 13' and 90 47'. Fig. 483 represents a doublet with (001) for twin-face. It can be imitated by placing two cleavage-rhombs side by side in twin-orientation. The drawing is made as follows. FIG. 482. TWINS OF CALCITE. Oil' The rhombohedron {100} is drawn in the way described in Chap, xvi, Art. 31. It is then bisected by the combination-plane g& lt e drawn through 3,, parallel to (001), 8,, being the middle point of the median edge /*,/i. The normal Or" to (001) meets the polar face-diagonal \ r p" at r", where Fr" : F/" = 3 : 8, nearly. For, from Fig. 484, Fr" = 0Fsiu/>, and .*. Fr" : V = 3sin 2 .D-M=3 : 8, nearly. For, D being 44 36'6', sin 2 D= 1-^-2 very nearly. The point r" is now found M" It. FIG. 483. Fro. 484. by proportional compasses, and Or" is produced beyond (001) to Q, where OQ. = 20r". Q F is the rotated triad axis OVj in direction and magnitude. The figure can be now completed, corresponding coigns lying on lines parallel to VV,. The twin represented in Fig. 485 can be drawn as follows The triad axes in Fig. 483 are produced both ways to points F 3 , F 3 , F 3 , and F 3 . Each half- rhombohedron in Fig. 483 is also completed in faint pencil beyond the combination-plane ; and the median coigns of each rhombohedron are then joined to the apices of the correspond- ing scalenohedron. The corre- sending polar edges intersect in the combination-plane; and the adjacent pairs of these points give the alternate salient and re-entrant edges. The angles between corresponding faces are, 2TO A (210) = -120 A (120) = 180"- 2 x 82 29' =--15 2'; 20T A (201) = 180- 2x103 56-5' = -27 53' =-021 A (021). Fio. 485. 518 TWINS OF THE RHOMBOHEDRAL SYSTEM. FIG. 486. To construct Fig. 486 the two rhombohedra in Fig. 483 are completely drawn as required for the previous twin, and the middle points 8 of their median edges are found. The lines through these points parallel to the respective triad axes are the prism-edges, corre- sponding pairs of which intersect in the combination-plane. The salient and re- entrant edges in this plane can then be drawn. A convenient length, corre- sponding fairly to that of the edge in the twin, is now cut off on one of the prism- edges of the fixed crystal, and the edges [me], [m t e'], &c., are drawn in the manner described in Chap, xvi, p. 407, for the combination {211} and {110}. The coigns on the rotated crystal are found by drawing lines parallel to VV_ t of Fig. 483 through each of the coigns of the fixed crystal to meet the triad axis OV lt the corresponding prism-edges, and lines through the middle points of [mm], &c., parallel to OV t . The coigns of the rotated crystal having been determined, the figure can be completed. The polar edges of the rotated crystal can also be found in the same way as those of the fixed crystal. 46.4- Twin-face f(\\\). Fig. 487 represents a scalenohedron v {210} twinned according to this law. In Chap, xvi, Art. 52, it was shown that the faces /of {HT} truncate the acute polar edges of the scalenohedron {210}, and in Art. 37 that the apices of {111} are at distances 2c from the origin, the equivalent symbol being - ZR. The inclination of the normal/" (1 1 1) to the triad axis is therefore 63 7'; and in the twin the two triad axes include an angle of 180 - 2 x 63 7' = 53 46'. Further, the pair of acute polar edges in the plane of the triad axes are parallel, and the opposite pairs of obtuse polar edges in the same plane include a salient and re-entrant angle of 97 54'. The faces which meet in the combination-plane form salient and re-entrant angles : these are 210 A (210) = - 120 A(l20)=180-2x71 23' = 3714' l 102 A (I02) = -0l2 A (012) =--180 -2x51 18' = 77 24'. TWINS OF QUARTZ. 519 This twin is rarer than the others. The several twins can be readily distinguished from one another by the positions of the cleavages which can be generally detected by means of flaws travers- ing the crystals. The position of the rotated triad axis is easily found when the rhombohedron {111} is drawn as in Fig. 370, p. 406; for the foot of the normal/" lies in V\" at a distance given by Vf" : K\" = 3sin 2 FO/''-^4 = 3sin 2 637'-7-4 = -669 = 2-:-3 nearly. If a section in the plane V\"V 2 of Fig. 370 is made similar to Fig. 484, we shall obtain similar relations to those established for the latter ; the angle VOf" replacing the angle VOr", and being at the middle point of VV a . The normal Of" is prolonged to where Oft=20/". OF is then the direction of the rotated axis OVy and gives a length 2c upon it. The rest of the drawing presents no special difficulty. Quartz. 47. Twin-axis [111]. 1. Dextroyyral twitis. Let Fig. 488 (a) represent a simple dextrogyral crystal of normal habit, having the faces of the forms, m {21 1], r {100}, z {122}, and x = a {412}; and let Fig. 488 (c) represent an exactly similar crystal, the position of Fio. 488 (a). Fio. 488 (b). Fio. 488 (c). which is such that a semi-revolution about the triad axis will bring it into the same orientation as the crystal (a). Let us now suppose three equal wedge-shaped segments to be cut out of each of the two crystals by planes inclined at 60 to one another and passing through the triad axis ; and suppose the wedges to include similarly placed alternate edges of both crystals. Let the segments of crystal (c) include the prism-edges having faces x at both ends, then the similarly placed wedges of (a) include no x faces. Now interchange the wedges and insert those from (c) in the vacant spaces in (a) and vice versa. 520 TWINS OF THE RHOMBOHEUKAL SYSTEM. When the three wedges of (c) are inserted in (a), a twin, Fig. 488 (6), is obtained in which the faces r of the one individual are co-planar with the faces z of the other, and the trapezohedral faces x occur on all the prism edges. These faces x all lie in zones which, proceeding from a face of the apparently hexagonal pyramid towards a prism-face, slant in the direction of a right-handed screw ; and the twin is, like the simple crystal, throughout dextro- gyral. The necessity for this is obvious, for it has already been shown that a dextrogyral crystal cannot be brought into the same orientation as a laevogyral crystal by rotation about any axis. By the insertion of the three wedges of (a) in (c), a twin is obtained which has no x faces, and is geometrically indistinguish- able from a simple crystal having the faces m, r and z. Even in this latter case the twin character can often be detected by the difference in lustre and markings of the r and z portions of the composite-faces. Further, the prism-edges bearing faces x are at the analogous poles ; consequently with falling temperature, the prism-edges in Fig. 488 (6) are all negatively electrified, and those of the apparently simple crystal are all positively electrified. Other similar twins can be formed by the interchange of a single segment cut from (a) and (c), so as to include a similarly placed prism-edge. Two twins are then produced, one of which has faces x on four prism-edges, the other only on two. When two wedges are interchanged, twins having five prism-edges or only one modified by x faces are obtained. We are thus able to explain the irregu- larity of distribution of the x faces observed on crystals of quartz. So far the wedge has been supposed to extend from apex to apex, but there is no necessity for this ; and it often happens that a portion at one corner, or in the middle of a face, has been, as it were, excised from a crystal and replaced by an equal amount of matter in twin-orientation. It is indeed frequently observed that a crystal has several patches of such interpolated matter; the presence of such patches in the r and z faces being recognized by the difference in lustre and corrodibility, and on the prism-faces by the different orientation of the figures produced by corrosion. Occasionally these twins and the twins given under 24 are i n juxtaposition with a face of the prism {211} for combination-plane. In these cases the individuals are sometimes sharply separated by the combination-plane, sometimes there is a greater or less amount of overlapping and interlocking. TWINS OF QUARTZ. 521 2. Lavoyyral twins. Figs. 489 (a), (6) and (c) serve to illus- trate the production by an exactly similar process of a twin of a hevogyral crystal having faces x t of a {421}, arranged in a similar FIG. 489 (a). FIG. 489 (I). FIG. 489 (c). manner on all the prism-edges. The preceding discussion holds for all the possible variations of the twins of laevogyral crystals, but in these crystals the tautozonal faces z', x,, m follow one another along a left-handed screw when we proceed from either of the extreme faces. Thus we may have Isevogyral twin-crystals showing no faces x ( , or faces x f only on one, two, fec. prism-edges; and we may have the crystals in juxtaposition along a face of {211}. 48. 8. Lcevo-dextrogyral twins a. Brazil-law. Composite crystals of quartz with coincident triad axes occur in which the different portions have opposite rotations. Thus the twin, shown in Fig. 490 (b), which was first described by G. Rose, can be obtained by the interpenetration of crystals of opposite rotations. Suppose wedges including similarly placed prism-edges to be cut from a dextro- and a laevo-gyral crystal, placed as in Figs. 490 (a) and (c) with their faces r parallel to one another ; and suppose the wedges including the faces x f of Fig. 490 (c) to be inserted in equal spaces in (a). A composite crystal represented by Fig. 490 (b) is obtained. Now Figs. 490 (a) and (c) are both in normal orientation, and no rotation of either individual has been assumed. The crystal falls under the class of symmetric twins; and it is symmetrical with respect to each of three planes which are parallel to the prism-faces truncating alternate edges of the prism {2lT}. Had the wedges from (a) been inserted in the spaces in (c) a similar twin would be produced which would show no trapezohedral faces, and would be 522 TWINS OF THE EHOMBOHEDRAL SYSTKM. indistinguishable, except by optical examination, from a simple crystal. Twins of this kind showing trapezohedral faces are very FIG. 490 (a). FIG. 490 (b). FIG. 490 (c). rare. The boundaries of the individuals and the number of faces x actually present on any crystal seem to be as variable as in the case of dextrogyral or Isevogyral twins. The rhombohedral faces r and z forming the hexagonal pyramid are in this twin each of one kind, and a face r of one individual is not co-planar with a face z of the other. The law of twinning is known as the Brazil-law. 4- Lwvo-dextrogyral twins (3. Another possible composite crystal of quartz into which both dextro- and laevo-gyral portions enter is shown in Fig. 491 (6). Suppose a Isevogyral crystal, Fig. 491 (a), and a dextrogyral crystal to be placed side by side with faces r parallel; and let the latter crystal be then turned through 180 about the triad axis, when its position is given by Fig. 491 (c). Suppose both crystals to be cut into similar wedges of 30 by planes passing through the middle point of each crystal ; one plane being horizontal, the others vertical ; of the latter six pass through the prism-edges and six are perpendicular to the prism-faces. Replace twelve alternate wedges in Fig. 491 (a) by twelve similarly placed wedges from Fig. 491 (c). A twin is then obtained represented by Fig. 491 (6). This is also a symmetric twin ; and it is geometrically and physically symmetrical to the equatorial plane and to each of the planes passing through opposite prism-edges, i.e. to planes parallel to the faces of {2ll}. The faces of the hexagonal pyramid are com- posite as in the twins of single rotation ; the prism-faces are composed of four portions, those placed diagonally belonging to a crystal of the same rotation. The composite nature of twins of laws 3 and 4 can De proved by TWINS OF QUARTZ. 523 the examination in plane-polarised light of plates cut perpendicularly to the optic axis. Such examination has shown that the purple FIG. 491 (a). FIG. 491 (b). FIG. 491 (c). variety (amethyst) consists of a number of thin layers alternately of opposite rotations and different colour ; the layers being parallel to the faces r, and sometimes both to the faces r and to two or three of the faces s of a {412}. The sections are therefore divided into three or six segments which are usually bounded by very definite lines, and they are sometimes separated by triangular spaces near those prism-edges on which the faces z stand. These triangular spaces are often again divided into two distinct triangles in which the rotations are opposite. 49. 5. In these twins the triad axes are inclined to one another at angles of 84 34' and 95 26' ; and, irrespective of the rotatory character of the individuals, we have two varieties. i. In the first, Fig. 493, the one individual may be supposed to have been turned through 180 about the normal to a pyramid-face which truncates the edge [r"z]. This face belongs to a trigonal bipyramid a {125}. ii. In the second, Fig. 494, the twin-axis is the zone- axis 07 T [210] parallel to the edge [r"*]. The twins of both kinds are united along a surface which is very nearly, if not exactly, parallel to the face (T25). i. Twin-face (125). Fig. 492 is a plan of a simple crystal of quartz projected on a plane parallel to (121). Suppose the crystal to be bisected by a plane through OT perpendicular to (121), and the lower half to be turned through 180 about 0. The two halves, being then joined in the plane of section, give the twin shown in Fig. 493. The rhombohedral faces r and of the one are 524 TWINS OF THE RHOM150HEDRAL SYSTEM. symmetrical to similar r and z faces of the other with respect to the combination-plane; but, if the crystals have the same rotatory FIG. 492. FIG. 493. character, the trapezohedral faces x would, if present, not be symmetrical to this plane. The twin is neither geometrically nor optically symmetrical to the combination-plane ; for a right-handed screw is reflected in a left-handed one. The prism-faces 121 and (T2T) are co-planar ; and the boundary traversing this face is jagged and irregular. The other prism-faces meet in definite salient and re- entrant edges ; the angles over them being IT2 A (112) = - 211 A (211) = 79 41'. If the individuals are of opposite directions of rotation, they are optically symmetrical to the combination-plane; and so are the faces r, z, m and the trapezohedral faces x : the twin is symmetric. Each individual is often again twinned according to one of the preceding laws ; for the faces r and z generally show patches of different character which indicate twinning. Whether the indi- vidual is in these cases dextrogyral or laevogyral or is Isevo-dextro- gyral can rarely be determined without sacrificing the specimen. ii. Twin-axis OT [210]. If now two crystals placed side by side in parallel orientation are bisected by the plane through OT of Fig. 492 perpendicular to 121, and the two upper halves are united together in the planes of section after one of them has been turned through 180 about OT 7 , a twin represented by Fig. 494 is produced. In this twin the faces r, z and x are not symmetrical to the combina- tion-plane, and the twin is asymmetric whether the crystals are of like or of opposite rotations. Fig. 494 represents a twin from Japan TWINS OF CINNABAR. in the Cambridge Museum, which, judging from the pyramidal faces of a {421} and the trapezohedral faces x t shown, consists of two Isevogyral crys- tals : the relative dimensions of these faces to the others are exaggerated. The faces r and z seem to be composite, and the individuals composing the twin are probably far from simple. As in twins of the variety described under i, the individuals united according to this law may be of like rotation, or twins of the kind described under 8 and 4- Fio. 494. 50. Cinnabar. Fig. 495 represents an interpenetrating twin, in which the forms of the individual in normal position are ; {100}, n{5ll}, X = u {13, 1,5}. The individual with barred letters is after a semi- revolution about the triad axis brought into a position in which the rhombohedral faces become coincident with the corresponding faces of the first crystal, but the trapezohedral faces \ come into the position of the enantiomorphous form a {13, 5,1}. Professor Tschermak (Min. u. Petr. Mitth. vn, p. 361, 1886) found a simple crystal having the forms r, n and X to be dextrogyral. The twin is therefore a symmetric twin of a dextro- and a laevo-gyral crystal, the latter being in an azimuth differing by 180 from that given to the simple crystal. The individuals are re- ciprocally symmetrical with respect to the equatorial plane, and to three vertical planes parallel to the faces of the prism {21 1}. This twin is similar to the twin of quartz according to the fourth law. Symmetric twins were also observed in which the faces of the rhombohedron are coincident, but all the edges [ra] are modified by faces of the enantiomorphous trapezohedra % an d x- S\ich crystals therefore have the geometrical development characteristic of crystals of class III of the rhombohedral system, and fall into the same group of twins as those of quartz twinned according to the Brazil-law. 51. Hematite. Fig. 496 represents a twin in which the triad axis is the twin-axis, and the combination-plane _. is parallel to a face of the hexagonal prism ^H_f_ {21 1}. As shown in the figure the faces ^^XjL^iS of the pinakoids are co-planar, and likewise Fio. 496. 526 TWINS OF THE RHOMBOHEDRAL SYSTEM. a pair of opposite faces a of the prism {10l}. The faces shown are c{001}, r {100} and a {10l}. The composite character of the face c is often revealed by barbed striae, which indicate precisely the position of the combination -plane. 52. Chabazite. The crystals are usually rhombohedra {100} which resemble cubes, for the angle rr' = 8514'. The twins are interpenetrant twins with the triad axis for twin-axis. The twins consist usually, as shown in Fig. 497, of a large individual from the faces of which small portions of others protrude in twin-orientation. The appearance is similar to that observed in many cubes of fluor. FIG. -197. FIG- 498. In Pig. 498 a twin of another habit is shown, which is common in the variety called phacolite (seebachite). This may be regarded as an interpenetrant twin of two rhombohedral crystals, crossing one another with fair regularity in the same way as the rhombohedi-a of cinnabar, Fig. 495. The forms present are r{100}, e{110}, s{Tll}. The faces r and e are striated parallel to the edges [re]; the striae on some of the faces r are however exaggerated in the figure to show the deep re-entrant angles more conspicuously. Plates cut perpendicularly to the triad axis have been found to consist of six segments similar to those of potassium sulphate, and the segments are often, if not generally, biaxal, the acute bisectrix being perpendicular to the plate. But the angle of the optic axes is not the same in all the segments of the same plate, and the double refraction also varies in strength. The substance loses some of the water of crystallization very easily, and the anomalous optical characters are probably due to the strains caused by the loss of water. Since different segments may be unequally affected, we have an explanation of the variation from segment to segment ; and this interpretation of the phenomena is supported by the fact that a further expulsion of water by heat strengthens the double refraction where it is already manifest, and brings it into prominence in segments in which it was at first inconspicuous. TWINS OF PYRARGYRITE. 527 53. Pyrargyrite. The crystals of this mineral belong to the acleistous ditrigonal class, and afford an instance of complementary twins having their triad axes and also their planes of symmetry coincident in direction. The twin shown in Fig. 499 may be supposed to consist of two similar halves of separate crystals united along the equatorial plane (the plane of section) after one has been turned through 180 about the normal to one of the faces a of the prism {101}. The forms present are a {101}, m = /*{2TT}, v = n {201} and <=/*{310}. A section of the simple crystal in the equatorial plane is a hexagon, having its alternate corners truncated by the traces of the trigonal prism p. {211}. A semi-revolution of such a hexagon about a line in its plane which bisects a pair of its opposite sides, interchanges the sides of the hexagon, but exchanges a modified with an unmodified corner. Consequently the truncated edges of the rotated hexagonal prism stand below the unmodified edge of the fixed crystal, and vice versa. The pyramid- faces on the two portions are pai-allel in pairs, and the like faces together build up each a scalenohedron characteristic of class III of the rhombohedral system. The small triangular faces, which appear at the points where the edge [act] overlaps the face of (211), belong to the trigonal pyramid p. {TOO}. Fio. 499. V. TWINS OF THE HEXAGONAL SYSTEM. 54. Twins of crystals with their hexad axes coincident in direction have been mentioned in Art! 19 of Chap. xvn. Such twins are complementary twins and are only possible in classes I and III. Their geometry offers no difficulty, and we need not dwell further on them. Arguing from the fact established in Chap, ix, Art. 21, that only a single hexad axis is possible in a crystal, we should expect that twins with inclined hexad axes would be inconsistent witli the possibility of crystal-growth. A single twin of apatite with inclined axes is described in the Am. Journ. o/Sci. [iii], xxxni, p. 503, 1887 : the axis of rotation is given as the normal to a face of {1121}. A similar composite crystal with the same twin-face is recorded as observed in a rock -section (Jour. Geol. in, p. 25, 1895). But in a 528 TWINS OF THE OBLIQUE SYSTEM. mineral so common and abundant as apatite, these instances seem hardly sufficient to establish a twin-law. VI. TWINS OF THE OBLIQUE SYSTEM. 55. Crystals of this system are most commonly twinned about a normal or zone-axis lying in the axial plane XOZ ; this plane being a plane of symmetry in classes II and III, and perpendicular to a dyad axis in classes I and III of the system. The plane of symmetry and the dyad axis have therefore the same directions in the several portions of the twin. When the portions are in juxta- position, and the twin-axis is a possible normal, the combination- plane is usually perpendicular to the axis ; the twin-law may then be shortly described by specifying the position of the twin- face. The orientation of these hemitropes may in many cases be also given by taking for twin-axis a line in XOZ perpendicular to that usually adopted. Thus, the twin of hornblende, Fig. 503, is usually said to have (100) for twin-face : it may also be described as having the vertical axis [001] for twin-axis, and (100) for combination- plane. In the Carlsbad-twin of orthoclase, we adopt the vertical axis for twin-axis : we might also refer it to the normal to (100) as twin-axis, but by doing so important relations of the twin are lost sight of. When there is no special reason for choosing the zone-axis rather than the normal, the latber is generally taken as the twin-axis. The directions of the axes of X and Z have usually been selected arbitrarily, so as to give ^simple indices to the most conspicuous faces. They are therefore dependent on the habit of the crystals first described, and it may prove that these directions have no apparent connection with those of the twin-axes. It follows that the twin-face may in some cases be (100), in others (001) and in others (hW). Twins having for twin-axis a normal or zone-axis inclined to the dyad axis or plane of symmetry at angles other than 90 or 0, are rare except in the case of orthoclase and the harmotome group. In such twins, the twin-axis may be a normal (MO), (Okl) or (hkl), according to the edges which have been selected to give the axes of X and Z. As a rule, however, the twin-axes have symbols with very low indices. TWINS OF GYPSUM. 529 Fio. 500. 56. Gypsum. 1. Tivin-faee (100). Fig. 500 represents a doublet of gypsum, which is common in many localities, or can be produced in the laboratory. The drawing has been made by bisecting the crystal represented in Fig. 140, p. 188, in the plane YOZ, and then turning the front half through 180 about the normal to this plane. The position of the twin-axis is the back-and-fore cubic axis in the projection which serves as basis for the construc- tion of the oblique axes (Chap, vi, Arts. 15 and 16). Similar coigns of the fixed and rotated portions lie on lines parallel to the twin-axis at equal distances from the combination-plane. The faces I of {111} meet in pairs in two re-entrant and in two salient edges, the angle over each of them being 70 51'. The angles between the edges [bl] in the common face b are 75 9' and 104 51'. The portions occasionally intercross, forming a twin which has common (010) faces and shows sometimes re-entrant angles at both ends, and sometimes salient angles. 2. Twin-face (101). Another twin of gypsum having (101) for twin- face and the face (010) common to _ both portions is shown in Fig. 501 (after ^\ *' V^ Hessenberg). In this twin the re-entrant angles are formed by faces /8 having the symbol {509}, and the faces I {111} of the two portions are tautozonal. In well-developed twins the distinction between the laws 1 and 2 is easily per- ceived; for the faces m are brighter and more even than the faces I, and the angle mm, = 68 30', whilst tt t is 36 12'. If sufficiently translucent for optical examination between crossed Nicols, the twins can be distinguished by the minimum angle between the directions of extinction in the plane of symmetry; and plates suitable for the purpose are easily obtained by Fio. 501. We have seen in p. 188 that Bx a is inclined to OZ at an angle of + 52 27'. Hence in twin (1) the angle between the acute bisectrices is 104 54', and the least angle between the directions of extinction in the two portions is 14 54'. In twin (2) the vertical axis is inclined to the combination-plane at an angle of 52 25', and L. o. 34 530 TWINS OF THE OBLIQUE SYSTEM. Bx a therefore at an angle of 52 25' +52 27' = 104 52' (or its supplement 75 8'). The two directions of extinction are then inclined at 150 16', and the least angle between them is 180 -150 16' = 29 44', which is almost identically double that in the first twin. Again in the first twin the interruptions in the perfect cleavage caused by the conchoidal cleavage 100 are parallel to the combination-plane, those due to the fibrous cleavage n {111} are inclined to this plane at angles of 66 10'. In the second twin both sets of interruptions are inclined to the combination-plane, those due to {100} making angles with it of 52 25' and those due to n {111} angles of 61 25'. 57. Hornblende. Twin-face (100). The simple black crystal, Fig. 502, of this substance resembles a rhombohedral crystal, for the angles bm = 62 15', mm, = 55 30', cm = 76 48' and br = 74 14'. If such a crystal is divided in the plane YOZ, and the front half turned through 180 about the normal to the plane of section, a twin is obtained, which would only differ from Fig. 503, inasmuch as portions of two faces c would meet in a re-entrant furrow modify- ing the highest coign made by four faces r, and at the other end there would be small indentations at the coigns where faces b meet (no) (110) FIG. 503. the two c faces. The top would then resemble the end of Fig. 504 which is directed to the front. Such re-entrant edges are very rare in the twins, which, as shown in Fig. 503, consequently simulate hemimorphic crystals of the prismatic system in which an acleistous pyramid is developed at one end and a gonioid (or hemidome) of two faces at the other. A thin section cut parallel to the common face 6(010) would between crossed Nicols act like a cleavage-plate of a twin of gypsum, MANEBACH-TWINS OF ORTHOCLASE. 531 and would extinguish the light in directions including a minimum angle of about 40. Orthoclase. 58. This mineral is stated to form twins according to eleven different laws, most of them rare and some doubtful. We shall describe some of the twins according to the three common and well established laws. These laws are (1) twin-face c(001), the law being often known as the Manebach-law, (2} twin-face n(021), the Baveno-law, and (8) twin-axis OZ, the Carlsbad-law ; these last twins occasionally have (010) for combination-plane, but more commonly interpenetrate to some slight extent. The Carlsbad and Baveno twins are the most common. 59. 1. Manebach-law. Turin-face c(001). The appearance of these twins, Figs. 504 and 505, differs much according to the faces which predominate. We shall describe them separately as (a) and 08). a. Good specimens of these twins have been found at Manebach in Thuringia, Four-la-Brouque in Auvergne, the Mourne Mts. in Ireland, and Pike's Peak, U.S.A. The faces 6(010) and c(001) are largely developed and form a rectangular prism, in which the parallel faces c belong each to a separate portion, the faces b are composite and are often traversed by striae or markings parallel to the edges [bm] of the respective portions. At one end four faces m of {110} are developed, and form, as do the faces r in the twin of hornblende, an apparent pyramid with an angle mm 44 26', and an angle mm^ 61 13'. Occasionally, as shown in Fig. 504, the apex of this apparent pyramid is replaced by a small cavity bounded by faces y(201), yf\y" being - 19 24'. The other end is formed by two faces y, making with one another a salient angle of 19 24'. In specimens from the Mourne Mts. the coigns at which these faces y meet the common faces b are modified Fl0t S04> by pairs of m faces parallel to those forming salient angles to the front in Fig. 504: they therefore form re-entrant angles of - 44 26'; and in one of the Irish specimens at Cambridge the faces m forming 342 532 TWINS OF THE OBLIQUE SYSTEM. FIG. 505. re-entrant angles are developed so as to obliterate the faces y : this end of the twin has the same appearance as that end of Fig. 505 which is shown to the front; the faces b and c are large, and the crystal is attached to the rock at the end at which the salient pyramid of faces m would be developed. (3. Fig. 505 represents a twin of adularia from the St Gothard with (001) for twin-face. In these twins the prism-faces m are largely developed, and form, on opposite sides of the twin, salient and re-entrant angles of 44 26'. The crystals are terminated by small triangular faces c(001) and a?(10T), the face c being easily distinguished by its pearly lustre. The twins of this habit sometimes intercross ; and the portions forming the twin are often also combined with portions of other individuals twinned to them according to the Baveno-law. The doublet has been drawn by finding the direction and length of the normal on the face c (001 ). The normal lies in the plane XOZ at an angle of 26 3' to OZ, and its direction is therefore found by the same con- struction as that given in Chap, vi, Art. 15 for finding OX. By drawing through C on OZ a line parallel to XX t to meet the normal, its length is obtained. Doubling this length, the origin Q of the rotated axes is determined. The prism-edges [mm'] are then drawn through the points in which the prism-edges of the fixed portion meet the combination -plane ; corresponding coigns on the rotated and fixed prism -edges lie on lines parallel to the twin-axis. Twin a, Fig. 504, is drawn from the same axes, and only differs from /3, Fig. 505, in the relative size of the faces present. The zone-axis XX /} Fig. 505, may be taken as twin-axis ; for a semi-revolution about it gives the twin-orientation correctly. We have given the twin-law by means of a twin-face for the sake of brevity in its enunciation. 60. 2. Baveno-law. Twin-face w(021). Fig. 506 represents the habit of these twins, as commonly observed on specimens from Baveno and elsewhere ; the faces b and c being largely developed. Since b f\n = 45 3*5', the pair of adjacent faces b include an angle of 90 7', and the pair of faces c, which complete the almost rectangular BAVENO-TWINS OF ORTHOCLASE. 533 FIG. 506. prism, are at 89 53' to one another. Opposite faces are of dissimilar character, and are not strictly parallel. At one end pairs of faces m and y meet in small salient angles, which can be computed from the angles given on p. 190. Faceso{Tll}, a{T01}, and z{ 130} are also often present. The angles are m Am = 180 - 2n'm = 10 34', y /\y = 180 - 2n'y = 1 3 42', and o f\o = 180 - 2n'o = 92 34'. The crystals are usually broken, having been attached to the rock at the end opposite to that shown in the figure, but occasionally specimens showing both ends are preserved. In the perfect speci- mens in the Cambridge Museum, the faces at the end opposite to that shown to the front in Fig. 506 make re-entrant angles. The individuals are rarely united along a plane passing, as in Fig. 506, through the edges [66], [cc], but parts of a face 6 of one individual overlap, and are practically co-planar with, a face c of the other, and vice versa ; and the boundaries of the two individuals are often very irregular. When portions of 6 and c faces are co- planar or nearly so, the distinction in their character is plainly seen, and much assists the student in tracing the boundary of the twin ; for the faces 6 are often marked in directions parallel to the edges [bm], which are inclined to the edge [be] at angles of 63 57'; the faces c have a pearly lustre, and are sometimes corroded in lines parallel to the edges [cxy] which are at right angles to the edge [be]. Adularia from the St Gothard twinned according to this law is frequently found in triplets and occasionally in quartets, which Figs. 507 510 serve to illustrate. In these plans the edge [be] is perpendicular to the paper, the edges [ex] are parallel to it, and the edges [bm] are inclined to it at angles of 26 3'. The plans are easily constructed ; for a square being drawn, it is only necessary to find the angle which the projected zone-axis [xm] makes with the trace c to be able to complete each of the figures. Now any face in the zone [6f] = [100] meets the paper in a straight line, which makes with the trace c t the same angle as the face makes with the face c,. Also the face which is common to the zones [*] = [m] and [6c] = [100] has the symbol (Oil). Knowing the angle en to be 44 56'5', the angle c A Oil is found to be 26 31'. The lines [mx], [m/c], &c., are then drawn each at 26 31' to 534 TWINS OF THE OBLIQUE SYSTEM. (001 the traces c,, c, &c., through the corners of the square. The edges [mm,] are parallel to the traces 6. Fig. 507 represents two crystals in contact along an edge [be], that labelled / being in the usual position of an oblique crystal, the second in an orientation differing from that of /by -' a semi-revolution about the normal to the face w(021). Suppose now the two crystals to be bisected by planes through the corners n t and n' parallel to the twin-face n, and the more remote halves to be united in the planes of section. We thus get the doublet given in the plan, Fig. 508. In this doublet the faces m t and m meet in a salient angle of 10 34', 001 c > FIG. 507. the small portions of m and m in a re-entrant angle of 77 42', and the faces x make a salient angle xx = 53 47 '5'. The faces m are usually striated parallel to the edge [mb]. If now at the corner n', Fig. 507, a third crystal is introduced in twin-orientation to, and touching individual II with (021) for twin-face, and the three crystals are cut down so as to form a rect- (oio) 6 c, {ool} c, c, 001 FIG. 508. c, FIG. 510. angular prism, we obtain the triplet shown in Fig. 509. In this case, three of the faces bounding the rectangular prism are c faces, and the fourth consists of two b portions inclined to one another at a salient angle of 14' : for the angles b i c i = bc i = 90, and c t c = cc, = 89 53'. The individuals I and III are very nearly, but not absolutely in twin- orientation according to the Manebach-law ; for the plane through MANEBACH-BAVENO-TWINS. 535 [mm] in which they meet is very nearly parallel to the faces c t and c,, but the faces m t and in meet in a re-entrant angle of - 44 16'. The edges \c t x\ of the triplet are all parallel to the paper, the edges \b,m^\ and [bm] are both inclined to it at angles of 26 3'. Triplets of this kind are fairly common, but the re-entrant angles mf\m and m'Am', are almost always obliterated as shown in Fig. 510 by the development of the faces x, and the filling up of the troughs. Fig. 510 represents a quartet of the same kind which may be supposed to be produced by associating a fourth individual in twin-orientation at n t in Fig. 507. Individuals III and IV are not then in strict twin-orientation, and the faces of the prism are not quite parallel. The twin approximates in habit and develop- ment to a tetragonal prism surmounted by a tetragonal pyramid. The twins are therefore of interest in showing how a crystal may be developed as a mimetic twin out of portions of crystals of much inferior symmetry. Specimens corresponding to each of the drawings of the ideal twins given above, are far from showing the regularity of development assumed in the description. The crystals are generally much corroded, and the faces are sometimes encrusted with chlorite. The faces c seem to be most easily corroded, and those of b to be most readily encrusted. 61. Manebacli-Baveno twins. In the collection of minerals recently presented to Cambridge by the Rev. T. Wiltshire, M.A., Hon. Sc.D., of Trinity College, is a remarkable twin of adularia from the St Gothard shown in Fig. 511. It forms a nearly rectangular prism bounded by composite faces b (010), and is strictly speaking an octet ; each segment, such as that having its faces labelled b^ , m^ , cCj, being twinned to an ad- jacent segment (that carrying suffixes 2) on one side accord- ing to the Baveno-law, and to the adjacent one on the other side (that lettered b' 3 , m 3 , x' 3 ) Fio. 611. according to the Manebach- law. The combination according to the two laws is not as regular as that seen in ordinary doublets, and there is a certain amount 536 TWINS OF THE OBLIQUE SYSTEM. of interlocking of the matter of adjacent portions across the com- bination-planes. Thus, as is slightly indicated in the composite faces b, the markings transgress the trace of the ideal combination- plane of the Manebach-twin. A similar irregular interlocking is observed at the back of the crystal, where two faces m meet at a small salient angle over an edge which should lie in the twin-plane (021) of the Baveno-law. The edges of the prism formed by the composite b faces are also modified, and are replaced by very narrow rectangular grooves having the faces c for sides. These grooves much resemble the well-marked grooves on twins of harmo- tome, Fig. 516, p. 540. The twin is not entirely free from attached matter, but sufficient of the back can be made out to show that the edges separating faces of different portions are all salient. The faces in Fig. 511 have been labelled to correspond to an in- terpretation of the twin as consisting of two intercrossing Manebach- twins ; the one having even suffixes, the other odd ones. Again adjacent wedges consisting of portions of odd and even labelled individuals are twinned according to the Baveno-law. Similarly labelled faces x give almost simultaneous reflexions of a bright signal. Hessenberg (Min. Not. n, p. 4) has described a similar twin. 62. 3. Carlsbad-law. Twin-axis [001]. The twins shown in Figs. 512 and 513 have the vertical axis for twin-axis with an angle of rotation of 180. Consequently in each twin the faces b {010} and m {110} of one individual are parallel respectively to faces b and in, of the other. The twins are often in juxtaposition, the Fio. 512. FIG. 513. combination-plane being parallel to (010) : in other cases there is, as shown in the figures, a greater or less amount of interlocking of the individuals. This interlocking is generally perceived when the crystals have the faces c {001} and y {201} well developed. CARLSBAD-TWINS OF ORTHOCLASE. 537 Since the portions of an oblique crystal on opposite sides of the plane of symmetry are antistrophic, two twins according to this law are possible which, depending on the relative positions of the components, are sometimes described as right- and left-handed. Suppose two exactly similar crystals, having the faces in Figs 512 and 513, to be placed side by side in the usual position of an oblique crystal, and suppose them to be bisected in their planes of symmetry. Suppose now the two portions to the left of the plane of symmetry to be united in this plane after one of them has been turned through 180 about the vertical axis [6wt] : a left- handed twin identical, save for interlocking portions, with Fig. 512 is produced. Fig. 513 is obtained by the similar union of the two portions to the right of the plane of symmetry, and may be called a right-handed twin. If the two twins are placed side by side with the faces b and the edges [&/,] parallel, the pair are reciprocal reflexions in a mirror parallel to their planes of symmetry, and cannot therefore be placed in similar positions. Good instances of right- and left-handed twins have been found in many localities, and have been recently obtained in great abundance at Bob Tail Gulch, Colorado. When the faces c and x {101} are the only faces associated with the prism {110} and pinakoid {010}, the juxtaposed twins resemble a simple prismatic crystal, for a face c of the one individual is as shown in Fig. 514 practically co-planar with a face x of the other. The inclinations of the two faces to the vertical axis are nearly the same, for, Fig. 142, p. 191, C A ^=26 3' and x A ^=24 13-5'. G. Rose, who carefully examined the twins, was however unable to detect any divergence of the two portions of the composite face, corresponding to the difference of 150', between their inclinations. Owing to the difference in physical character and corrodibility of the faces c and x, the composite character of the faces in the twin is well seen. This has been indicated in the figure by marking the faces x with dots. A repetition of the twinning according to this law is sometimes indicated by narrow interruptions traversing the faces c and * parallel to the edges [be] and \bx\ of an otherwise simple crystal. A rotation of 180 about the normal to the face (100) will satisfy the relations of this twin. The prism-faces of the rotated Fio. 514. 538 TWINS OF THE OBLIQUE SYSTEM. individual would in that case have different indices, for after a semi-revolution about the normal each of them is brought into a position coincident with that of a different homologous face of the prism. But such an interpretation of the twin will not satisfy the relations of the similar twins of the closely allied anorthic felspars. For in these latter substances the angle 100 A 010 differs from 90, and the faces (010) of the two portions would not be parallel, as has been found to be the case in the twins. 63. J. D. Dana pointed out in the 5th edition of his Mineralogy a remarkable approximation in the angles of orthoclase and the anorthic felspars to those between faces of the more common forms of a cubic crystal. Thus 6m = &,?,= 59 23'5' and mm, = 6ri3' are all comparable with 60, the angles between the faces of the rhombic dodecahedron parallel to a triad axis. Hence the axis OZ of orthoclase corresponds to a triad axis of a cubic crystal, and may be regarded as a pseudo-triad axis. The faces z in the zone [bm] make 29 24' with b and 29 59'5' with adjacent faces m : they correspond to faces of {211} which truncate the edges of the rhombic dodecahedron. Again, n being (021), the angles 6?i = 45 3'5' and cn = 44 56'5' are very nearly 45. The zone [bnc] is therefore comparable with a tetragonal zone, and the zone-axis [be] (the axis OX) is a pseudo-tetrad axis. But from the zone [bmm,], the face b is seen to correspond to a face of the rhombic dode- cahedron, and its normal is a dyad axis. Hence the face c also corresponds to a face of the rhombic dodecahedron and its normal to a pseudo-dyad axis. The faces n then correspond to the tautozonal faces of the cube, and their normals to pseudo-tetrad axes ; the relation in this zone being similar to that described in the zone [be] of staurolite (p. 511). The third face of the cube has to be sought in the zone \cxy\ at right angles to c and n. The nearest important face to this position is y (201), for cy=80 18', ny = 83 9'. Knowing the angles ex and cy, we find that the indices of the face (hQl) at 90 from c must satisfy the equation l-43A = 3-86. Hence (201), (301), (803) and (27,0,10) are successive approximations to the position : they correspond to angles 001 A 301 = 93 50-5', 001 A 803 = 8939'5' and 001 A 27,0, 10 = 90 0'. By the aid of the relations just given, many of the twins of orthoclase can be explained from considerations as to the probable arrangements of the particles in the different classes of crystals. Amongst the regular arrangements possible, there are some in which groups of the particles about one point can be brought into the position of similar groups by translation through some finite interval. When the crystals have also axes of symmetry, pairs or triads or tetrads of like groups can be inter- changed by rotations of 180, 120 and 90 about the respective axis. Such arrangements of the groups of particles may be said to be conformable. A TWINS OF THE HARMOTOME GROUP. 539 rotation through the respective angle about an axis which is only approxi- mately one of symmetry will not bring a group into coincidence with a similar group ; but the relations of the rotated group to the matter in itw neighbourhood may so nearly conform to those required for the attachment of the group as to allow growth to proceed, and the more nearly the axis approximates to a true axis of symmetry the greater the probability of twinning and the more stable is the twin likely to be. From this point of view, the Manebach-twin is due to the normal to the face c(001) being a pseudo-dyad axis; the Carlsbad-twin to the axis OZ being a pseudo-triad axis, and to the structure about this axis having much the character which is connected with the twinning about a triad axis in cubic crystals. The Baveno-law may, as in the twins of pyrites and the rectangular twins of staurolite, be given in two ways : (a) a rotation of 180 about the normal (021) which is a pseudo-tetrad axis, or (/3) a rotation of 90 about the zone-axis OX (the same as [6nc]) which is also a pseudo- tetrad axis. Further, this hypothesis suggests that the normals (110), (201) and (111) should also be twin-axes ; for each of them is nearly in the direction of a pseudo-axis of symmetry of even degree. Twins with each of these normals for twin-axis the angle of rotation being 180 have been described, but they are very rare. Since these lines are not so nearly in the directions of axes of symmetry as the axes OX, OZ and the normals (001), (021), the conditions which would give rise to twinning about them are not likely to be as frequently met with as in the case of the three important laws. Harmotome Group. 64. The twins of orthoclase and their relations to the pseudo- cubic symmetry of the crystals throw light on the remarkable twins of harmotome, phillipsite and wellsite, in none of which has the simple crystal been observed. Fig. 515 represents a crystal of harmotome from Scotland, which was long believed to be a simple prismatic crystal. Des Cloizeaux showed, however, by optical examination between crossed Nicols, that sections cut parallel to the face marked b and b t are composed of four quadrants, such that the directions of the extinctions in opposite quadrants (those marked by similar letters 6) are the same, whilst those in adjacent quadrants (marked b and b t respectively) are inclined to one another. One of the directions of extinction in each quadrant is inclined, as shown in Fig. 515, at an angle of nearly 60 to the edge [be]. This crystal resembles a Manebach twin of orthoclase, in which, however, the two individuals intercross so that the two ends show a similar pyramid of four like faces. Occasionally the 540 TWINS OF THE OBLIQUE SYSTEM. apex of this pyramid is modified by a small uneven face f, the in- clination of which to c cannot be distinguished from 90. Des Cloizeaux took the upper pair of faces m to be those of the prism {110} of an oblique crystal, in which b (010) is parallel to the plane of symmetry, c is the base (001) and / is (10T). The faces are very uneven and striated, so that accurate measurements are not possible. He adopted the following values, viz. mm, = 59 59', cm = 60 21', and c/= 90 0'. From these angles we can compute the elements of the crystal, and the angles which a face e(011) truncating the edge [be] makes with the faces on the crystal. The elements of the crystals of the other minerals can be found from similar data. We thus have for : FIG. 515. Harmotome 55 10' Phillipsite 55 37 Wellsite 53 27 703 709 768 1 : 1-231 1 : 1-256 1 : 1-245 be 44 42' 43 58 45 me 45 18-5' 45 25 43 8 90 25' 50'. 91 27 2 54. 90 45 1 30. 65. The crystals are, however, for the most part complex twins similar to Figs. 516 and 517. The former is common in harmotorne, the latter in phillipsite and wellsite. The striated faces in Fig. 516 correspond to those marked b in Fig. 515, and are, as far as measurements can be made, at right angles to one another ; so also are the faces c which form the sides of the grooves running the whole length of the edges of the apparently rectangular prism. Similar crystals are often observed in phillipsite, but the faces c are 51*5. FIG. 517. FIG. 518. TWINS OF THE HARMOTOME GROUP. 541 very often those forming the prism, and the faces b form the sides of the groove. Regarding the intercrossing doublet, Fig. 515, as if it were a simple crystal, the complex twin is formed of two such individuals having e(011) for twin-face; or it may be treated as consisting of four wedges cut from the doublets and, like the wedges of orthoclase in Fig. 510, united in succession along a twin-face (Oil). Or, again, as was pointed out in the case of orthoclase, we may suppose the wedges to be turned through 90 about the zone-axis [be] (a pseudo-tetrad axis), and each wedge to be united to the adjacent ones more or less regularly along a face (Oil). The twin agrees most closely, however, with the octet of adularia, Fig. 511, in the Wiltshire collection, save that in the twins of the harmotome group of minerals the ends showing salient angles are alone developed. Since em = 89 35', the faces m separated by a line joining the apex to the end of the groove are almost co-planar. On the assumption of a semi- re volution about e (Oil), they make in harmotome a salient angle of 50'; and on the assumption of a quarter-revolution about [be], one of 25'. The composite character of the m faces is however most clearly demonstrated by the barbed striae on them, the striae being parallel to the edges [bm] of the respective individuals. These complex twins are very regularly developed in harmotome, and the grooves along the edges of the prism are well marked. In phillipsite the faces composing the rectangular prism are free from striae and belong to c{001}; the grooves are bounded by faces b, which sometimes show patches of striae similar to those on the prism- faces of Fig. 516. But in phillipsite and wellsite the grooves are for the most part obliterated, and the twins closely resemble a tetragonal prism {100} terminated by a tetragonal bipyramid {111}. Examination between crossed Nicols of plates cut parallel to the faces of the pseudo-tetragonal prism shows that combination does not take place regularly along the plane (Oil), but portions of a face c are co-planar with a face b of the adjacent individual. This is indicated in Fig. 517, where the irregular lines of dots show the boundaries between the individuals, labelled I IV, in a twin of wellsite described by Messrs Pratt and Foote (Am. Jour, of Sci. [iv], in, p. 443, 1897). They observed that the boundary between the portions b forming part of the doublet is a definite line corre- sponding to combination parallel to c(001). 542 TWINS OF THE OBLIQUE SYSTEM. Wellsite 1 was found by the authors to form other octets (or intercrossing quartets), Fig. 518, in which the faces a (100) are largely developed. The faces a of the doublet, having (001) for twin-face, make with one another salient angles of 73 6' (an angle used in the computation of the crystal-elements) ; and adjacent faces a of different doublets, such as a and a', make re-entrant angles of 49 48'. Similar twins of harmotome from Kongsberg, St. Andreasberg, and Scotland, and of phillipsite from Asbach have been described. 66. Applying to these twins the views as to pseudo-cubic symmetry already propounded in the discussion of the twins of staurolite and ortho- clase, we see that the zone [be] is one of pseudo-tetragonal symmetry, the normals b and c being the one a dyad, and the other a pseudo-dyad, axis. The doublet, Fig. 515, is then due to crystal -growth being possible when the particles or groups of particles are deposited in orientations differing from that of conformability by a semi-revolution about a pseudo-dyad axis (the normal to (001)). The same orientation is obtained by a semi- revolution about the zone-axis [be]. Again, the complex twin is due to [be] being a pseudo-tetrad axis ; a quarter-revolution about this line bringing the groups of particles sufficiently near to a position of conformability for growth to be possible. Further, the edge [6mm,], Fig. 515, is a pseudo-triad axis ; and a rotation of 120 about it should leave the particles in approximate conformability with their first orientation. This axis is inclined to [be] at the angle ft, which is in all the crystals very nearly equal to 54 44' the angle between a triad and tetrad axis of a cubic crystal. Still more complex twins, such as that of phillipsite shown in Fig. 519 (after Kohler, Pogg. Ann. xxxvn, p. 561, 1836), are therefore possible, which may be explained as due to this pseudo-triad axis. The same twin is arrived at by regarding it as the result of twinning about the pair of pseudo- tetrad axes which are the normals to the faces e of {Oil}, and are at right angles to [be] ; the angle of rotation about each of these normals being 90. 1 Wellsite is shown by Messrs. Pratt and Foote to be a member of the group of zeolites formed by phillipsite, harmotome and stilbite ; the composition being : Wellsite (Ba, Ca, K 2 ) Al 2 Si 3 O 10 . 3H 2 0. Phillipsite (Ca, K 2 ) Al 2 Si 4 12 . 4H 2 0. Harmotome (Ba, K 2 ) Al 2 Si 5 14 . 5H 2 0. Stilbite (Ca, Na 2 ) Al 2 Si 6 16 . 6H 2 0. Seeing that the determination of the true state of hydration is very difficult, and that the analyses of phillipsite vary considerably, it is probable that the number of molecules of water of crystallization should be 4 and not 4. The authors point out also that a lower member of the series, having two molecules of water, may exist. Such a member would be a dimorphous form of natrolite. TWINS OF THE HARMOTOME GROUP. 543 The upright individual in Fig. 519 is placed as if the largely developed faces were {110} of a tetragonal crystal and the normals to the faces e were the axes of X and Y. By a rotation of 90 about the back-and-fore normal (0 X], the vertical axis is brought into the position of the axis of Y. By joining two quartets related to one another in this manner, an intercrossing rectangular twin is obtained, such as is sometimes found in phil- lipsite : it may be represented by the two individuals of Fig. 519 which have their axes [be] parallel to the paper. If now a third quartet is joined to the two, after it has been turned through 90 from the upright position p about the axis OY, the twin, Fig. 519, is pro- duced. The composite faces m of Fig. 516 are in Fig. 519 arranged in sets of two which are very nearly co-planar, and are parallel respectively to faces of the rhombic dodecahedron. If, further, the matter is deposited so as to fill up the re-entrant spaces between the prisms, and if the faces m are developed so as to obliterate all the other faces, a twelve- faced figure is produced, which is very nearly a rhombic dodecahedron, each of the faces being composed of segments which belong to four separate individuals. A mimetic dodecahedron of phillipsite of this kind has been described by Strong (N. Jahrb. f. Min. 1875, p. 585). Langemann (N. Jahrb. f. Min. 1886, n, p. 83) has shown that plates cut parallel to the face cc of the doublet, Fig. 515, consist of four segments, such that, between crossed Nicols, opposite segments extinguish the light simultane- ously, but adjacent segments have different directions of extinction. The face b is not therefore parallel to a plane of symmetry, or its normal a dyad axis ; and the simple crystal of harmotome and phillipsite must be regarded as anorthic. The intercrossing doublet, Fig. 515, is then a mimetic octet or a mimetic intercrossing quartet, the individuals being separated along planes parallel to b and c passing through the edges mm, and mm. The relations to pseudo-cubic symmetry apply to the crystals as before, but the normal to the face b is now only a pseudo-dyad axis. vii. TWINS OF THE ANORTHIC SYSTEM. 67. We shall limit our discussion to a few of the twins of the felspars, microcline, KAlSi 3 O 8 , albite, NaAlSi,O 8 , and anorthite, CaALjSiaOg, mentioning in the intermediate members of the group a few cases which are pertinent to the discussion. The plagioclastic felspars are all characterised by having two perfect cleavages inclined to one another at angles of nearly 90 and by the prom- inence of faces parallel to these cleavages : they also have faces 544 TWINS OF THE ANORTHIC SYSTEM. making with one another and with the cleavages angles nearly the same as those which in orthoclase the prism-faces m{110} and the pinakoids x {101} and y {201} make with one another and with the cleavages. The crystals are therefore placed so that the cleavages and the faces have nearly the same situations as the cleavages and corresponding faces in orthoclase. Thus in the ideal simple crystal of albite, Fig. 520, the less facile cleavage contains the axes of Z and X, and the parallel faces M are {010} : the faces P of {001} are parallel to the best (pearly) cleavage; and the axis of Y is parallel to the edge [Px], x being 101. The crystals are always placed so that the normal-angle 001 A 010 is less than 90 : in albite it is 86 24', in anorthite 85 50'. The angle between the positive directions of the axes of Z and Y is greater than 90, varying from 94 5' in albite to 93 13-5' in anorthite. In albite the faces ^{110} and {110} take the place of the prism-faces ra of orthoclase; in anorthite and the other felspars the letters T and I are interchanged, T being {iTO} and {110}. The other faces given in the following figures are labelled and have the same symbols as the crystals discussed in Chap, xi, Arts. 17 and 27. 68. Albite-law. Twin-face Jf(OlO). Twins according to this law are very common in all the plagioclastic felspars, and are scarcely ever absent in crystals of albite, with the exception of those of the variety known as pericline : the law is therefore called the albite. law. If now a crystal like Fig. 520 is bisected by a plane parallel to M, and if the half to the right is turned through 180 about the normal to M and the two portions are then united in the plane of section, the ideal twin, Fig. 521, is produced. In this twin we have no overlapping of portions in the combination plane, for the figure made by the plane of section is bounded by pairs of equal parallel ALBITE-LAW OF TWINNING. 545 lines, so that the figure is self-congruent, i.e. exactly fits, after it has been turned through 180 about the normal to its plane. Fig. 521 gives a good representation of a doublet of oligoclase; but in albite the thickness of the twin perpendicular to the faces M is much less in proportion to the other dimensions, and the twinning is often repeated. The pairs of faces P, x and y (when present) which meet at the upper end of the vertical axis include re-entrant angles, the opposite pairs equal salient angles. The angles are P/\P = r\2', o;A=7 20' and yAy = 441'. Notwithstanding the small difference in the angles, the pearliness of the faces P renders it easy to distinguish them from the others ; and they can often be recognised by the directions of the cleavage-flaws, which traverse the crystals parallel to P. Fig. 522 represents an intercrossing doublet of albite from Roc Tourne in Savoy which was first described by Rose (Pogg. Ann. cxxv, p. 456, 1865). The faces indicated by letters only are /{130}, *{lTO}, p' {III}, o' {III}. In these twins the faces P make re-entrant angles on opposite sides of the centre, and the faces y salient angles ; further, the largely developed faces M are traversed by narrow troughs bounded by faces / The twin is composed of four portions separated by the combination- plane and by the plane through the troughs perpendicular to M, adjacent por- tions being in twin-orientation. By breaking the crystal along the pearly cleavage /*, a fragment shown in Fig. 523 was obtained. Rose found that the angle between the cleavages changed at the troughs from a re-entrant to a salient angle. The cleavage portions labelled P are parallel, and so are those labelled P. Twinning according to the albite-law is often repeated, giving rise to polysynthetic twin-lamellae of varying thickness. A second semi-revolution of 180 about the same line brings any face to its original position. Hence the faces P, y and x FIG. 522. L. C. 35 546 TWINS OF THE ANORTHIC SYSTEM. of alternate lamellae will be parallel, and faces of the same form will in equably developed twin-crystals always meet in the combination- planes. The presence of such twin-lamellae can be often detected in cleavage-fragments of the plagioclastic felspars by the presence of a series of parallel striae due to the alternating salient and re-entrant angles at which the pearly cleavages of the several portions meet. Such striae, when present, serve as a means of distinction between these minerals and orthoclase ; for in the latter the normal to (010) is a dyad axis, and the faces 010 A 001 are at 90 to one another. Even when the above test fails, polysynthetic twinning according to this law can be often recognised by examining a thin cleavage- flake parallel to P between crossed Nicols ; for, except in the case of some specimens of oligoclase and andesine, the directions of extinction in the twin-lamellae of the cleavage-flake are inclined to the combination-plane at very appreciable angles. 69. Twin-face P(001). Fig. 524 is a sketch, which serves to show the unequal development of the faces and individuals in a twin with twin-face (001) of the green variety of microcline called Ama- zon-stone from Pike's Peak, U.S.A. The figure is a plan with [P ( J/] perpendicular to the paper, and is made in the same way as the plans of the Baveno-twins of orthoclase, Figs. 507 510. Owing to the close approxi- mation to 90 of the angle P A Af the value adopted by Des Cloizeaux being 89 45' the faces M and M_ t are practically co- planar, the theoretical angle being 30'. The edge [y'y] is interrupted by a dimple having for sides faces parallel to I and T. A portion of the rotated individual projects on one side beyond the plane M t of the other ; and at the irregular line the crystal is attached to a fragment of another. Crystals of pericline twinned according to this law have been observed, the faces M on the different portions of the twin being inclined to one another at angles of 7 12'. 70. Tunn^axis XX t = [100]. The twin of microcline, Fig. 524, may like the similar twin of orthoclase, Fig. 504, be referred to XX t as twin-axis. In this cause it should be regarded as composed TWINS OF MICROCLINE AND ANDESINE. 547 of two like portions of separate crystals, and the faces {110} and {1TO} meeting in the combination-plane belong to different forms. Thus I would meet a face T, and a face 2 of {130} would meet a face / of {130}. The faces M and M t would be exactly co-planar ; but the traces of / and T, z and / in the plane XOT would not be accurately congruent. The crystals are too irregular and the faces too uneven for such minute differences to be determined. A twin of andesine having XX ' t for twin-axis has been described by vom Rath. His drawing of the twin has a fairly close resem- blance to that of orthoclase shown in Fig. 504, but several additional forms are given. The composite faces M are co-planar and the faces P parallel ; whereas, if the twin-axis were the normal (001), the faces M would meet in the combination-plane at an angle of 7 32'. The angle between the axes of X and 7 is 89 59', so that the traces of {110} and r{lIO} in this plane make practically a rhombus, and after a semi revolution about XX t a face I meets a face T in a line lying in the plane XOY as combination-plane. A twin of pericline having XX ] for twin-axis and />(001) for combination-plane is given by Des Cloizeaux, but it generally occurs associated with the albite-law in complex twins. In the simple twin the faces M are co-planar, but the traces I and T in the plane XOT will, after a semi-revolution about XX t , be inclined to one another at angles of 2 59' : there is therefore a certain amount of overlapping. If this overlapping is avoided, then combination must take place along a plane inclined to the base in a manner similar to that described under the pericline-law. 71. Twin-axis ,[001]; combination-plane (010). This twin occurs in several of the felspars, e.g. in anorthite and andesine from Japan, and it is fairly common in albite, although, from the continual occurrence of twinning according to the albite-law, its recognition in this last mineral is often difficult. The doublet according to this law of the ideal crystal, Fig. 520, will so nearly resemble Fig. 521 that the difference is not perceptible in a drawing ; for, taking the elements of albite adopted by Dana, the angle ZOP=26" 42' and Z0x= 24 10'. But the re-entrant angles at one end and the equal salient angles at the other are each made by a face P and a face x of different individuals. When the faces are wide enough to be distinctly seen, their difference of lustre makes it easy to dis- criminate between them, and therefore to determine the twin-law. 352 548 TWINS OF THE ANORTHIC SYSTEM. Another easy test for distinguishing between twins according to this and the albite-law is afforded by the cleavage-cracks which so frequently traverse the crystals of albite. When these cracks occur and the crystal is sufficiently translucent for us to get the reflexions from them of light traversing the crystal, the cleavages are perceived to cross one another at angles of 52 58' in this twin, whilst they are parallel in the albite-twin. This latter test often proves an albite- twin to consist of several lamellse united together successively according to the albite and the present laws. When faces y are also present, portions of the individuals overlap in a manner similar to that observed in the Carlsbad-twins of orthoclase. Peridine-law. Twin-axis YY t - [0 1 0]. 72. This twin called after the variety of albite in which the faces P{001} and x {101} are largely developed will be best under- stood if the corresponding twins of anorthite represented in Figs. 526, 528, and 529 are first described. The Vesuvian crystals of anor- thite, having smooth and bright faces, admit of accurate measurement ; and, though variable in habit, generally show the forms in Fig. 525. The crystals and twins of anorthite are described by vom Rath, in two classical memoirs (Pogg. Ann. cxxxvin, p. 449, 1869; CXLVII, p. 22, 1872), from which the figures have been copied with slight modifications. The forms shown in Fig. 525 are given on p. 159, and the positions of their poles and the axial points in Fig. 121, p. 158. We need only to know that P is {001}, M {010}, I {110}, T {ITO}, 7i{100}_ that the elements and angles are : a= YOZ=93 13-3', p = ZOX=I1555-5', y = XOY=9l 11-6'. a : b : c= '63473 : 1 : -55007. J/P=8550', PA=6357', hM=8T&, IT=59 29', 7W, = 6226'5'. Suppose two crystals like Fig. 525 to be placed in similar positions and to be bisected by planes parallel to the base. If now the upper halves are united in their planes of section, after one of them has been turned through 180 about YY t , with this axis in common, a twin is produced in which the faces M to the right make FIG. 525. and PERICLINE-LAW OF TWINNING. 549 a salient angle with one another, and those to the left a re-entrant angle. An actual twin of this kind is shown in Fig. 526. If the lower halves are united in a similar way, the twin produced has the re-entrant angle between the faces M on the right and the salient angle on the left. The two mo- difications resemble the similar modifications of the Carlsbad-twin of orthoclase; and, like them, the pairs of twins composed from the same two crystals of anorthite can be placed so that they are reciprocal reflexions in a vertical plane ZOX : this relation can be fairly well seen by comparing the twins shown in Figs. 528 and 529. FIG. 526. 73. T/te rJwmbic section. Any plane through YY t of Fig. 530 meets the two pairs of faces I and T, produced beyond M to meet in vertical lines, in a parallelogram having YY t for one diagonal and for the other a line OR in the plane ZOX which contains the vertical edges [IT]. If the plane of section is JOY, the two diagonals are XX t and YY t , the angle between them being the crystal-element y ; and the parallelogram is BXBX t marked by the continuous lines I and Tin Fig. 527. When this parallelogram is turned through ISO'about.&e ,itis brought into the position shown by the inter- rupted lines. The traces M parallel to OX cross one another at an angle XOX= 2y- 180 ; and the pairs of traces I and T f , &c., do not exactly fit. An actual twin, Fig. 526, having XOY tor combination-plane, has been observed in anorthite. In Figs. 526 and 527 the overlapping of the parallelo- grams and of the individuals in the twin has been considerably exaggerated so as to show with distinctness the lack of congruence. But, generally, the individuals meet without any overlapping, as is seen in Figs. 528 and 529 showing actual twins of anorthite. The trace of the plane of section on M and the parallel diagonal OR in the plane XOZ must then be at right angles to YY t , for such a line FIG. 527. 550 TWINS OF THE ANORTHIC SYSTEM. is the only one which retains the same direction after a semi- revolution about YY r Now the parallelogram having its diagonals FIG. 528. FIG. 529. at 90 is a rhombus, and the plane of section through YY t in which the traces I and T are all equal is known as the plane of the rhombic section of the plagioclastic felspars. The pericline-law is therefore that the twin-axis is YY t , and that combination takes place in the plane of the rhombic section. We proceed to determine the position of this plane. Let OR, in Fig. 530, be perpendicular to OY, and the plane ROY be that of the rhombic section. About as centre describe a sphere, meeting OR and the axes at R, X and Y. Then from the spherical triangle XRY, cos RY=cos RXcos XY+sin RXsin XYcos RXY. But f\RXY=l\ PM, and, R Y being 90, cos R Y= ; . . cot R X = - tan X F cos RXY= - tan y cos PM. Also sin XYR = sm RXsin RXY=sin RXsin PM, gives the inclination of the plane to the base (001). Introducing the angles y and PM given for anorthite, we find f\RX=Wl', and f\XYR=lb 58'. The angles of albite are very variable and the values of y and PM to be used for any par- ticular crystal are very uncertain. If the values adopted by vom Rath for those from Schmirn are taken, viz. y = 8751'5' and PJf= 86 30' ; then RX=3l 29'4', and XYR = 3l 25'5'. If Dana's values, y = 888'6' and PM=86 24', are taken ; then RX=1T 17'5' and XYR=W 14'. If Breithaupt's angles for pericline, viz. y = 8913-3' and PJf=8641'; then ^JT=1312-3' and Z7^=13ll'. From the cases given above it is seen that, FIG. 530. PERICLINE-LAW OF TWINNING. 551 since the tangent of an angle of nearly 90 changes very rapidly for small increments, the angles RX and XYR vary much more widely than the elements of the crystal. Whilst, moreover, the angle a = YOZ is in all the felspars > 90 (varying from 93 13' to 94 5-3') the angle y is less than 90 in albite and greater than 90 in anorthite. Hence, if a plane, passing through YY t in Fig. 530, is turned about YY t from the position ZOY to XOY and then to ZOY, the angle which the front half of the line of intersection with the plane ZOX makes with OY diminishes continuously from a>90 to its supplement. The angle becomes 90 only when the plane coincides with that of the rhombic section. But in albite y = XOY is less than 90 ; and the line OR, where A ^07=90, lies between OZ and OX: the plane of the rhombic section is then inclined to /*(001) on the same side as o;(T01). In anorthite y is greater than 90, and the line OR at 90 to OY must lie between OX and OZ, : the plane of the rhombic section is inclined to P on the same side as (201). Since the combination-edges in most of the twins of anorthite, and, as far as can be judged, in all those of pericline, are congruent, the edge [MM] in which the faces M meet is parallel to OH : the edge is therefore in albite inclined to the edge [/*J/] in a direction lying between [/M/] and [aj^/J, and in anorthite it is inclined the other way so as to lie between [PM] and [tM]. The former direction is sometimes indicated by affixing a plus sign to the angle, and the latter by affixing a minus sign. Thus in albite the angle between the edges [JO/] and [PJ/], computed respectively from the values of y and PM given in the preceding paragraph, is + 3129-4', + 27 17-5' and +13 12-3'; in anorthite it is - 16 1'. In albite y is less than 90, in anorthite it is greater than 90, and it has been shown that the position of OR, and therefore of the parallel edge [MM], depends mainly on the value of y. Hence, if for the intermediate felspars the angle y changes more or less regularly with the chemical composition, the angle \P.M]h\MtC\ will serve as a means of discriminating between them. Thus vom Rath was led, by observation of striae on the face M of a crystal of esmarkite, to think that an error had been made in placing it near anorthite. The striae on M, due to polysynthetic twinning according to this law, are inclined to [PM] at an angle of about + 4, and such an angle requires y to be less than 90 and that the substance should be an oligoclase, not far removed from 552 TWINS OF THE ANORTHIC SYSTEM. albite. Analysis confirmed this surmise, for the mineral was found to contain 61*9 of silica and only 4-45 of calcium oxide. Knowing the angles hP, Px of the felspars, it is easy, from the anharmonic ratio of four tautozonal planes, to calculate the indices of possible faces which are approximately parallel to the plane of the rhombic section. In albite such a face will require a different symbol for values of y differing by small increments. The plane can therefore have no place amongst possible crystal-faces of the substance. In anorthite the face (307) is inclined to P at an angle of 15 59 - 5'; and it may be that this approximation to the position of a possible crystal-face with fairly low indices accounts for the great uniformity of the trace on M of the rhombic section in crystals of this substance. 74. The twins of anorthite sometimes intercross as shown in Fig. 531, so that the faces M make re-entrant angles at both ends. This figure shows clearly the enan- tiomorphous character of the pair of doublets which, being formed from two simple crystals, have, like the doublets in Figs. 528 and 529, re-entrant angles M f\M FIG. 531. only at one end. The twin, Fig. 531, can be composed by bisecting in the plane XOZ two such doublets, and uniting in the plane of section the portions to the extreme right and left ; i.e. the portions on the left may be composed of upper parts on the left of two like crystals, and the portions on the right of lower parts on the right of the same crystals. A similar inter- crossing twin with salient angles M f\M at both ends, though le, has not been observed. 75. Pericline. The twins of pericline, when both ends can be perceived, are seen to be always intercrossing doublets similar to that of anorthite, Fig. 531, and the angles Mf\M are invariably re-entrant. The general habit is fairly well given by Fig. 531, but the trace of the combination-plane on M is inclined the other way to the edge [3/7^]. Further, the combination along the rhombic section does not seem to be very regular, as is shown in the sections Figs. 532 and 533. The first shows the natural face, in which the exact division between the twinned individuals is partly TWINS OF PERICLINE AND MICROCLINE. 553 masked by a deposit of albite substance on the pericline-twin. Fig. 533 shows the appearance of a section of the same crystal obtained by cleavage parallel to M. Fio. 532. Fio. 533. 76. The twinning is often repeated, giving rise to striae on the faces M, I and T. Such is the origin of the striae shown traversing the faces M, /, I in Fig. 525. The lines of interrupted strokes indicate twin-lamellae according to the albite-law, and vom Rath observed that the lamellae of different kinds never intersected one another in crystals of anorthite. The crystals of microcline are traversed by numbers of extremely thin lamellae which cross one another nearly at right angles. The lamellae of one series are parallel to [^-^/J and are combined accord- ing to the albite-law. The others are due to twinning parallel to YY t , combination taking place parallel to a plane making a very con- siderable angle with the base. The direction of extinction on the pearly cleavage P is + 15 30'. Hence a cleavage-flake parallel to this face shows between crossed Nicols a characteristic division resembling a grating which is due to the fine lamellae extinguishing the light in different azimuths. 77. The views as to pseudo-cubic symmetry propounded to explain the common twins of orthoclase receive considerable confirmation from the kws of twinning observed in the plagioclastic felspars ; for the angles being nearly the same in all the felspars, we have a similar approximation to a tetragonal zone in the cleavage-zone [PJf], and its axis XX, is one of pseudo-tetrad symmetry. Thus, e being (021) and n (021), we have in albite eP=43l =.4646 / . The relations of the particles about the normal to Jf(OlO) an d about the zone-axis YY t may both be considered to be nearly the same as those about a true dyad axis ; and suggest a reason for the occurrence of twinning about both the.se axes, i.e. according to the albite- and pericline- laws. Several crystallographers indeed regard orthoclase and microcline 554 TWINS OF THE ANORTHIC SYSTEM. to be merely varieties of the same anorthic mineral, and the former to be a mimetic twin composed of ultra-microscopic lamellae which are combined according to the albite- and pericline-laws. Again, the normal to (001) is a pseudo-dyad axis, and is the twin-axis adopted in microcline. We have also twins of microcline, and very rarely of albite, which may be referred to the same law as the Baveno-twin of orthoclase, i.e. twin-face (021) or (021) : in microcline it makes little difference whether the twin- face is e or n, or whether the twin-axis is Jf X t with a rotation of 90 ; but in albite and the other felspars there would be considerable differences according to the twin-axis and rotation adopted. In fact from the theoretical views under discussion three twins are possible, (i) a twin in which e is the twin-face, (ii) a twin with n for twin-face both with a rotation of 180 and (iii) a twin in which XX , is the twin-axis with an angle of rotation of 90. Further, a second rotation of 90 about the pseudo-tetrad axis XX t gives the twin-law in andesine, &c. Again, the faces M, I and T are nearly at 60 to one another, and the vertical axis ZZ t is a pseudo-triad axis: hence we have an explanation of the twins with this line for twin-axis. Twin-axis a line in a face perpendicular to one of its edges. 78. A twin-axis of this kind has been proposed as the twin-axis of the pericline-law, and for certain complex twins of albite and anorthite ; and single instances of its occurrence in labradorite and cyanite have also been given. The line has no crystallographic significance for it is neither a zone-axis nor the normal of a possible face ; and it is difficult to under- stand how such a line can have so important a relation to the crystalline structure. Kayser (Pogg. Ann. xxxiv, p. 109, 1835) proposed the line in the base (001) perpendicular to the edge [PM] for the axis of semi-revolution in the pericline-twin ; he adopted this line as the twin-axis rather than \_Px\ = YY lt for he perceived that the combination-edge [MM] must with the latter twin- axis be inclined to the edge [PM] at an angle 0, computed from Breithaupt's measurements to be 13 12 '3' (see Art. 73), in order that the individuals should meet in congruent edges. He points out that, in spite of the imperfections of the faces M on the crystals of pericline, the combination - edge is on the whole parallel to [PM], and that this is the more apparent the more perfect the face. Consequently, since the faces M meet in con- gruent edges parallel to the edges [MP] of the two individuals, and the faces P are parallel, the twin-axis must be a line in P perpendicular to [PM]. Rose (Pogg. Ann., cxxix, p. 1, 1866) adopts Kayser's twin-axis, but points out that the combination-edge is not often parallel to [PM ] and that it is always in a direction intermediate between [PM] and [Mx] : his careful drawings serve indeed to strengthen the argument in favour of [Px] as twin-axis. Kayser supports his view by the fact that, in the twins of oligoclase from Arendal in Norway, the combination-edge is parallel to DOUBTFUL ANORTHIC TWINS. 555 We have already shown in Art. 73 that the angle changes rapidly for very small increments of y, and it follows that, since y is very nearly 90 in the crystals of oligoclase and andesine, the combination-edge is, in their twins, substantially parallel to [PM]. Further, the twins of anorthite according to the pericline-law were unknown, and the evidence supplied by the well-marked inclination of the combination-edge to [PJf] was wanting. Seeing then that, except when the faces M meet in a jagged line indicative of combination along a very irregular surface, the direction of the combination-edge is generally in conformity with that required for congruence with \Px\ as twin-axis, and, further, that Kayser himself insists on the identity of the law of twinning in the several felspars, we must reject his interpretation of the law, and adopt YY, as the twin-axis. In the complex twins of albite and anorthite we may possibly have to deal with lamellae twinned according to different laws ; and the presence of an extremely thin lamella twinned on one side according to the albite-law and on the other side according to the Carlsbad-law (extending this term to embrace the plagioclase-twins described in Art. 71 which have [001] for twin-axis) will give rise to a composite crystal, one portion of which seems to be turned through 180 about a line at right angles to the two axes of the separate laws. For, by Euler's theorem, successive rotations of 180 about two lines at right angles to one another are equivalent to a single semi-revolution about a line perpendicular to the two first axes. In a complicated group and in an ill-developed crystal, the presence of a thin lamella can easily escape detection. Too little is known of the crystals of labradorite, to justify the acceptance of a twin-axis which is neither a normal nor a zone-axis ; and more especially as the specimen itself to which such a twin-axis has been ascribed did not admit of accurate determination. In cyanite the crystal-elements are not capable of determination with great precision, and possibly differ slightly from one crystal to another. In this mineral the edge [ac], Fig. 118, p. 155, is an accepted twin-axis, combination taking place parallel to (100). The angle between the edges [ac] and [a&] is very nearly 90 ; the value computed by vom Rath varying from 90 0' to 90 5', that by Professor Bauer being 90 23'. Hence, with the latter angle and [ac] as twin-axis, the prism-faces of the rotated indi- vidual are slightly out of the zone formed by the fixed individual, and this was observed to be the case. But in a single instance (Zeitsch. d. Deutsch. geol. Ges. xxx, p. 306, 1878) the faces of the prisms of both individuals were found to be exactly in one zone. Professor Bauer therefore inferred that rotation had taken place about the line at 90 to [ab] of Fig. 118 ; this axis of rotation making with the edge [ac] an angle of 23'. Considering the untrustworthiness of the angles on twin-crystals, as indicated, for instance, by the apparent coincidence of the faces c and x in the twin of orthoclase, Fig. 514, the evidence supplied by this observation is in- sufficient for the acceptance of the twin-axis. 556 TO FIND THE POSITION OF THE TWIN-AXIS. On the method of determining the position of the twin-axis. 79. The method employed to determine the position of the twin-axis and, when the portions are juxtaposed, of the combination-plane will depend on the development of the crystal ; and in many cases the law is perceived without difficulty. Thus, after the student has examined a few twinned octahedra of spinel, he will easily recognise the same twin-law in the case of octahedral crystals of other substances. The general method to be followed may be given as follows : (i) Determine the crystal-forms present on each portion ; and by measurement the elements of the crystal and the symbols of the faces, (ii) The positions of the like faces on the two portions having been determined, the student must consider round what line one portion must be turned through 180 to bring like faces into coincidence or parallelism. The perception of the direction of the twin- axis is much aided when certain faces or edges on the two portions are seen to be parallel to one another, or to be coincident in direction, (iii) A semi-revolution of a normal about any origin-line brings it again into the plane containing the normal and axis of rotation, and the latter bisects the angle between the initial and final positions of the normal. Thus, if TT' in Fig. 534 is a twin -axis with an angle of rotation of 180 and P and P' are the poles of corresponding faces, then a semi-revolution about TT' brings P to p', the opposite extremity of the diameter through P', and f\TP= /\Tp'. Also M being the point midway between P and P', the arc MT=90. So far as these two faces are con- cerned the axis may equally well be the diameter through T or M. If, however, the angle between any other two corresponding poles Q and Q', not in a zone with P and P', is also measured, the position of TT' is fixed. For T'Q = T'gf = TQ'. If the twin is combined along a plane per- pendicular to the twin-axis, then P and Q' are known poles belonging to one individual. These being projected in the way usual for a simple crystal of its system, the position of T can be found by the construction given in Chap, vu, Art. 19 ; and its relation to the axes of the crystal can be then determined. If P and Q belong to one individual, then the triangle TPQ is known, and can be projected in the way given for the previous case, and the point T determined. In a juxtaposed twin the relations of the individuals to the combination- plane must be recognised in order to determine the twin-axis ; and, except in the anorthic system as already illustrated in the discussion of the peri- cline-law, seldom present any serious difficulty. CHAPTER XIX. ANALYTICAL METHODS AND DIVERS NOTATIONS. 1. SEVERAL of the problems examined in the preceding Chapters, and a few which were then omitted, are capable of simple treatment by the methods of co-ordinate geometry. 2. To find the equation of the origin-plane (see p. 35) parallel to a face (hkl). Let the co-ordinate axes OX, OY, OZ, Fig. 53o, be parallel to any three edges of the crystal which are not parallel to one plane. Let, also, a, b, c be three finite lengths giving the distances from the origin at which OX, OY, OZ are respectively met by a definite face of the crystal. Let any other face (hkl) meet the axes at points H, K, L, so that OH=a + h, OK=b + k, OL = c + l; h, k, I being rational numbers. Then the equation of the face is Replacing the intercepts by their values in terms of the para- meters and indices, we have *5 + *l + ^ = 1 (1) ' Now a parallel face will meet the axes at points //', K', L\ where OH'=p.OH=pa+h; OK'^p.OR OL' =p. OL pc-r-l. The equation of the plane H'K'L' is x OH' OK' 558 ANALYTICAL METHODS. which, after the introduction of the values of OH' ', &c., becomes an equation only differing from (1) in the constant term. When, however, the plane passes through the origin, equation (2) must be satisfied by the values x = y z = ; .'. p 0. Hence, a plane through the origin parallel to the face (hkl) is given by the equation - b (3). This plane will be called the origin-plane (hkl). Since the angles measured by the goniometer give only the relative directions of the faces, and the computations into which these angles enter are not concerned with the actual distances of the faces from the origin, it follows that all the angular relations bjetween the faces of a crystal will be the same as those of the origin-planes parallel to them. 3. To find the equations of the straight line through the origin parallel to two non- parallel faces (hjkjl^, (h z k. 2 l 2 ) ; i.e. of their zone-axis. The equations of the two origin-planes are : b But for the line of intersection the two equations must be satisfied by the same values of x, y and z ; hence the equations of the zone- axis are If, as in Chap, v, the differences of the products of the indices occurring in the denominators of (4) are called the zone-indices, and expressed by W 12 , V 12 , W 12 ; then equations (4) can be written ~ This line is drawn by constructing a parallelepiped having for edges in the axes of X, Y and Z lengths equal to {/ 12 , &K 12 , cW 12 , TAUTOZONAL RELATIONS. 559 respectively. The diagonal OT through the origin is the zone-axis [u l2 K ]2 W 12 ] ; and any three numbers in the ratios au n : bv 13 : cw n may be called its direction-ratios. 4. To find the condition that any face (hkl) shall be in a zone with the two faces (/i^i)> (hjcj,.^). From the definition of a zone (p. 1) the line of intersection of the faces (hjcj,^ (hjej.^ must be parallel to the face (hkl). Hence the origin-plane (hkl) must contain the zone-axis [Ai&i^, hjcj^\, the equations of which are given in (4) and (5). Hence the equations h- + A: 7 + I - = 0, a b c x _ y _ z must both be satisfied by the same values of a;, y and z. Introducing the ratios for x, y and z given in the latter equations into that of the plane, we obtain the required condition ; which is hu lz + kVn + lWw = (6). This is the relation which in Chap, v, Art. 8, we designated as Weiss's zone-law. It was there pointed out that it can be written as a determinant, involving the indices of the three faces in an exactly similar manner. 5. To show that the plane containing two zone-axes is parallel to a possible face. Let [l/!?!*,], [tf 2 K 2 W 2 ] be the zone-axes, and let h- + k%. + I* = a b c be the origin-plane containing them. The zone-axes have the equations x y z and = -r = - W*)^ + (i- y*)f+ (UiV,- Kitf.,) * = (8). As stated in Chap, v, the indices of a face common to two zones [tfiKilVi], [fAj^a] are obtained from the zone-indices by the same rule as that by which zone-indices are deduced from the indices of two non-parallel faces. 6. To change the axes of reference to another set of axes, so that [Wi^^] becomes OX', [u?V?W^\ becomes 07', and [w 3 K 3 W 3 ] becomes OZ' ' ; supposing the parametral face (111) to retain the same indices. Let the parametral face (111) meet the new axes at the points A', B', C' respectively. Now the equations of the parametral face and of the axis [Wi^ivj, referred to the original axes, are nil hv fW " ... v ~-,. G>UI O'l CWi The co-ordinates of the point A', Fig. 536, are obtained by introducing the ratios of x : y : z from the latter equations into that of the plane. Let them be x, y', z' : then x' y' z ^ + ~b + c 1 aU l '. &Ki ~ CW 1 ~ U l + V l + Wi ~ U l + K]. + The new parameter OA' on OX' is the diagonal of the parallele- piped OYA'Z, having for edges along the original axes the co- ordinates of = NY, y' = OY, z' = NA', TRANSFORMATION OF AXES. 561 The co-ordinates of B' and C" are obtained in an exactly similar manner by combining equation (9) with those of each of the axes [u a v a w s ], [U 3 v t w,]i the co-ordinates may be represented by (x"y"z") and (x'"y'"z") respectively. The parameters OB' and OG" are the diagonals of parallelepipeds having these co-ordinates for edges. Fio. 536. Let a face (hkl) meet the original axes at H, K, L ; and the new ones at H' t A", L' ; and let its new symbol be (h'k'V). When referred to the original axes, the equation of the face is .(12). To find the co-ordinates of the point H' in which the face meets the new axis OX', we must combine equation (12) with (10). Let the co-ordinates of H' be a;,, y,, z,, then _ _ au l bv l cW! hUi + kV! + lw^ hu l + &K, + lw l "' By combining (12) with the equations of the axes OY\ OZ' t similar expressions are obtained for the co-ordinates of K\ L' in which the face meets these axes. But the new indices are: And by similar triangles OA'N, OH'F, we have OA' :OH' = A'tf:H'F=z':z 1 = y':y l =x'i Similarly, .(14). 36 562 ANALYTICAL METHODS. In the above transformation the axes OX', OT', OZ' are not necessarily zone-axes ; for any lines through the origin may be represented by equations of the form employed in (10), and the lines will only be possible zone-axes when the indices u it K u &c., are all commensurable. In a similar manner equations of transformation can be obtained in which the symbol of the parametral plane is also changed. The equations have been found in Chap, vm, Art. 20, on the assumption that the new axes are zone-axes, and have the same form whether this is the case or not. 7. To show that the face-indices are rational only when the axes of reference are zone-axes. In the transformation given in Chap, vm, Art. 20, the new axes were zone-axes, for the formulae were obtained by means of an- harmonic ratios which involved face- and zone-indices ; these being all necessarily commensurable numbers. The new indices were therefore also commensurable. But in the analysis given in the last Article, the new axes may be any lines whatever; and they will only be possible zone-axes when the ratios u l : v l : w lt &c., are all commensurable. For other lines these ratios will be irrational. From the first of equations (14) we have (A'-A)W 1 + (A'-^)K 1 + (A'-^)IV 1 = (15) The original axes are supposed to be zone-axes, and for a possible face the indices A, k, I are then rational (Chap, iv, Art. 9). If in (15) ti is rational, the coefficients of U 1} V lt W 1 are all rational, and the ratios U l : W 1 and V^ : H^ must also be rational. For take any other possible face (pqr), and suppose it to become (p'q'r) when referred to the new axes ; p', q', r being also supposed to be rational. Re- placing the face-indices in (15) by those of the new face ; we have (p'-p)u 1 + (p'-q)Y 1 + (p'-r)w 1 = (16). Solving equations (15) and (16), we find k (17). (h'-h)(p'-q)-(h'-k)(p'-pY The numbers in the denominators of (17) are all rational : therefore #1, VD W 1 are also rational, and the axis OX' is a possible zone-axis. THE ANHARMONIC RATIO OF FOUR FAOES. 563 Hence if the face-indices referred to the new axes are to be rational, the new axes must be possible zone axes. 8. To find the value of the anharmonic ratio of four tautozonal faces in terms of their indices. Let OT, Fig. 537, be the line of intersection of four origin-planes TOP,, TOP.,, TOP 3 , TOP., parallel respectively to the four tauto- zorial faces (h&lj, (W), (Ws) (W^, no two of which are parallel. Let the origin-planes meet the plane through perpendicular to OT in the lines OP lt OP a , OP,, OP 4 ; and the plane XOY in the lines OZ/j, 0Z 2 , OL 3 , OL 4 . Through any point T on the zone-axis draw a plane TP t P 4 parallel to OX; and x .. let it meet OY in K and the origin " planes in the straight lines TL l /\, TL.f^ TL 3 P 3 , TL 4 P 4 , respectively. Two planes necessarily intersect in a straight line, so that the points P l , P.,, P s , P 4 are co-linear; and so are the points K, L lt Z 8 , L 3 , L 4 . Then using the symbolical notation for the anharmonic ratio given in Chap. VIH, Art. 9, and { L } f^L 3 L 4 } to represent that of four lines meeting at a point we have Fio. 537. . t + BinP 4 OP t = Tf, - Pf t ' where PiP 2 , &c. are the lengths on the line /^/^ intercepted between the several planes. The relation between the sines of the angles and the corresponding lengths on P 1 P 4 can be easily proved from the known properties of plane triangles, in a manner similar to that adopted in proving a corresponding relation in spherical triangles in Chap, viu, Art 19. In a similar manner we can show that the A.B. of the intercepts on a line is equal to that of the pencil of four lines drawn to any point whatever lying outside the line. Hence, we have But And, 362 ; ,/,, = KL t - KL 4 ; Ac. 564 ANALYTICAL METHODS. The lines OL^, OL 2 , &c. are given by the following equations : for OL* , h, - + k, T = 0, z = a b OL 2 , OL 3 , A 3 - + & 3 f = 0, 2=0 a o (18). Moreover the line KL^...L^ is parallel to OX, and the lengths KL^, KL 2 , &c. are the ordinates x in equations (18) when y is made equal to OK. b 7ij b h 2 T T T T VT VT V T KT X/j LJ 2 J 4 LJ 2 A 2j 2 I\. .L/i A Z/2 A -^4 ^KL, KL.-KL, 9. To find the angles which a line PQ makes with the axes in terms of its direction-ratios e, /, g ; also the length intercepted between any two points the co-ordinates of which are given. Let a, ft, y be the angles YOZ, ZOX, XO Y, respectively ; and let A., fj., v be the angles which PQ makes with the axes of X, Y and Z. Construct the parallelepiped PNLQM, Fig. 538, having for edges PL, PM, PN parallel to the axes and PQ for diagonal. Then PL = e.PQ, PM = LF=f.PQ, PN=FQ=g.PQ (20). Again, if P is supposed to be a fixed point (xji/fa) and Q the moveable one (xyz), the equations of PQ are FIG. 538. f .(21). DIRECTION-RATIOS, ANGLES AND LENGTH OF LINE. 565 Now the orthogonal projection of PQ on the axis of X is equal to the sum of the orthogonal projections on the same axis of the lines PL, LF and FQ. Hence, PQ cos \=PL + LF cos y + FQcosfi; similarly, by taking the orthogonal projections of the same lines on the axes of Y and Z, we have L ......... (22). PQ COSIJL = PL cosy + LF+ FQ cos a PQ cos v=PLcos/3 + LFcos a + FQ Introducing into (22) the values of PL, LF and FQ from (20), we have cos A. = e +fcos y 4- g cos ft ; cos p. = e cos y +/+ gr cosa; ......................... (23). cos v = e cos ft +/cos a + g. The angles X, /A, v are therefore known in terms of the direction- ratios e, f, g and the angles between the axes. Again, since the sum of the orthogonal projections of the lines PL, LF and FQ on PQ is equal to PQ, we have PQ = PLcos\ + LFcosn + FQcosv ............ (24). Introducing the values given in (20) and (23), we have 1 = e (e +/COS y + g cos ft) +/( cos y +f+ y cos a) + g (e cos ft +/cos a + g) = e * +/2 + g* + 2fg cos a + 2ge cos ft + 2efcos y ......... (25). If the three equations in (22) are multiplied respectively by PL, LF and FQ, and the equations are then added, we have PQ (PL cos X + LF cos /* + ^ cos v) = (from (24)) PQ 1 = PZ 2 + Z.^' a -1- /V* + 2Z.F . FQ cos a + 2FQ . PL cos /3 + 2PL.LFcoay. But, PL = x- Xl , LF=y- yi , FQ = z-z 1 -, + 2 (2 - zj (x- Xl )cosp+2 (x- Xl ) (y - yi )cosy ...... (26). 10. To find the direction-ratios e, f, g in terms of the angles which the line makes with the axe?. Multiply the second equation of (23) by cos n cos /3, and the third by cosy cos a, and add the products to the first equation; then cos A + cos p cos a cos /3 + cos v cos y cos a = e (I + 2 cos a cos /3 cos y) +/(cos y+ cos a cos + cos y cos 2 a) + g (cos + cos 8 a cos ft + cos y cos a). 566 ANALYTICAL METHODS. Again, multiply the first equation of (23) by cos 2 a, the second by cos y and the third by cos/3, and add the resulting equations; then cos X cos 2 a + cos fj, cos y + cos v cos )3 = e (cos 2 a + cos 2 j8 + cos 2 y) + /(cos 2 a cos y + cos y + cos a cos /3) + g (cos 2 a cos /3 4- cos a cos y + cos /3). Subtract the second equation from the first, and we have e ( I - cos 2 a - cos 2 )3 - cos 2 y + 2 cos a cos /3 cos y ) = eN 2 (say) This equation gives the direction-ratio e in terms of the angles which the line makes with the axes. By a similar process the direction-ratios / and g can be shown to be fN 2 = cos fi sin 2 $ + cos v (cos /3 cos y - cos a) " + cos X (cos a cos j3 cos y) gN 2 = cos v sin 2 y + cos X (cos y cos a cos /3) + cos n (cos /3 cos y - cos a) where N 2 is written for 1 cos 2 a cos 2 ft cos 2 y + 2 cos a cos /3 cos y. But from (25), ecosX+/cos/Li + <7cosi/=l ; .'. from (27) and (27*) cos X {cos X sin 2 a + cos /* (cos a cos |3 cos y) -f cos v (cos y cos a cos /3)} + cos fi. {cos /x sin 2 + &c.} + cos v {cos i/ sin 2 y + &c.} = TV 2 collecting the terms, we have cos 2 X sin 2 a -|- cos 2 /i sin 2 + cos 2 v sin 2 y + 2 cos /* cos i/ (cos /3 cos y - cos a) + 2 cos i> cos X (cos y cos a cos /3) + 2 cos X cos /* (cos a cos /3 cos y) = 1 - cos 2 a - cos 2 /3- COS 2 y + 2 cos a cos /3 cos y (28). Equation (28) gives a relation connecting the angles which any line makes with the axes. When the axes are rectangular cosa = cos/3 = cosy = 0, and equation (28) becomes cos 2 X+cosV + cos 2 j/=l (29). 11. To find the angle between two lines the direction-ratios (efg) and (e'fg'} of which are known. Let PQ, Fig. 538, be one of the lines having the direction-ratios e, f, g, and making the angles X, ft, v with the axes ; and let PQ be the second line having the direction-ratios e, f, g' and making with the axes the angles X', /*', v. Take on PQ any definite length PQ, and with PQ as diagonal construct, as in Art. 9, a parallelepiped having PL, PM, PN for edges parallel to the axes, their lengths are given by equations (20). Let them all be now projected orthogonally on the second line P'Q', The projected lengths are : PQcosO, PL cos X', PMcosp and THE ANGLE BETWEEN TWO GIVEN LINES. .567 Now the orthogonal projection of PQ is equal to the sum of those of the three others. Therefore ......... (30). And from (23), cos X' = e' +f cosy+g cos ft and similar expressions for cos //, cos v. Introducing these values, and those of PL, &c., given in (20) into equation (30), we have cos = e (e +f cos y + g' cos ft) +f(e' cosy+f + g' cos a) + g (e' cos ft +f cos a + g') = ee> +ff + 99' + (fg' + gf} cos a + (ge' + eg') cos ft + (ef+fe) C osy ............... (31). When the axes are rectangular, equation (31) reduces to coa6 = ee'+jy"+gg' ..................... (32). Again, from (31) we can show that sin 2 6 = 2 (fg' - gf)* sin 2 a + 22 (cos ft cosy - cos a) (ge' - eg') (ef -fe 1 ). . .(33) ; where 2 (// - 9f? sin 2 a = (fg 1 - gf}* sin 2 a + ^ef-eg 1 ^ sin 2 ft + (ef -fe')* sin 2 y ; and 2 (cos ft cosy -cos a) (ge 1 - eg 1 ) (ef -fef) = (cos ft cos y - cos a) (ge 1 - eg 1 ) (ef -fe") + (cos y cos a - cos ft) (ef -fe') (Jg 1 - gf) + (cos a cos ft - cos y) (fg' - gf) (ge 1 - eg 1 ). 12. To find in terms of x, y, z the equations of the normal OP to a face (hkl). Let the normal be inclined to the axes at angles A, /n, v. Then from equations (1) of Chap, iv, (34). If a parallelepiped is constructed with edges in the axes and OP for diagonal, similar to that made in Art. 9 with PQ as diagonal, then the equations of the last Articles apply to the normal and to the angles A., /x, v. Hence, if P has the co-ordinates a:, y, z, OP cos A = x + y cos y + z cos ft, OP cos p. x cos y + y + z cos a, OP cos v = x cos ft + y cos a + z 568 ANALYTICAL METHODS. Introducing these values in (34), we have for the equations of the normal to (hkl) a(x + y cos y + zcosfi) _ b (x cos y + y + z cos a) h k When the axes are rectangular, cos a = cos y8 = cos y = 0, and the equations of the normal become ....... (37), k* P where r is the distance of any point P (x, y, z) from the origin. 13. To find the angles X, p., v which the normal to a face (hkl} makes with the axes of reference, and also its direction-ratios e,f,g in terms of the indices h, k, I and of the angles between the axes ; and hence the angle 9 between any two normals. For the normal OPp (say) we have from equations (34) and (23), cos X =p =e +/cos y + g cos ft j k O I cos v = p - = e<. r c Introducing the first values of cos X, cos /a, cos v into (27) and (27*), we have N % =p J - sin 2 a + r (cos a cos ft - cos y) + - (cos y cos a - cos ft) [ N 2 f=p - =p \- (cos y cos a - cos/3) -f 7- (cos /3 cos y cos a) -|- - sin 2 y > ...(39). From (25), (38) and (39) the value of p can now be determined : it is given by y(cosaCOS/3-coSy) + - - - sin 2 a + y(cosaCOS/3-coSy) + - (cosyCOSa - cos$)-f &C. = 2 - g sin 2 o + 22 ^(<*>s/3 cosy- cos a) ........................... (40); where 2 - - 9 sin 2 a = , sin 2 a + yr, sin 2 + - sin 2 y, a 2 a 2 /> 2 c- /; THE ANGLE BETWEEN TWO NORMALS. 569 and 2 ^ (cos /3 cos y - cos a) = T- (cos /3 cosy -cos a) -\ (cos y cos a COS /3) + -T (cos a cos /3 cos y). Introducing the value of p into equations (38), we have OOBX-* k COS M = I _ h* kl a 2 - 2 sin 2 a + 22 f- (cos /3 cosy -cos a) y *2 / y/2- 2 sin 2 a+&c. A T kN ! bt ...(41). A/ 2 - 2 si Let /j, be the normal on any other face (hjkfo ; then from (30) and (34) cfn*-4 f ^+/j<$+p,! ..................... (42). Hence from (39) os^ h, (h . k, I \ = ^ -j-sm 2 a4- j(cosaCOS/3-cosy) + - (cosy cos a- cos /3)> |M-(co8acos/3-cosy) + rsin 2 /3 + - (cos /3 cosy -cosa)> co8^cosy-cosa)+-sin 2 y> t (cos & cos y - cos a) .................. (43). Therefore, introducing the values of p and p, from (40), cos 6 = 2 sin 2 a -J- 2 ' (cos cos y - cos a) ./ |2-2si where ) J2 -^sin 2 (44); in 2 a = ~ sin 2 o + W sin 2 /3 + -5' sin 2 y, a* o* c* and ^""^T ^ (cos /3 cos y cos a) = '-j- '(cos^cosy OM) ' (cos y COS a - cos j8) + '' (008 a COS ^ - 008 y). 570 ANALYTICAL METHODS. 14. To find cos 0, cos X, cos p., cos v of the last Article in terms of the face-indices and the angles a t , (3 t , y t between the poles A, B and C of the axial planes. To obtain the required expressions we have to replace, in the expressions of Art. 13, sin 2 a, &c. by their equivalents in terms of a,, 8 t , y t . From the polar triangles XYZ and ABC, Fig. 539, we have trv' j 7Y & 7? "F" V C 1 From the triangle ABC, , + sin 8, sin y t cos (A = n a). cos 8, cos y, cos a, sin 8 t sin y, and FIG. 539. _ 1 cos 2 a, cos 2 j3, cos 2 y, + 2 cos a, cos 8 t cos y, _ JV, 2 sin 2 a, sin 2 /3, sin 2 y, ~ sin 2 a, sin 2 /3, sin 2 where ^V 7 2 = 1 cos 2 a, - cos 2 8 t cos 2 y, + 2 cos a, cos 8 t cos y, . Similarly, sin 2 a, sin 2 /3, sin 2 y, ^ ( 2 sin 2 y -r- ' -. ~j-^- . , sin'' a, sin-* /3, sm'' y, .(45). Again, from the triangle XYZ, we have cos 8 cos y - cos a = sin 8 sin y cos X= sin /3 sin y cos a, /r , A E \\ ^/ 2 sni ft g i n y/ cos a / = (from (45)) * * " . similarly, cos y cos a - cos 8 = cos a cos jg- cos y= sin 2 a^ sin 2 /3, sin 2 y, ' #,* sin y, sin a, cos /3, sin 2 a y sin 2 /3, sin 2 y x N? sin a, sin /3, cos y, sin 2 a, sin 2 8, sin 2 y / (46). Introducing into (44) the values of sin 2 a, cos 8 cos y - cos a, &c., given (45) and (46), we have COS0 = 2 g-' sin 2 a, + 2 ^ - sin /3, sin y y cos a r / J 2- 2 sin 2 a, + 22 ^- ^osal J2-| sin 2 a / + 22-^- / sin 8, sin y cos ol ....(47). PLANE PERPENDICULAR TO A ZONE- AXIS. 571 Again, ^ a = 1 - 2 cos 2 a + 2 cos a cos /3 cos y = sin 8 /3 sin 2 y - (cos cos y - cos a) 1 = (from (45) and (46))=^- ' sin 4 a, sin 2 0, sin 2 y, sin 4 a / sm 8 /3 / sm :l y / vt (48). sin 2 a, sin 2 0, sin 2 y, Introducing the values of N, sin 2 a, cos cos y - cos a, and those of the similar factors into equations (41), we have hN,\ a <]>, . h COS A = - a / A* H . . / 2 2 sin 2 a, -|- 22 r- sin /3, sin y, cos a, /* = ! .V / A 2 . . / 2 -, s V a + &c. ...(49). 15. To find the equation of an origin-plane perpendicular to a zone-axis OP having the direction-ratios all, bv, cw. Let the zone-axis make angles A, p., v with the axes, and let it meet the perpendicular plane at distance OP = p at the point P, Fig. 540: in this plane take any point T(x, y, z) t and draw the co-ordinates TF, FM parallel respectively to OZ and OX. Then, since TP is perpendicular to OP, the sum of the pro- jections of OM, MF, FT on OP is equal to OP. .'. p xcos\ + y cos p. + ZCQ&V; and the parallel origin-plane is x cos X + y cos p. + z cos v = 0. Introducing from (23) the values of cos A, &c., in terms of the direction- ratios, we have Fio. 540. x(au + by cos y + cW cos ft) + y (au cos y + bv + cW cos o) + z(au cos /? + &K cos a + CW) = ... (50). 16. To find the condition that a zone-axis should be perpen- dicular to a possible face. 572 ANALYTICAL METHODS. If the plane given by (50) is parallel to a face (hkl), the coefficients of x, y, z in (3) and (50) must be proportional. Hence a (all + bv cos y + cw cos (3) _ b (au cos y + bv + cW cos a) h k c (au cos ft + bv cos a + cw) /K1 . I ......... ( &1 )- The zone-indices tf, v, W being integers, h, k, I can only be integers in very special cases; and a zone-axis is not, as a rule, perpen- dicular to a possible face. We shall give a few of the exceptional cases in which the zone-axis is a possible normal. i. In the cubic system, every zone-axis is perpendicular to a possible face ; for a - b - c, and cos a = cos ft = cos y = ; h_k__l_ u ~ v~ w' Hence the zone-axis [WKW] is perpendicular to a face (uvw). ii. In the tetragonal, rhombohedral and hexagonal systems every zone-axis perpendicular to the principal axis (p. 112) is perpendicular to a possible face. Thus, in the rhombohedral system a = /? = y, and a = b = c. Equations (51) then become , 52) U + (V + W) COS a K + (W + U) COS a W + (U + V) COS a For the principal axis, U V W , and h - k - I. When the zone-axis is in the equatorial plane, it may be taken to be the intersection of a face (pqr) with (111). But the indices of this zone are : u = q - r, v = r-p, w=p-q. Equations (52) then become _ J^_ _ k I q - r + (r - q) cos a ~~ r -p + (p - r) cos a ~ p - q + (q -p) cos a ' h k I q-r~r-p~ p-q' the plane (hkl) is therefore a possible face. A zone-axis having any general position in these systems and in the three other systems is not perpendicular to a possible face. iii. We have already proved that a plane of symmetry is perpendicular to a zone-axis, and that an axis of symmetry of even degree is perpendicular to a possible face. The student can easily verify this for (say) (010) in the oblique system. RHOMBOHEDRAL SYSTEM. 573 Some propositions relating to the rhombohedral system. 17. To find the equations of the faces of the fundamental rhombohedron and of the pinakoid, taking/^ F M H- 3 for parameter on the Millerian axes. Let Fig. 541 represent a section of the rhombohedra R and mR by the plane 2 containing OX, V t M and the triad axis, and let us adopt the same letters and notation that were adopted in Chap, xvi in the discussion of class III. Let F/A the polar face-diagonal of (100) meet OX in *; and let 0V be the linear element c measured on the principal axis. From the similar triangles VOs, Os : V t p. = OV: FF= 1 : 2. From the relations established in Chap, xvi, Art. 31, Vjt= 2V M^ 3. /, Os= F,J/-r3=/...(53). But the face of the fundamental rhombohedron through Vp. is parallel to the axes of Y and Z. Hence the equation of (100) is x=f\ (010) y./l (54). (001) z=f) Again, taking the face (111) through V, its equation is x + y + z=3f. (55); for it meets the axis OX at X, where OX '= VM= 3/, and similarly it meets the axes of Y and Z at distances 3/from the origin. 18. To find the equations of the faces of \htt\ = mR (p. 378). We shall suppose the faces of all rhombohedra to be drawn through pairs of the points A, A,, &c., of Fig. 542 in which the faces of the fundamental rhombohedron R {100} meet the dyad axes. As shown in Chap, xvi, these points coincide with the points 8, S p &c., of the figures of rhombohedra and scalenohedra of class III. The face (hll) of the rhombohedron mR has for its equation hx + ly + Iz = tf 574 ANALYTICAL METHODS. where t is an arbitrary constant, which is determined when the face passes through a known point or line. Now this face meets the face (100) in the line A'A ti lying in the equatorial plane, i.e. in the origin-plane parallel to (111). But (100) is given by *=/, and the origin-plane (111) by x + y + z = Q (56). Introducing the values which satisfy these two equations into that of the face (hll), we have nf The equation of the face (hll) passing through A'A it is therefore hx + I (y + z) = (h - l)f, j similarly (Ihl) is hy + l(x + z) = (h- l)f, \ (57). To find the parallel faces, we have only to change the sign of the constant term. Thus (hll) is hx + I (y + z) = (I Ji)f. 19. To find the relation between the equivalent symbols (hll\ and mR, and the length of an edge of the rhombohedron. Let the face (hll) passing through A'A it meet the triad axis 0V at V m , where OV m =mc. Now the co-ordinates of V at a distance c on the triad axis are seen from equations (54) and (55) to be (/,/,/) ; and the equations of the triad axis are x = y = z = t,f. (58); t t being an arbitrary constant depending on the distance of the point from the origin. For the point V m lying in (hll) the values of x, y and z given by (58) must satisfy each equation in (57) ; .'. t t (h + 2Z) = h-l (59). Again, the apex V m being at m times the distance V, then by similar triangles the co-ordinates of V m are (mf, mf, mf). Hence in (58) for the point V m , t / = m. RHOMBOHEDRAL SYSTEM. 575 The two values of t t corresponding to F m must be equal, M = /T74 ........................ < 60 >- This is the same equation as (33) of Chap, xvi, Art. 35. When m is negative, the rhombohedron is said to be an inverse one, and is denoted by -mR. Again, since the polar edges join V m to the points M in the equatorial plane ; and, Figs. 541 and 542, OM=20B = 2acos3Q = aJ3 ............... (61). Also OV m = OV = mc. 2 c 2 ............... (62). But the orthogonal projections of the edges of the rhombohedron on the triad axis are all equal. The coigns X lie therefore on hori- zontal lines parallel to OM, OM\ &c. through the points of trisection of V m V m . By similar triangles V m \T t , V m OM, Fig. 541, we have V m \ : V m M= F W T, : 7 m O = 2 : 3. Hence, Fig. 351, 20. To find the length of a line OP to any point (a;, y, z), and expressions connecting the parameter f and the angle a between the axes with the linear elements c and a. From equation (26) we have, by making (3 = y = a, for the length of any line PQ Hence, making x t = y, = z t = 0, we have OP a = x* + y* + z a + 2cosa(yz + zx + xy) ......... (65). Therefore, since F, where OV=c, has the co-ordinates (//,/), we have OF 8 = c 2 = 3/ 1 (1 + 2 cos a) .................. (66). But from the right-angled triangle OVM, Fig. 541, F,3/ a = c 8 + 3a 8 = (from (53)) 9/ ............ (67). /. from (66) and (67), Sc 8 = (c + 3a') (1 + 2 cos a) ; and 2cosa = ..................... (68). 576 ANALYTICAL METHODS. The scalenohedron {hkl} = mRn (p. 387). 21. To find the equations of the faces of the scalenohedron {hkl}. The equation of the face (hkl) is hx + ky + lz = tf; t depending on the distance of the face from the origin. Its value can be found when the face is taken to pass through A', Fig. 542. Now A' is the point in which (100) meets an origin-line which lies in the equatorial plane (56) and is parallel to (010). Its co-ordinates are therefore given by x =/, y-0, and x + y + z = ; and they may be expressed shortly by (/, 0, -/). Introducing these co-ordinates into the equation of (hkl), we have (h-l)f=tf; and * = />-. Similarly the face (hlk) is given by hx + ly + kz = tj: where t t is found by drawing the face through A fi , the co-ordinates of which are (/, -/, 0). .-. (h-l)f=tj; and^-A-Z. The equations of all the other faces can be shown to have the same form, the coefficients of x, y and z being taken in the two cyclical orders, the constant term remaining unchanged. We therefore have for (hkl), (hlk), . ............... ( D y ). (klh), kx + ly + hz = (h-l)f ' The parallel faces passing through A t , A", - 2* (A + f)} = 3a 2 {*" + 2& (A + 1) + (h + 1)*} + c 2 {4/fc 2 - 4/fc (A + /) + (A + /) 2 } m 2 c 2 } ................................. (72); Now the edge meets the face (Ihk) in the coign X, and for X the co-ordinates x, y, z in (70) must satisfy the equation of the face lx + hy + kz = -(h-l)f. .................... (74). The corresponding length of p is 8'X : it is most easily found by multiplying the numerator and denominator of the first term in (70) by I, those of the second by A and of the third by &, and then adding the numerators and also the denominators. Hence for p = 8'X, we have (A-*)(A + t + /)~ (from < 72 * ...... (76) - The minus sign in (75) arises from the fact that X is the lower coign of the edge : a plus sign is obtained when X (/ in (khf) is taken, In (60) it was seen that the edge of the rhombohedron mR is given by f N/3a 2 + m 2 c 2 . Hence, equation (73) gives the value of m for the auxiliary rhombohedron mR. It is the same as that given in (60) of Chap. xvi. 23. To find of mRn in terms of A, k, I. By the method of construction given in Chap, xvi, Art. 40, the apex V is at distance nO V m = mnc from the origin, m having the i. r.. 37 578 ANALYTICAL METHODS. value given in (73); and, the apex being on the triad axis, its co- ordinates are found by similar triangles to be (mnf, mnf, mnf). Introducing these into the equation of one of the faces (69), we have ......... (77). Equations (73) and (77) are those already found in Chap, xvi, for the conversion of the Millerian symbol {hkl} of the scalenohedron to the corresponding Naumannian symbol mRn. By a slight trans- formation, they suffice also to determine h, k, and I, when the Naumannian symbol is given. When m is negative, both the scalenohedron and the auxiliary rhornbohedron mR are said to be 24. The trapezohedron, a {hkl}. To find the length of the median edge 2S'y. Geometrically the trapezohedron a {hkl} consists of the three pairs of faces of the scalenohedron {hkl} which pass through the alternate median edges interchange- able by rotation about the triad axis.' The faces are therefore hkl Ihk klh Ikh hlk khl. One pair of the faces which are common to the trapezohedron and scalenohedron being supposed to pass through 8', V n and V n , the equations of the median edge S'-y are the same as those given in (70) for S'X. This edge has in the trapezohedron to be ex- tended to meet (khl), Fig. 543, at y ; and the length is found, in the same way as that of S'A, by multi- plying the numerators and denominators of the first, second and third terms of equations (70) by k, h and I respectively. Therefore P = *7 -k(x-f)-hy-l(z+f) FIG. 543. (k-h)(h HEXAGONAL AXES. 579 If 2 and 2-rj are the angles over the obtuse and acute polar edges of the scalenohedron {hkl}, then from equation (74) of Chap, xvi, k-l ^sing h - k ~ sin i) ' " ...... < 79 >- This is the relation (90) used in p. 413 for drawing the trapezo- hedron. Hexagonal axes. 25. The correlation of the four axes to which hexagonal and rhombo- hedral crystals are often referred is fully given in Chap, xvn, Arts. 5 14. We shall therefore assume Miller's (100) to be (OlTl) in hexagonal symbols. Then a and c being the parameters on the equatorial and principal axes, let (hkil) be the symbol of a face, which in Millerian notation is (hkl). As already stated in Chap, xvn, the fourth axis is superfluous when we require to determine the zone-indices, the anharmonic ratio of four tautozonal faces or other relations between the faces and edges. The faces are therefore referred to three axes OX, OY, and OZ ', where the first two axes are inclined to one another at an angle of 120 and are i>erpendicular to the principal axis OZ. Hence the equation of the face (hkil) is (80); it a c the constant t depending on the distance from tho origin. The equation of the parallel origin-plane is h-+k2 + l*=0.... ...(81). a a c 26. To find the formula) of transformation from Miller's to hexagonal notation. Miller's (hkl) becoming (hkil), we have to find the intercepts made by (hkl) on the hexagonal axes OF, OS, Ob', Off'. Referred to Millerian axes, the equations of these lines are : = (from (66))^ (82); 371 580 ANALYTICAL METHODS. p,p,, &c. being the distances of any points on the lines from the origin ; and from (66) and (68) If now the values of x, y and z which satisfy each line are introduced into the Millerian equation of the face we find the intercept made on each of the hexagonal axes. Hence multiplying the numerator and denominator of the first term in each of equations (82) by h, those of the second by k and of the third by I, we have OV n hx + k + lz h-l ~\ h+k+l p t _ hx +ky + Iz _ h I a = (l-k}f = l^k p lt ^/uc+ty + lz_h-l on OZ, OX, OY, OU, The intercepts F", p n p in p llt are in the ratios c-hl : a-=-h : a,-7-k : 7/ ^-i. 1 JL = _^^ J__ ' h + k+l~l-k~ h-l'~ k-h These are the equations found in Chap, xvn, Art. 10. (83). (84). h+k + i +1^ of this last expression is zero, the numerator must be so too ; Each term in (84) is equal to -- y 7 f y -. Since the denominator (85). 27. To find the equations of the six faces of r {hkil}. Let the pole P of (hkT) = (hkil) be so situated that Pr are for all possible values of h, k and I at right angles to one another. Since equations (92) and (92*), which connect the parameters in the two systems of axes, involve these parameters in an exactly similar manner, it follows that the plane (referred to Grassmann's axes) x y z is perpendicular to the zone-axis (referred to zone-axes) =^. = . ... (96). au bv cw This can be easily proved from (89). For let two faces (AA^i), (hjc 2 l 2 ) have for zone-axis the line which has the symbol [uvw] given in Art. 3. Then (96) are the equations of the zone-axis. Now the ray perpendicular to the face (/h&i^i) is at right angles to [tfkw], and so is the ray perpendicular to (h^l^. The plane containing these rays is therefore perpendicular to the zone-axis [uvw]. The ray (hjkj^ is given by jL --L - I_ and the ray (h y k z l 2 ) by x _ y - z ah^W^cl,' WEISS'S AND NAUMANN'S SYMU>|.s. 585 The origin-plane containing these two rays is (kA-Wl + V^-hJ^+^-MJ^U (97). This latter plane is that of the zone- circle containing the poles of tautozonal faces: as was shown in Art. 16, it is not generally a possible face. WeisJs, JVaumann's and levy's notations. 32. Crystallographers have represented the forms of crystals in several ways, and the Milleriau notation is not even yet universally adopted ; but, with hardly an exception, crystallographers who still use other symbols now give the Millerian equivalents also. We shall briefly indicate the principles of three of the most important notations. 33. Weiss represented a face by its intercepts on the axes. Assum- ing the axes to be the same, the Millerian symbol is found by dividing by the common multiple, and writing the denominators (expressed as integers) in the order of the axes X, F, Z. In the hexagonal system four axes were used, and the further transformation given in Chap, xvu, Art. 10, has to be used if Millerian axes are adopted. Weiss's notation is used in Rammelsberg's Handbuch d. Kryst.-Phys. CAemie, 18812. 34. Naumann sought to make the symbol indicate the shape of the form, and made it depend on the characteristic holohedral form of the system. Thus in the prismatic system he takes as fuiidamental pyramid P that which has its apices A, B and C at distances a, b and c on the axes of X, Y and Z. Every other pyramid is supposed to bo drawn through A or B on the axes of X and Y, when its intercepts are given (i) by a : nb : me, or (ii) by na : 6 : me. The pyramid (i) he represents by mPn, (ii) by mPn. The first index in gives the intercept on the vertical axis OZ. The letter P is affected with the long or short sign according as the second index n refers to the makro-axis OY or the brachy-axis OX. By giving to m and n the special values oo and 0, all the special forms of the system can be represented. In the tetragonal and hexagonal systems P carries no sign, for the horizontal axes are interchangeable and the para- meters equal. In the cubic system P is replaced by 0, for all the parameters are equal, and the fundamental pyramid is an octahedron. In the oblique system the same method is employed, but when the index n gives the intercept on the inclined axis OX a stroke is drawn slant- wise across P, and when it refers to OF a horizontal stroke is drawn across it ; it is left unmodified when the indices h and k are equal. To represent pinakoids (hOl) a minus sign must be placed before the symbol, 586 NAUMANN'S AND LEVY'S SYMBOLS. if the face, meeting OX on the positive side of the origin, meets OZ on the negative side ; and a similar artifice has to be adopted in representing the different four-faced forms which we may call plinthoids. In the anorthic system P is affected with the long and short signs and with dashes placed on various sides of it which serve to indicate the directions in which the intercepts on the several axes are measured. The perplexing arrangement of these various modifications in the oblique and anorthic systems renders the symbols ill-adapted for the purpose they were intended to serve. Naumann's notation in the rhombohedral system has been already explained in Chap, xvi ; and its utility for comprehending the relations of, and for drawing, scalenohedra and rhombohedra was there demonstrated. 35. Levy's notation, still used by French crystallographers, is based on the modifications of the edges and coigns of a parallelepiped or of a hexagonal prism which may be regarded as Haiiy's primitive form. According to the system the primitive forms are a cube, a tetragonal prism terminated by the base, a rhombic prism terminated by a pinakoid, a rhom- bohedron, a hexagonal prism and pinakoid, an oblique prism terminated by a pinakoid, and an anorthic parallelepiped. The anorthic parallelepiped, from which all the others (save the hexagonal prism) are derived by making certain of the angles and edges equal, has four different coigns denoted by the vowels a, e, i, o, a being that at the back of the upper face ; three different faces p, m, t (from the word primitive}; and six pairs of different edges, four 6, c, d,f in the base, and two g and h vertical. The back edges of the upper face meeting at a are b and c, b being to the left ; the front edges are d and /. Again, b and d meet at e on the left, c and / at { on the right. The vertical edges are measured downwards, h passing through a and o, g through e and i. Any face is denoted by the intercepts measured on the edges meeting at the coign which it modifies, these lengths being indicated by indices attached to the corresponding edges. Thus the face #(241) of anorthite modifies the coign e at which b, d and g meet ; and the intercepts are in the ratios b-S-2 : d+6 : g-S-1 : the face is given by b l/2 d i/6 g l . The parametral length on a vertical edge g or h is that intercepted by some definite face (arbitrarily chosen) which is parallel to one of the diagonals of the base. Thus x (101) of Fig. 121 may be taken to give the parameter in anorthite : it modifies the coign a and may be drawn through the diagonal ei, when it has the intercepts b, c and h ; it is represented by a 1 . The face y (201) may be drawn through the same diagonal to modify the same coign a and to meet h at distance 2h. Its intercepts are b, c and 2A, or 6-j-2, c-r2, h ; and it is represented by a 1/2 . Any other face parallel to ei modifying a may be given by b+n, c+n, h-^z: it is represented by a eln . If a face in the same zone modifies the coign 0, it is represented by o z ' n ; the intercepts being d+n, f-t-n, h-^-z. Thus o 1/2 is the face Z (201) of LEVY'S SYMBOLS. 587 Fig. 120. In the same way a face through the other diagonal ao inter- cepting on g a length ng+z is represented by e* /w or i* 1 *, according as it modifies the coign on the left or on the right; for instance, e(021) of Figs. 120 and 121 is P and n(021) is e" 2 . The face is also represented by a single letter and index z/n when it is parallel to and modifies the edge to which the letter is attached. If the edge is in the base, z gives the intercept on the vertical, and n on a basal edge. Thus m(lll) of Fig. 120 is/'/*,p(Tll) is c 1 / 8 , and generally/'/" has the intercepts d+n and hz on the edges meeting / at o. Similarly for all the other basal edges. When the modified edge is one of the vertical edges, the symbol is h ylx or g lx if the face is to the right of the observer ; it is v ' x h or vlx g if to the left ; y/x being always greater than unity. Thus /(130) of Figs. 120 and 121 is y a ; for its intercepts are c-j-2, f+1 and #-7-0, since a line through o bisecting the edge c meets the diagonal ei at the centre of gravity of the triangle aio. Similarly z (130) is *g. With the introduction of planes and axes of symmetry some of the angles and edges become equivalent, and fewer different symbols are needed. Since the symbols denote forms, it is always necessary to know the system and class of the crystal to perceive the character of the form. Although the symbol, such as b l/x d l ' g l ", resembles Millerian intercepts a-^-h, b-k, c+l, the indices in the two notations are only in exceptional cases the same. For Miller's axes of X and Fare, as a rule, parallel to the diagonals of the base of LeVy's parallelepiped, XX, being parallel to ao and YY, to ei. The conversion of LeVy's symbols to Millerian involves there- fore a transformation of axes in all systems except the cubic and rhombo- hedral. In the rhombohedral system the parallelepiped becomes the funda- mental rhombohedron p ; the two apices are denoted by a, the median coigns by e, the polar edges by b, and the median edges by d. Assuming the same fundamental rhombohedron, p is {100} ; and a face of any form is given by intercepts which are (i) b llk b lllt b l!l , or (ii) b ljk d l ' k d 1 ' 1 , according as the face modifies an apex or a median coign. i. In these forms the intercepts are the same as Miller's, for we may suppose the origin to be shifted to the apex. Hence ftWftWftW is {Ml}, all the indices being positive. An obtuse rhombohedron {Ml} is a*', and it is direct or inverse according as A^J : a 1 is {111}. A face (AOJ) parallel to a polar edge is 6* /J ; the form is a scalenohedron, except when h=l or h = 2l: in the former case b l is the rhombohedron {101}, in the latter ft 2 is the hexagonal bipyramid {201}. ii. In b l/lt d l/tc d l/l the Millerian indices are numerically the same, but the signs of one or two of the indices have to be changed according to the coign modified by the face. If the face is that of a direct scalonohedrou modifying the coign /*, Fig. 545, it meets the edges V,p, /*/*, ftp, at distances b+h, d+k, d+l respectively. Suppose the origin to be at P,, 588 LEVY'S SYMBOLS. and the face to be shifted to a point between V, and /x, where the intercept measured from F, is b-z-h. In its new position the face meets F,/x' and V tf i." produced beyond F,, i.e. on the negative sides of the origin ; and its symbol is (hkl), where h k l>0. The pole lies in the spherical triangle ra'a,, of Fig- 546. When therefore the scalenohedra are direct and the superior faces modify inferior median coigns of {100}, b llh d l/k d l/l is {hkl}. Let now a superior face d llh b l/k d l/l meet a superior median coign \t (say) : it belongs to a direct or inverse scalenohedron according as its pole lies in the spherical triangle ra,,c or in a ll cm l of Fig. 546. When the face is shifted so as to meet the axes at distances from the origin F, equal to the intercepts on the parallel edges meeting at fi t , the Millerian symbol is (hkl), where h k + l>0. The direct form is distinguished from the inverse by writing the intercept b l/k first or last. A particular case is given when the face passes through a polar face-diagonal, when the symbol d klh b l d l is abbreviated to e k/h equivalent to {hkk}. When the intercepts on the median edges are equal, the form is a rhombohedron et/ l ={hli} : it is direct or inverse according as h- 2^0. When the face is parallel to a median edge, the form is given by d hl1 and is the same as {hQl}. The particular cases e 2 and d l are the hexagonal prisms {211} and {101} respectively. CHAPTER XX. ON GONIOMETERS. 1. THE angles of crystals are measured by instruments called goniometers, of which there are several kinds in use. The contact-goniometer invented by Carangeot, and used by Rome de 1'Isle and Haiiy, is still employed for the approximate measurement of the angles of large crystals, and more especially when the faces are rough or deficient in lustre, or the crystal is attached to a large piece of matrix. The essential part consists of two flat straight bars often called limbs (1,1' of Fig. 547) of steel Fio. 547. or brass connected by a pin and screw k. One edge * of each bar is made accurately straight; and these edges are placed in contact with the two faces the angle between which is required. If the limbs are also perpendicular to the edge in which the faces in- tersect, the angle between them is the Euclidean angle and ite 590 THE CONTACT-GONIOMETER. supplement that between the normals (Chap, u, Art. 2). To secure accurate contact the crystal and limbs are held between the eye and the light; and when the limbs are correctly adjusted, the screw is tightened and the limbs carefully removed from the faces. The limbs are now placed on a graduated circle or semicircle in the centre of which the pin fits, and the included angle is read off. The circle may be divided to degrees, half- or quarter-degrees. But to give correct readings on the circle the continuations of the bars must be cut down, as shown in Fig. 547, to straight edges which are radii of the circle and necessarily parallel to the edges s placed in contact with the crystal-faces. Or one limb may be so adjusted FIG. 548. that its central line passes through the zero of the scale as shown in Fig. 548. The edges at which readings are made are usually bevelled to a thin edge. To be able to measure the angles between faces partly en- veloped by matrix or other crystals, slots are made in the bars as shown in the figures, and the bars are sloped to points at the ends placed in contact with the crystal. The bars being shortened, their ends can be introduced into shallow cavities, and their tips brought into contact with the faces. 2. The, reflexion-goniometer. In 1809 Wollaston (Phil. Trans. 1809, p. 253) invented this goniometer by which the angle between the normals of two faces is determined by the reflexion of a well-defined signal S (say) from each face in succession ; and all modern goniometers suitable for accurate measure- ment depend on the laws of reflexion. Thus, \^ suppose Fig. 549 to represent a section of a FIG. 549. THE REFLEXION-GONIOMETER. 591 crystal by a plane perpendicular to the edges of tautozonal faces ; and let the signal S, lying in the plane of section to the left of the face A B, be reflected from AB in a fixed direction. The incident and reflected rays both lie in the plane ABC, and the normal ON bisects the angle between them. If ON Z is supposed to be equal to ON lt a rotation about through f\iV 1 ON. i = I\CBE brings the face EC exactly into the position of AB, and S will be reflected from BC in the fixed direction. A second rotation through N t ON 3 brings the face CD into the position of AB, and S is reflected from CD in the fixed direction; and so on for all the faces in the zone. If then is in the axis, perpendicular to a graduated circular disc, which can be turned about this axis, the angles N^ON^ N t ON it (fee. between the face-normals can be determined. The graduated circle may be either vertical or horizontal, and the instrument is accordingly known as a vertical-circle or horizontal-circle goniometer. We shall describe one of each kind now in common use. The axis has been supposed to be equally distant from the faces, and then the angles turned through are accurately equal to those between the normals. But if, as in the figure, is unequally distant from AB and BC, a rotation about through N V ON Z brings BC parallel to AB but not into coincidence. If then S is at a short distance from the crystal, the angle of incidence on BC in its new position is ^**m than that on AB, and the reflected ray is not in the fixed direction. If ON 2 is less than ON V , the crystal must not be turned quite so far, and we get an error in the measured angle due to defective centering. If ON^ is equal to ON l , the total angle N^ON^ is determined with accuracy, for CD is brought into coincidence with AB. Hence in the measurement of N Z ON Z , where we have an increase in the radius, an error is made of the opposite kind to that made in measuring N^N.^. These errors dis- appear if *S' is at a very great distance, or, as it is shortly expressed, at infinity : the rays falling on the crystal from are then all parallel, and the angles of incidence on the successive faces are equal when the latter are brought into parallel positions. The error of centering is therefore eliminated by the use of a very distant signal. It can also be eliminated by bringing each edge into the axis of the instrument. Thus, if the crystal is turned about B through the angle CBE=N 1 ON 2 , BC is brought into the con- tinuation of AB, and the portions of the faces used in reflecting the signal are those near B. To get N 2 ON 3 , the crystal must be displaced BO as to put C in the centre, and so on for all the angles in the zone. The use of signals at a short distance from the crystal has therefore serious dis- advantages. Consequently, when considerable distances cannot be secured, collimators giving parallel rays are used. 592 THE VERTICAL-CIRCLE GONIOMETER. The vertical-circle goniometer. 3. The instrument shown in Fig. 550 was designed by Professor Miers (Min. Mag. ix, p. 214, 1891). The circular disc L, graduated on the edge to half-degrees, is screwed to the axle which is accurately fitted in a bush in the vertical portion of the stand. The stand rests on levelling screws, by means of which the axle can be placed horizontally. The vernier, reading to minutes, is engraved on the edge of a disc V, which is fixed to the stand ; and the two discs L and V are kept in contact by a circular steel spring. G is a screw which clamps L to V, and D is a slow motion screw for the purpose of fine adjustment. The axle and disc L are rotated by the milled head B. FIG. 550. The crystal-holder consists of (a) a bar of metal bent at right angles, (6) a quadrantal arc turned by an axis Q, and (c) a second axis of rotation R at right angles to Q. (a) One arm of the bent bar is forked and slides on the disc L about a screw which, with the aid of a washer, clamps it firmly to L when the crystal has been adjusted : this movement is used to centre the edge or, approximately, the middle of the crystal when the latter is small, (b) In a bush at the end of the bent bar remote from L the axis Q works, THE VERTICAL-CIRCLE GONIOMETER. 593 and it is rigidly attached to the quadrantal arc. Rotation about this axis brings the image of S from a crystal-face into coincidence with that of S seen in the mirror (Art. 4). (c) The second axis R (a stout pin) works in a bush at the other end of the quadrantal bar, and gives rotation about an axis at right angles to Q. Rotation about R brings the image of S in a second crystal-face (one as nearly as possible at 90 to the first is the best to take) into coincidence with that of S in the mirror. The crystal is attached by stiff wax to a small plate, which rests in a slit in R; or when very small, the crystal is stuck on a pin by a solution of shellac in alcohol and the pin fixed to R in any convenient way. After the edge has been brought by the axes Q and R into parallelism with the axis, the crystal is centered accurately by sliding the holder in the slot on the disc L. To test the centering the following simple apparatus may be used. A vertical knitting-needle is supported on a stand, and carries a cork which moves stiffly on it. Through the cork a second needle passes horizontally, the end of which is brought close to the crystal edga The disc and crystal are then turned through 180. If the edge is not in the centre, it will describe a semicircle, and in the second position will be separated from the needle by double its distance from the centre. The cork is then moved so as to bring the needle half-way to the edge, and the crystal-edge is moved by a translation of the holder along the slot until it is close to the needle. The operation is repeated until the edge remains close to the needle during the whole of a revolution. 4. The signals. To give the bright signal a diamond-shaped hole is cut in a board, which is placed, with a diagonal of the diamond horizontal, across the lower part of a window at the opposite side of the room to that occupied by the goniometer-table. On a stand outside the window a mirror is placed, which reflects the brightest portion of the sky and directs the light through the hole on the crystal : a heliostat and screens for regulating the in- tensity of the light may be used instead. For the faint signal 2 a white horizontal line on the wall below the bright signal may be used, but it is inconvenient. A small mirror of blackened glass is therefore attached to the stand of the goniometer ; and the faint signal is the image in this of a narrow slit cut in a board, which is placed across the upper part of the window with the slit horizontal. The mirror. On the cylindrical axis A of the stand a tube E slides which can be clamped to A in any desired position by the 38 594 THE VERTICAL-CIRCLE GONIOMETER. screw-head e. To E another tube F is fixed at right angles, in which a cylinder carrying the mirror can slide without rotation. It is clamped by a screw-head hidden behind the mirror. The mirror inclined at a convenient angle is attached to the end of the cylinder by a screw having a capstan-head, and can be turned into any position round the cylinder F. It should, by rotation round the axis A, be first placed so that the image of S is seen near the bottom and that of 2 near the top by an observer looking across the crystal. The mirror is adjusted so as to reflect the bright signal S in a plane parallel to L, and is then securely clamped by the capstan-head //. When once adjusted the mirror should as far as possible be left in the same place. The crystal is easily brought into the plane of reflexion by moving the plate carrying it, away from or towards the disc L. To be able to adjust the crystal accurately, the image of S should be visible in the mirror when the observer looks across the crystal. The image of S is sometimes used as second signal 2, but such a use has many grave disadvan- tages and is quite unnecessary. 5. Adjustments. The plane of reflexion through the crystal must be parallel to the disc L and vertical. It is desirable, though not essential, that the plane of reflexion should be perpendicular to the faint signal and therefore to the window. i. A line in a vertical plane through S perpendicular to the window is traced as accurately as possible on the table. The goniometer is then placed on the table so that the crystal is in the vertical plane through this trace, and the disc L very nearly parallel to it. Let the average distance of the crystal from L be n mm. A cardboard is now attached to the window having two narrow parallel slits in it 2n mm. apart. One slit is placed accurately in the position of the vertical diagonal of the rhombus forming the bright signal S, the other 2n mm. to its right, with regard to the observer. If the surface of L is a bright smooth plane the image of the slit covering S is seen by reflexion in it ; and when the disc is correctly adjusted the image in L should exactly cover the other slit seen by direct vision across the edge of Z. But since the disc seldom gives a distinct image, a plate of blackened glass is cemented to it by wax at some convenient position near the edge ; the wax being sufficiently plastic to enable us to adjust the plate at right angles to the axis by gentle pressure. By slightly moving the goniometer (the crystal being maintained in the vertical plane through the trace) the image THE VERTICAL-CIRCLE GONIOMETER. 595 in the glass is brought to cover the second slit seen directly. The disc is then turned through 180 by the milled head Ii, and the coincidence of the image and second slit is tested by looking over the opposite edge of the disc. Any divergence is halved by shifting the goniometer. The glass is then pressed until the images become coincident ; and the disc is then turned back to its original position, and the coincidence of the images again tested. The divergence, if any, is halved by a change in the direction of the goniometer- axis, and the remainder corrected by again pressing the glass ; the disc is now turned through 180, and the adjustment tested. The process is repeated until the image always coincides with the second slit as the disc L is turned, when the plane of reflexion through the crystal is parallel to that of L. A strong rod is now screwed to the table pressing gently against the milled heads of the two levelling screws to the left; so that, when once adjusted, the goniometer can be immediately put in position by bringing these milled heads into contact with the rod. ii. A porcelain dish full of clean mercury is now placed on the floor so that the signal S, still limited to the narrow slit in the card, is seen by reflexion in the mercury by an observer looking across the crystal. If the crystal has bright parallel faces for instance, a good cleavage-fragment of calcite we proceed by means of the axes of rotation of the holder to bring the image of the slit from one of the faces into coincidence with that seen in the mercury. The crystal is now turned by means of the milled head B through 180, when the image of the slit from the parallel face should coincide with that in the mercury. If the image is displaced to the right or left the axis is not horizontal, and the levelling screws must be turned until the deviation is halved. The remaining half is corrected by rotating one of the axes of the holder; and the crystal is then turned so as to bring the image from the original face into coincidence with that in the mercury. Any deviation is again corrected in the same way, and the process continued until the adjustment is correct. The first adjustment must now again be tested, and any error produced by the levelling must be corrected by the process described under (i). The two processes have to be carried out alternately until the goniometer is adjusted. iii. The mirror on A is now adjusted so that the images from the crystal-faces (known from testing by reflexion from mercury to be all in a vertical plane parallel to L) coincide with 382 596 THE VERTICAL-CIRCLE GONIOMETER. the image of S reflected in the mirror. The instrument is now ready for use; and with careful handling it will remain in good adjustment for a considerable time. The adjustments should however be occasionally tested; and any defects corrected in the manner described under (i), (ii) and (iii). 6. Measuring a crystal. A freehand sketch of the crystal is made, and if necessary two or three from different sides. The faces are lettered in any way deemed convenient, and any marks on the faces noted. Such examination will generally enable the observer to perceive the symmetry ; and this can be tested, if the crystal is translucent, by examining the directions of extinction between crossed Nicols. The observer then judges of the zones which it is desirable to measure. The crystal is now attached to the plate by stiff wax with one face of the zone to be measured as nearly as possible parallel to the plate. The plate is then attached to the holder in such a way that the crystal is in the plane of reflexion and one edge is nearly in the direction of the axis. The crystal is then turned by the milled head B until the image from the face parallel to the plate is seen near that of S in the mirror. By turning the milled head R the two images are brought into coincidence. The crystal is now turned by means of B until the image of S from another face (preferably one nearly at 90 to the first face) is seen near that from the mirror. The two images are brought into coincidence by rotation of the screw-head Q. This adjustment generally affects the first one, and the first face has to be again adjusted; and the process is repeated until both faces are accurately adjusted. The images of *S from all the other faces in the zone can by rotating L be then brought in succession to cover that in the mirror. The edge to be measured is now to be centered, or if the crystal is very small its middle point is put as nearly as possible in the axis. The adjustment of the zone must be checked, and any slight error caused by the change of position of the holder on the disc L corrected. A screen is then interposed, which cuts off all light from the mirror except that from the faint signal. The crystal and L are next turned by means of B until the image from an easily recognised face is bisected by the faint signal. The corresponding angle is read off', and recorded against the letter used to denote the particular face : any peculiarity of the image, such as its being double or elongated, being noted at the same time. THE VERTICAL-CIRCLE GONIOMETER. 597 Distinctly double images should be separately determined. The disc and crystal are now turned until the image from the next face is bisected by the faint signal, and the corresponding angle entered below the first one. If the crystal is not re-centered for each edge, the process is continued without interruption until the first reading is again obtained. The difference between each pair of successive readings gives the angle between the corresponding faces. 7. The author gave in the Proceedings of the Canib. Phil. Soc. iv, p. 243, 1882, an analysis of the error due to defective centering which shows that the error can never exceed half the angle subtended at either signal by the extreme positions of the edge when the readings are made. If the signals are, as at Cam- bridge, at a distance of about 7 -25 metres from the crystal, a dis- placement of the edge through 4 mm. will not cause a greater error than 1' in the reading. The error arising from defective adjustment of the zone, or of a slight deviation from zonality, is one of the second order, and is inappreciable within the limits which occur with an instrument which only reads to minutes. The faces of few crystals admit of measurement to half-minutes ; and, in passing from one crystal to another, divergences quite beyond those due to errors of centering are continually met with in the angles between corresponding faces. Considerable experience and judgment is needed in selecting the angles to be used in the computation of elements and the theoretical angles. A fixed telescope supplied with cross-wires may be used to determine the positions of the images from successive faces, and may replace the mirror and faint signal. Vertical-circle gonio- meters provided with a telescope, with or without a collimator, are fairly common ; but the adjustments of the telescope necessary for its axis to be strictly in the plane of reflexion are somewhat trouble- some : they are made with two plates of glass in the way described in Art. 10. The telescope may however be employed in conjunction with the mirror, when it merely serves to focus the two signals and to give the position of the faint one when it is concealed behind the crystal. The necessity for very careful adjustment of the telescope is then avoided. The horizontal-circle goniometer. 8. Fig. 551 represents a vertical section through the axis of the goniometer, Model n, made by Herr Fuess of Berlin to whose 598 THE HORIZONTAL-CIRCLE GONIOMETER. courtesy I am indebted for this figure and for Figs. 547, 548, 552, and 553. The stand consists of a thick metal-plate o supported on three legs which are provided with levelling screws. The axis con- sists of three concentric conical shells fitted the one within the other, and working in a conical bush in the plate o. The outer cone b carries a circular disc d, to which are fixed the verniers and the FIG. 551. standard B which carries the observing telescope. The verniers are at opposite extremities of a diameter and read to 30". The cone b is clamped by the screw a, so that the telescope and verniers can be fixed in any convenient position. For fine adjustment in determin- ing refractive indices a slow motion screw is attached, the head of which is just visible behind the leg. The second conical shell e fits into b and carries the graduated disc f. This cone terminates in the milled rim g for turning the axis. The cone e and graduated disc are clamped by the screw (3, and the corresponding slow motion screw is just visible behind the leg. The third cone h works within e and is useful for turning the crystal during adjustment without turning the graduated disc : it ends in the milled head i and is THE HORIZONTAL-CIRCLE GONIOMETER. 599 clamped to the cone e by the screw I. Within h is a cylindrical steel axis which carries the crystal-holder : it terminates at the lower end in a screw working in k, so arranged that the crystal can be raised or lowered for the purpose of bringing it into the centre of the illuminated field 9. The crystal-holder consists of two plane slides m and m at right angles to one another and two arrangements for circular motion in planes also at 90 to one another. A movement of translation for centering the edge can be given to the slides m and m by screws a and a. To rn is attached a 'felly' r having on its inner (concave) side a groove in which a plate t of equal curvature and with toothed edge can move. Motion is given to the plate by the screw x, the thread of which works in the toothed edge of t. Any line in the plane of t can be therefore inclined to the vertical at any desired angle within the range of the felly. On t is fixed a similar arrangement of felly and toothed plate, which gives circular motion in a plane at right angles to the first The centres of the two circular motions are nearly coincident and lie at a short distance above the plate u which carries the crystal. This holder can be used with the vertical-circle goniometer de- scribed in Art. 3, but here the back-lash in the screws a, x, &c. which cannot be prevented after the instrument has been used for some time renders the holder untrustworthy (Dauber, Pogg. Ann. cm, p. 107, 1858). 10. The collimator is carried on a fixed standard C, and its axis can be adjusted by screws above ft perpendicular to the axis of the goniometer. With it different forms of signal can be used, the most useful being one formed by two circular plates which have their centres on opposite sides of the slit and very nearly touch in the centre of the field. For determining refractive indices a straight slit, like that of a spectroscope, should be used. The slits can be illuminated in iiny convenient way; e.g. for measuring a crystal, by an incan cent burner, and for refractive indices, by a Bunsen's burner flame coloured by sodium, lithium, 9 Chalcopyrite, 240, 492 Chrysoberyl, 507 Cinnabar, 417, 525, 146 Circular polarization, 146 oblique crystals, 175 prismatic, 204 tetragonal, 259 267 cubic, 301, 326 rhombohedral, 359, 415, 523 hexagonal, 446, 459 twin - crystals, 480, 523 Cleavage, 1, 11 Clinographic drawings, 65, 49 Clinohedral class (oblique), 177 Clinohedrite, 178 Closed form, 230 Cobaltine, 333 Coign, 17 Combination of forms, 150 -plane of twin-crystals, Combinations, 202 Complementary forms, 148, 180 twins, 463, 477 Composite crystals, 461 Constancy of angle, 10 Constituent molecules, 11 Co-polar edges, 360, 427 faces, 427 Copper, 308, 471 pyrites, 240, 492 Cordierite, 208 Corrosion-figures, 147 of twin-crystals, 465 Corundum, 403 Crossed dispersion, 144 Crystal, definition of, 1 Crystallization, 3 Crystallographic axes, 23 Crystallometric angles, 21, 121 Cube, 285, 314 Cubic coigns, edges and faces, 285 crystals, drawing, 55, 340 optical characters, 140 planes of symmetry, 303 - system, 275, 139 formulae, 283 symmetry - axes, 132, 276, 311 twins, 466 Cuprite, 302 Curie (J. & P.), 135, 147, 332 Cyanite, 155, 554, 62 Cyclical order, 291 Cyclograph, 158 Czapski (S.), 601 Dana (E. S.), 510 (J. D.), 509, 538 De lisle (Rome), 10, 461, 589 Deltoid dodecahedron, 321, 334 Derivative forms, 10 Des Cloizeaux (A.), 146, 177, 190, 193, 252, 502, 539, 546 Development, equable, 6, 22 unequable, 6, 8, 22 Dextrogyral, 176 twins of quartz, 519 Dextro-tartaric acid, 176 Diacetyl-phenolphtalein, 268 Diametral zones, 79 Diamond, 337, 467, 468, 481 Dihedral angle, 8 pentagonal dodecahedron, 316 Dihexagonal-bipyramidal class, 426 -pyramidal class, 426 Dihexagonal prism, 345, 369, 452 Dimetric system, 139 Dioptase, 362 Diplohedral, 15, 205 dihexagonal class, 455, 426 INDEX. 607 Diplohedral.ditetragonal class, 243, 224 ditrigonal class (cubic), 275 hexagonal class, 448, 426 tetragonal class, 260, 225 trigonal class (rhombo- hedral), 360, 344 trigonal class (cubic), 275 Direct and inverse rbombohedra, 378 Direction-ratios of line, 564 of zone-axis, 559 Dirhombohedral forms, 357, 439 Dispersions of optic bisectrices, 143 Disphenoid (tetragonal), 233 Ditetragonal-bipyramidal class, 224 -pyramidal class, 225 prism, 228 Ditrigonal bipyrainidal class (rhombo- hedral), 424, 345 pyramidal class (rhombo- hedral), 344 scalenohedral class (rhom- bohedral), 344 coign, 292, 289 prism, 410, 419 Dodecahedral planes of symmetry, 303 Dodecahedron, deltoid, 321, 334 dihedral pentagonal, 316 dyakis-, 329, 341 rhombic, 286, 315, 334 tetrahedral pentago- nal, 324, 343 Dolomite, 514 Domatic class (oblique), 177 Doublet, 464 Drawing crystals, 48 cubic crystals, 340, etc. rhombohedral crystals, 371, 376, etc. twin-crystals, 468, 486, 517 implements, 67 reduction of scale, 68 Dufet (H.), 196 Dyad axes, 19, 113, 134 conditions for, 111 in cubic system, 278 Dyakis-dodecahedral class (cubic), 328, 275 Dyakis-dodecahedron, 329, 341 Edge, possible, 2 Electrical phenomena, 146, see Pyro- electricity Elements of a crystal, 29 of symmetry, 21 relations be- tween, 108 Enantiomorphous, 146, 149, see Cir- cular polarization Epidote, 193, 181 Epsomite, 202 Equable development, 6, 22 Equations to normal, 29, 567 Equatorial axes, 430 plane, 365, 427 Error of centering, 591, 597 Erythroglucine, 261 Esmarkite, 551 Ethylene-diamine sulphate, 267 Kulrr's theorem of rotations, 123 Eulytine, 477 Face angle, 9 -symbol, 27 common to two zones, 43, 559 Faces, 1 Fahlerz, 339, 473 False faces, 2 Fedorow (E. von), 601 Felspars, plagioclastic, 543 see Orthoclase Fletcher (L.), 491, 493, 495 Fluor, 309, 474 Foote (H. W.), 178, 420, 542 Form, 21 -symbol, 28, 149 Forms, closed, 230 complementary, 148, 180 open, 179 special and general, 173 Formation of crystals, 3 Fresnel's wave-surface, 141 Fuess' goniometers, 597, 601 Fundamental pyramid, 349, 585, etc. rhombohedron, 361, etc. Oadolin (A.), 119 Galena, 467, 472, 307 Garnet, 810, 8, 17, 307 Gauss (J. K. F.), 90 General forms, 174 Glaucodote, 507 Gold, 467, 807 Goldschmidt (V.), 601 Gonioid class (oblique), 177 Gonioids, 210 Goniometer-adjustments, 594, 600, 602 contact-, 689 horizontal-circle, 597 reflexion, 590 theodolite-, 601 three-circle, 604 signals, 598, 599, 602 vertical circle, 692 Goslarite, 203 Grassmann's method of axial repre- sentation, 581 Greenockite, 420 Groth (P.), 269, 359, 480 Guanidine carbonate, 267 Gypsam, 186, 529, 143 608 INDEX. Habit, 5 Haidinger (W.), 55, 241, 493, 515 Harmonic ratio, 108 Harmotome group, 539 Haiiy (B. J.), 10, 461, 589 Haiiyne, 481 Heat, effect on crystal- elements, 247, 347, 484 optical characters, 145, 189, 192 Hematite, 405, 525 Hemi-domes, 210 Hemihedral, 16, 149, 180 class (prismatic), 197 - (oblique), 177 with inclined faces class (cubic), 276 Hemihedrism, 259 Hemimorphic axes, 175, see Pyro- electricity class (oblique), 173 (prismatic) , 209, 197 (rhombohedral), 344 -hemihedral class (hex- -hemihedral class (te- tragonal), 224 Hemitrope, 461 Hemitropic twins, 462 Herschel (J. F. W.), 143, 146 Hexad axis, 19, 120, 127, 131, 136, 426 conditions for, 112 Hexagonal axes of reference, 430, 579 crystals, drawing, 62 optical characters, 141 bipyramid, 400, 448 pyramid, 428, 581 acleistous, 444, 452 prisms, 345, 367, 432, 444, 452 symbols, transformation of, 436, 441, 580 system, 426, 139 twins, 527 zone, 345 Hexagonal-bipyramidal class, 426 -hemimorphic class, 426 -pyramidal class, 426 -trapezohedral class, 427 Hexakis-octahedral class (cubic), 303, 275 -octahedron, 304, 340 -tetrahedral class (cubic), 333, 276 -tetrahedron, 335, 341 Hidden (W. E.), 491 Holohedral, 16, 180, 259 Holohedral class (oblique), 178 (prismatic), 197 (tetragonal), 224 (cubic), 275 (hexagonal), 426 Homologous faces, 21 Horizontal dispersion, 143 Hornblende, 530, 181 Hydrogen potassium dextro-tartrate, 203 trisodic hypophosphate, 178, 196 Ice, 403 Icositetrahedron, 296 pentagonal, 298 Idocrase, 251, 248 Imbedded crystals, 4 Inclined dispersion, 143 Indices of faces, 25 optical refraction, 142 zone-axis, 34 law of rational, 26, 562 Intercepts, 25 Intercrossing twins, 463 Interpenetrant twins, 462, 463, 474 Inverse and direct rhombohedra, 378 scalenohedron, 394 lodyrite, 454 Isogonal zones, 121 Isometric system, 139 Isotropic crystals, 140 Juxtaposed twins, 462 K {hkl}, 149 Kayser (G. E.), 554 Klein (C.), 479 Labradorite, 554 Lssvo-dextrogyral twins of quartz, 521 gyral, 176 twins of quartz, 521 -tartaric acid, 176 Lang (V. von), 58, 119 Langemann (L.), 543 Lead-antimonyl dextro-tartrate, 447 Lead-antimonyl dextro-tartrate + po- tassium nitrate, 460 Lead nitrate, 327 Levy's notation, 586 Line, direction-ratios of, 564 inclination of, to axes of refer- ence, 565 length of, 565 Linear elements, 161 projection, 70 Lithium potassium sulphate, 446 Liveing (G. D.), 135 fi{hkl}, 149 INDEX. 609 p{hkl}, 210 Made, 461 Macro-axis, 198 -diagonal, 198 -dome, 201 - -pinakoid, 202 Magnetite, 309, 466 Mallard (E.), 510 Manebach-twins of orthoclase, 531 -Baveno twins of orthocla.se, 535 Marbach (H.) 146 Maskelyne (N. S.-), 119 Mean lines, optic, 142 Measuring crystals, 596, 603, 604 Median edges of rhoinbohedron, 360 Meionite, 261 Merohedral, 16, 149 Merohedrism, 259 Metastrophic, 18 Meyer (0.), 491 Microcline, 543, 546, 553 Miers (H. A.), 302, 592 Miller (W. H.), 26, 50, 76, 90, 102, 105, 107, 139, 148, 582, 601 Millerian axes in hexagonal crystals, 427 Mimetic twins, 464, 478, 501, 506, 510, 543, 553, etc. Mimetite, 451, 506 Mispickel, 507, 207 Mitscherlich (E.), 145 Model of axes, 25, 36, 151 Mohs (F.), 55, 139 Monoclinic system, 138 inB, mRn, Millerian equivalents, 380, 389, 577 Multiple twins, 464, 488 Naumann (C. F.), 140, 485 Naumanu's axes, 65 notation, 379, 387, 585 Nepheline, 446 Neumann (F. E.), 76, 143, 145, 190 Nigrine, 490 Normal-angle, 9 equations of, 29, 567 inclination of, to axes, 568, 571 Normals, 9 angle between two, 568 Norrenberg (J. G. C.), 143 Oblique crystals, drawing, 59 _ optical characters, 143, 175, 188, 192 system, 172, 138 formula, 181 twins, 528 Obtuse bisectrix, 142 Octahedral system, 139 Octahedron, 285, 6, 13, 15 Octant, first, 342 Octants, 24 adjacent, alternate and op- posite, 299 Oligoclase, 169 Open form, 179 Opposite octants, 299 Optic axis, 141 Optical anomalies, 479 characters, 140 examination of crystals, 145 extinctions, 144 phenomena, variation with temperature, 145, 189, 192 see Circular polarization Origin, 1, 23 -plane, 35 equation of, 557 Orthoclase, 190, 531, 60, 99, 143, 145, 181 Orthogonal projections, 49 Orthographic drawings, 55, 49 Orthorhombic system, 138 ir{hkl}, 149 Parallel- faced hemihedral class (cubic), 275 hemihedral class (rhom- bohedral) 344 Parallelism of faces, 1 Parameters, 24 Parametral plane, 24 ratios, 28 Pediad class (anorthic), 148 Pediou, 148, 174, 210, etc Penfield (8. L.), 147, 178, 241, 420 Penta-erythrite, 270 Pentad axis inadmissible, 19, 121, 134, 136 Pentagonal dodecahedron, dihedral, 316 dodecahedron, regular of geometry, 319, '2 dodecahedron, tetrahetlral, 324, 343 -icositetrahedral class (cu- bic), 275 icositetrahedron, 298, 342 Pericliue, 548, 552 -twins, 548 Phacolite, 526 Phenakite, 363 Phillipsite, 539 Piezo-electricity, 146 Pinakoid, 155, 174, 210, 220, 367, etc. Piuakoidal class (anorthic), 154 Piresonite, 211 Plagihedral class (cubic), 284, 275 Plagioclastic felspars, 543 Plane perpendicular to zone-axis, equation of, 571 39 610 INDEX. Planes of Symmetry, see Symmetry- planes Plans and elevations, 49 Plinthoid class (oblique), 178 Polar axes, 175 edges, 112 diagonal of rhombohedron, 366 triangles, 151 - ditrigonal class {cubic), 276 (rhombohedral), 418 tetragonal class, 254 - trigonal class (cubic), 275 Poles of faces, 76 of zone-circles, 81 Polysynthetic twins, 464, 545, 515 Possible edge, 2 face, 2 Potassium-antimonyl dextro-tartrate, 203 sulphate, 503 tetrathionate, 178 Pratt (J. H.), 211, 491, 541 Primitive circle, 77 form, 10 Principal axis, 112, 344 optical, 141 Prism, 180, 201 dihexagonal, 345, 369, 452 ditetragonal, 228, 50 ditrigonal, 410, 419 hexagonal, 345, 367, 432, 444, 452 tetragonal, 226 trigonal, 345, 351, 353, 409 Prismatic class (oblique), 178 crystals, drawing, 59, 198, 203, 221, etc. optical characters, 143, etc. system, 197, 138 formulae, 211 twins, 497 Projections, clinographic, 49, 65 linear, 70 orthogonal, 49 stereographic, 76 Pseudomorphs, 2 Pseudo- symmetry, see Mimetic twins Pyramid, acleistous, see Acleistous see Bipyramid Pyramidal class (prismatic), 197 (tetragonal), 224 -hemihedral class (hexa- gonal), 426 -hemihedral class (tetra- gonal), 225 system, 139 Pyrargyrite, 527, 420 Pyrites, 332, 476 Pyro-electricity, 146 Pyro-electricity, oblique crystals, 175, 178 prismatic, 210 tetragonal, 255, 259, 268 cubic, 328, 333 rhombohedral, 359, 409, 416, 418, 421, 424, 520 hexagonal, 445, 451, 460 twin-crystals, 465, 520 Pyromorphite, 451 Pyroxene, 181 Quadratic system, 139 Quartet, 464 Quartz, 413, 519, 9, 17, 146, 463 Quercite, 177 Bath (G. vom), 477, 547, 548 Ratio, anharmonic, see A. R. of two tangents, 97 Rational indices, law of, 26 conditions for, 562 Redruthite, 509 Regular system, 139 Rhombic dodecahedron, 286, 315, 8, 13 section of plagioclases, 549 system, 138 Rhombohedral axes of reference, 346, 366, 427 class, 344 crystals, drawing, 52, 62, 371, 405, 413 crystals, optical cha- racters, 141 -holohedral class, 344 system, 344, 139, 573 system, analytical re- lations, 573 system, relations be- tween indices, 354 system, equivalent Mil- lerian and Nauman- nian symbols, 380, 389, 574 system, formulsa, 354, 395, etc. system, twins, 513 Rhombohedron, direct, and inverse, 378 equations to faces, 573 drawing, 376, 384 Rose (G.), 489, 521, 537, 545, 554 Rotation of plane of polarization, see Circular polarization Ruby, 404 Rutile, 482 INDEX. 611 Sadebeck (A.), 472, 494 Salt, 302 Sanidine, 143 Sapphire, 400, 404 Scacchi (A.), 505 Scalenohedral class (rhombohedral), 365, 344 (tetragonal), 224, 225 Scalenohedron, rhombohedral, 387, 576, 12 tetragonal, 233 Scapolite, 261 Scheelite, 261, 496 Secondary faces, 11 Seignette salt, 203 Silver fluoride, 269 Simple crystal, 461 Smith (G. F. H.), 604 Smithsonite, 211 Sodalite, 475 Sodium chlorate, 327, 480 lithium sulphate, 420 periodate, 359, 463 potassium dextro-tartrate, 203 ammonium dextro-tartrate, 203 Spangolite, 147, 420 Special forms, 173 Sphenoid, 199, 231, 270 Sphenoidal class (oblique), 175 (prismatic), 197, 198 - (tetragonal), 225, 270 -hemihedral class (tetra- gonal), 224 -tetartohedral class (tetra- gonal), 225 Sphere of projection, 76 Spinel, 446, 307 Staurolite, 509 Steno (N.), 9 Stereogram, 79 Stereographic projection, 76 Stereo-isomers, 176 Strife, 2, 465 Strontium nitrate, 327 -antimonyl dextro-tartrate, 446 Structure of crystals, Haiiy's theory, 10 uniformity of, 134 Struvite, 211 Strychnine sulphate, 268 Succin-iodimide, 269 Sugar, cane-, 176 Sulphur, 207 Supplementary twins, 477 Symmetric twins, 462, 463, 493, 522 Symmetry, 15 relations and conditions, 108 Symmetry - axes, 18 angles between, 281 combinations of, 124, 128, 130, 276 conditions for, 110 dyad, see Dyad hexad, see Hexad in cubic system, 276, 311 intersections of sym- metry-planes, 126 of odd degree, 112 - perpendicular to n-fold axis, 128 - possible, 121 tetrad, see Tetrad - triad, see Triad uni terminal, 132, see Pyro-electricity centre of, 15 -planes, 17 conditions for, 110 cubic, 303 dodecahedral,303 least angles be- tween, 117 Systems, 138 r{hkl}, 149 r 9 {hkl}, 270 Tartar-emetic, 203 Tartaric acid, 175 Tartrates, 203, 259, 446, 447, 460 Tautomorphous, 210 Tautozonal faces, 20, 38, 87, 559 Tetartohedral, 16, 149, 180, 259 class (cubic), 275 (rhombohedral), 344 Tetrad axes, 19, 120, 127, 131 conditions for, 111 in cubic system, 276 Tetragonal crystals, drawing, 59, 50, 243, 253, etc. optical characters, 141 prisms, see Prism pyramids, see Pyramid Boalenohedron, 233 sphenoid, 231, 270 system, 224, 139 formula, 285 twins, 482 trapezohedron, 264 Tetrahedral - pentagonal - dodecahedral class (cubic), 275 -pentagonal-dodecahedron, 324, :m class (cubic), 311, 275 Tetrahedron, 313, 334, 15 612 INDEX. Tetrakis-hexahedron, 288, 335 Theodolite-goniometer, 601 Thermo-electricity, 332 Topaz, 221, 208 Tourmaline, 420, 146 Transformation of axes of reference, 104, 560 Trapezohedral class (tetragonal), 263, 225 (rhombohedral), 408, 344 (hexagonal), 457, 427 -hemihedral class (hex- agonal), 427 Trapezohedron, cubic, 296 hexagonal, 458 rhombohedral, 411, 578 tetragonal, 264 Traube (H.), 446, 460 Triad axes, 19, 126, 132, 345 conditions for, 114, 117 in cubic system, 277, 311 Triakis-octahedron, 292 -tetrahedron, 319, 334 Triclinic system, 138 Trigonal bipyramidal class (rhombo- hedral), 422, 345 -pyramidal class (rhombohe- dral), 344 -trapezohedral class (rhom- bohedral), 344 bipyramid, 410, 422, 425 coign, 292 prism, 345, 351, 353, 409 pyramid, 345, 350, 419 Trimetric system, 138 Triplet, 464 Truncated edges, 52 Tschermak (G.), 525 Twin-axis, 462 determining position of, 556 - a line in a face perpen- dicular to one of its edges, 554 -crystals, 461 drawing, 468, 486, 499, 512, 517, etc. -face, 462 Twin-lamelJffi, 464, 500, 515, 545 law, 464 determining, 467, 483 Twinned individuals, relations between indices, 485 Twinning, pseudo-symmetry produced by, see Mimetic twins tests of, 464 theory of, 511, 538, 542, 553 Twins of anorthic system, 543 cubic system, 466 hexagonal system, 527 oblique system, 528 prismatic system, 497 rhombohedral system, 513 tetragonal system, 482 Unequable development, 6, 8, 16, 22 Uniaxal crystals, 141 Uniterminal axis, see Symmetry-axes Vesuvianite, 252, 248 Wave-surface of light, 141 Weiss (C. S.), 40, 139, 585 Weiss's zone-law, 39, 559 Wellsite, 539 Whewell (W.), 26 Witherite, 501 Wollaston (W. H.), 590 Wulfenite, 2 259 Wulff (L.), 327 Zircon, 250, 491 Zonal point, 70 Zone-axis, 1, 33 equation of, 558 equation of plane perpen- dicular to, 571 direction-ratios of, 559 -circle, 77 indices, 33, 37 law, Weiss's, 39 symbol, 38 Zones, 1, 33 diametral, 79 face common to two, 43, 559 hexagonal, 345 relations of, 33 Zwilling, 461 CAMBRIDGE : PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. 1 4 1965 FEE 25 1978 S/S; Form L9-12c-7,'63(D8620s8)444 The RALPH D. REED LIBRARY DEPARTMENT OK GBOIXXJY UNIVERSITY of CALIFORNIA LOS ANGELES. C,\UF. UC SOUTVON flEGOML UVUftv ftOUTt lllllllllllll A 000838333 3