REESE LIBRARY 
 UNIVERSITY OF CALIFORNIA 
 
CRYSTALLOGRAPHY 
 
 STORY-MA SKELYNE 
 
Bonbon 
 
 HENRY FROWDE 
 
 OXFORD UNIVERSITY PRESS WAREHOUSE 
 AMEN CORNER, E.G. 
 
 ({ten? 
 
 MACMILLAN & CO., 66 FIFTH AVENUE 
 
CRYSTALLOGRAPHY 
 
 A TREATISE ON THE 
 
 MORPHOLOGY OF CRYSTALS 
 
 BY 
 
 N. STORY-MASKELYNE, M.A., F.R.S. 
 
 PROFESSOR OF MINERALOGY, OXFORD 
 HONORARY FELLOW OF WADHAM COLLEGE 
 
 AT THE CLARENDON PRESS 
 1895 
 
PRINTED AT THE CLARENDON PRESS 
 
 BY HORACE HART, PRINTER TO THE UNIVERSITY 
 
PREFACE 
 
 THE establishment of Crystallography as a science at 
 the hands of Haiiy dates back a hundred years from the 
 present time; and midway in that period (in 1839) Pro- 
 fessor W. H. Miller published his now classical Treatise. 
 
 The notation by index-symbols, the representation of 
 the relative positions of crystal faces by the distribution 
 of their ' poles ' on a sphere, xhe stereographic projection 
 of the sphere and of the poles lying upon it, and some 
 of the methods that Miller employed in dealing with the 
 symbols of crystal faces, may indeed be shown to have 
 been introduced in various memoirs by earlier geometers. 
 Thus Bernhardi (1809), Neumann (1823), Frankenheim 
 (1826), and Grassmann (1829) had all represented crystal 
 faces by their normals ; the three latter had projected 
 the poles of faces and of zones on the great circles 
 of a sphere : Neumann further represented a face by 
 a symbol that was a first approximation to the simple 
 form suggested by Whewell (1824-5) an d afterwards by 
 Justus G. Grassmann (1829) before it became in the 
 hands of Miller an effective implement by which to rear 
 the structure of modern Crystallography. Dr. Whewell, 
 Miller's immediate predecessor in the Cambridge Chair, 
 had indicated the method of deriving the symbol of a face 
 from those of two other faces truncated by it. 
 
 But, after the labours of previous crystallographic writers 
 
vi Preface. 
 
 have been thus recognised, the methods and systematic 
 treatment of Crystallography in Miller's Treatise remain 
 indisputably his own. The rationality of the anharmonic 
 ratio of any four tautozonal planes was his, independently 
 of a similar result afterwards published in the memoirs of 
 Gauss ; and to him belongs the merit of combining the 
 methods of his predecessors with the stereographic pro- 
 jection into a complete and elegant system, in which the 
 power of the indicial symbols in grouping the faces of 
 the forms of crystals, and in colligating tautozonal faces 
 as well as tautohedral zones, was manifested, and the 
 foundation laid on which the laws of crystal symmetry 
 have been well established. 
 
 The system thus identified with the name of Miller 
 has met at length with an almost universal acceptance. 
 Adopted forty years ago in Vienna by the school of 
 crystallographic physicists in which Grailich and V. von 
 Lang were pioneers, it has in recent times formed the 
 crystallographic language employed in such complete trea- 
 tises as those of Liebisch and of Mallard, in the encyclo- 
 paedic Mineralogy of Dana, and in that great storehouse 
 of crystallographic information, the Zeitschrift of Groth. 
 
 The growth of Crystallography during the last sixty years 
 has resulted, notably, from the recognition of the limitations 
 imposed on its symmetry by the homogeneity of a crystal. 
 By various methods, Hessel in 1829, Bravais, von Lang, 
 and Gadolin in 18667, and, among later investigators, 
 more especially Sohncke, Schonflies and Fedorow, have 
 defined the varieties of symmetry which can be presented 
 by crystals. 
 
 The present Treatise, dealing solely with the Morpho- 
 logy of crystals, represents the substance of courses of 
 
Preface. vii 
 
 lectures in which it was often found necessary to treat 
 Crystallography in the simplest form compatible with strict 
 geometrical methods ; the classes, in such cases, consisting 
 of students from other departments of science in which 
 familiarity with mathematics was not demanded. On 
 other occasions students of physics with high mathematical 
 training formed the class. 
 
 I trust that, while endeavouring to fulfil the requirements 
 of readers of the former kind, this book will not be found 
 lacking in demonstrations that may satisfy those of the 
 latter class, for whom indeed it may be said to commence 
 at Article 53, p. 65. Several chapters, those for instance 
 on Symmetry, are written with more detail than a geo- 
 metrical student might recognise as needed ; but this 
 seemed unavoidable if the usefulness of the book was 
 not to be confined to such readers. 
 
 Among my old students there have been some who 
 have taken a great interest in the subject and in my book : 
 to their relations towards it and towards myself I may 
 only advert here in a few too inadequate words. To 
 a remoter time belongs Professor Lewis, formerly my 
 colleague at the British Museum, and now Professor Miller's 
 successor in the Mineralogical Chair at Cambridge. 
 
 More especially are my cordial thanks due to the friend 
 who succeeded me in the Keepership of the Mineral 
 Department of the British Museum, Mr. L. Fletcher, for 
 numerous and valuable suggestions and additions to the 
 original MS. and to the proofs which he has so carefully 
 revised. 
 
 To Mr. Miers' skilful hand I am indebted for numerous 
 figures, the drawing of which I found, with failing eyesight, 
 to be increasingly difficult. 
 
viii Preface. 
 
 Various features that I have adopted from the remark- 
 able Treatise (1866) of my old friend Viktor von Lang, 
 formerly, for a time, my colleague at the British Museum, 
 justify my numbering the distinguished Vienna Professor 
 among those to whom I am as much indebted for the 
 matter of my book as I am for their sustained friendship. 
 
 It is intended in a future volume to deal with the 
 physical problems necessary to the practical crystallo- 
 grapher. It being my purpose in that as in the present 
 work to bring crystallography into more familiar use by 
 students of chemistry, mineralogy, and petrology, it was 
 necessary that the treatment of the subject should not 
 demand the advanced mathematics needed for reading parts 
 of the master-work of Liebitsch. That the principles of 
 crystallographic optics can be thus treated without the 
 surrender of exact geometrical method is evidenced by 
 Mr. Fletcher's Tract on the Optical Indicatrix ; and 
 there are other parts of crystallographic physics capable 
 of being represented with a like simplicity. 
 
 The greater part of the present Treatise was long ago 
 written, and indeed in print ; and for the delay in its publi- 
 cation its author is alone responsible. The causes have 
 been numerous, partly arising from the distractions of 
 other unavoidable and not unimportant duties, but mainly 
 from a certain indecision habitual to the workman, who has 
 felt how ' easy it is to begin but to finish how difficult.' 
 For the acknowledgement of the longsuffering and kindly 
 treatment he has experienced at the hands of that unique 
 institution, the University Press, he has not words. 
 
 N. S.-M. 
 
CONTENTS 
 
 CHAPTER I. 
 
 ON THE GENERAL PROPERTIES OF CRYSTALS. 
 
 PACK 
 
 The crystalline condition i 
 
 Morphological characters 3 
 
 Physical characters 5 
 
 CHAPTER II. 
 
 MODES OF EXPRESSING AND REPRESENTING THE RELATIONS 
 BETWEEN THE PLANES OF A SYSTEM. 
 
 Elementary considerations as to the methcds of estimating the mutual 
 
 inclinations of the planes of a system 15 
 
 Expression for the direction of the edge formed by two planes . . 22 
 
 A crystalloid system. Principle of rationality of indices . . . 25 
 On the sphere of projection, and the principles of its stereographic 
 
 representation 27 
 
 Expressions for determining the position of a pole on the sphere . . 41 
 
 CHAPTER III. 
 
 ON ZONES AND THEIR PROPERTIES. 
 
 Expressions for a zone 44 
 
 Relations connecting three tautozonal planes ..... 46 
 On t-he signs of the indices of a plane as determined by the position of 
 
 the plane in respect to a given zone . . . . . . 51 
 
x Contents. 
 
 PACK 
 
 Relations connecting four tautozonal planes . . . . . 55 
 
 Analytical investigation of the zone-law ...... 65 
 
 On isogonal zones 75 
 
 CHAPTER IV. 
 
 THEOREMS RELATING TO THE AXES AND PARAMETERS OF 
 A CRYSTALLOID SYSTEM. 
 
 On changing the axial system to which a crystalloid plane-system is 
 
 referred . So 
 
 The axes of a crystalloid system are necessarily origin-edges or face- 
 normals ........... 91 
 
 CHAPTER V. 
 
 ON THE VARIETIES OF SYMMETRY POSSIBLE IN A CRYSTALLOID 
 SYSTEM OF PLANES. 
 
 Application of the principles of geometrical symmetry to crystals and 
 
 crystalloid plane-systems . 97 
 
 Conditions for a crystalloid system of planes to be symmetrical to one 
 
 of its planes 106 
 
 Conditions involved in a crystalloid polyhedron being symmetrical to 
 
 more than one of its planes 109 
 
 The type of symmetry in which <f> = . . . . . . 121 
 
 The type of symmetry in which < = . . . . . . 126 
 
 4 
 
 The type of symmetry in which <f> = 1 30 
 
 Q 
 
 The type of symmetry in which <p = ->- 134 
 
 Three heterozonal planes of congruent symmetry 144 
 
 General discussion of the systematic triangle . . . . . 149 
 
 CHAPTER VI. 
 
 CRYSTALS AS CRYSTALLOID POLYHEDRA. 
 
 Mero-symmetry 156 
 
 On composite- and twin-crystals 172 
 
Contents. xi 
 CHAPTER VII. 
 
 THE SYSTEMS. 
 
 PACK 
 
 The Cubic (or Tesseral) System 188 
 
 Holo-symmetrical forms ........ 188 
 
 Mero-symmetrical forms 204 
 
 Combinations of forms 219 
 
 Twinned forms 232 
 
 The Tetragonal System 245 
 
 Holo-symmetrical forms . . 245 
 
 Mero-symmetrical forms . . . . . . . . 252 
 
 Combinations of forms ........ 264 
 
 Twinned forms . 268 
 
 The Hexagonal System 273 
 
 Holo-symmetrical forms . . 273 
 
 Mero-symmetrical forms 283 
 
 Combinations of forms ........ 309 
 
 Twinned forms ..... ..... 318 
 
 The Ortho-rhombic System . ........ 331 
 
 Holo-symmetrical forms 331 
 
 Mero symmetrical forms 336 
 
 Combinations of forms 339 
 
 Twinned forms 345 
 
 The Mono-symmetric (or Clino-rhombic) System . . . . 352 
 
 Holo-symmetrical forms 352 
 
 Mero-symmetrical forms ........ 358 
 
 Combinations of forms 360 
 
 Twinned forms 364 
 
 The Anorthic System . 370 
 
 Holo-symmetricai forms 370 
 
 Mero-symmetrical forms 373 
 
 Combinations of forms 3/3 
 
 Twinned forms .......... 375 
 
 CHAPTER VIII. 
 
 THE MEASUREMENT AND CALCULATION OF THE ANGLES OF 
 CRYSTALS. 
 
 The goniometer 388 
 
 Crystallographic calculation 417 
 
xii Contents. 
 
 PAGE 
 
 The Oblique Systems 427 
 
 The Anorthic System ..... .427 
 
 The Mono-symmetric System ..... 436 
 
 The Rectangular-axed Systems 441 
 
 The Ortho-rhombic System ........ 446 
 
 The Tetragonal System ........ 448 
 
 The Cubic System 453 
 
 The Hexagonal System ....... .463 
 
 CHAPTER IX. 
 
 THE REPRESENTATION OF CRYSTALS. 
 
 Parallel projection ....... 474 
 
 Orthogonal projection 475 
 
 Gnomonic projection .......... 492 
 
 Description of the Plates 
 INDEX 
 
CRYSTALLOGRAPHY. 
 
 CHAPTER I. 
 
 ON THE GENERAL PROPERTIES OF CRYSTALS. 
 
 
 1. The Crystalline Condition. 
 
 Most substances, whether chemical elements or compounds, 
 assume the crystalline condition when they become solid under 
 circumstances favourable to the gradual and unconstrained deposi- 
 tion of their particles. Under conditions conducive to the growth 
 of separate individuals, crystals are polyhedra with plane faces and 
 without re-entrant angles. 
 
 The processes by which substances pass into the crystalline 
 condition belong generally to one or other of the following 
 classes : 
 
 (a). From sublimation : in this case the substance passes directly 
 from the gaseous to the solid condition. 
 
 Iodine, arsenic and camphor are familiar examples of crystal- 
 lisation from sublimation. 
 
 (&). From fusion : when the substance passes directly from the 
 liquid to the solid condition. 
 
 Both bismuth and sulphur afford examples of crystallisation from 
 a fused mass during cooling ; and water, on freezing, assumes the 
 crystalline condition, though the individual crystals of ice are in 
 general difficult to distinguish. 
 
 (c). From solution-, the deposition of crystals may result from 
 
2 On the general properties 
 
 diminution of the solvent capacity of a liquid holding the substance 
 in solution : and this may arise from change of temperature, or of 
 osmotic pressure ; or from removal, by evaporation or otherwise, 
 of the solvent liquid ; or, again, from that liquid losing an ingre- 
 dient that imparts to it higher solvent power, or receiving a fresh 
 ingredient that diminishes that power. 
 
 These various methods of crystallisation from solution are uni- 
 versally employed in the processes of the laboratory, and, on a 
 larger scale, in the operations of technical chemistry. 
 
 (d}. By change in the solid condition. It sometimes happens that 
 a substance passes from a non-crystalline to a crystalline condition, 
 or from one crystalline type to another, without passing through 
 the liquid or gaseous state. 
 
 Non-crystalline (vitreous) arsenious anhydride in lapse of time 
 developes octahedral crystals ; again, the transition of (tetragonal) 
 crystals of the red mercuric iodide into those of the yellow (ortho- 
 rhombic) type, and the reconversion from the latter type into the 
 former, as the temperature is successively above and below i26C., 
 is a familiar example of a numerous class of changes. Of such 
 transformations the remarkable optical researches of Mallard and 
 others have contributed illustrations; prominent among these are 
 the passage of boracite at a temperature of 265 C., and of leucite 
 at about 500 C., in each case from a pseudo-cubic symmetrically 
 grouped aggregate of crystals (orthorhombic in the case of boracite) 
 to a single crystal of cubic type, and the reconstruction of aggre- 
 gates of pseudo-cubic type below .those temperatures (see Article 166, 
 p. 187). 
 
 Further, under any of the preceding processes, new compounds 
 produced as a result of the mutual decomposition of two or more 
 substances may assume the crystalline condition. 
 
 Many of the largest and finest crystals met with among minerals 
 have resulted from natural processes involving mutual decompo- 
 sition, and numerous illustrations of artificial processes of the same 
 character are afforded by the microscopic crystals produced in the 
 course of micro-chemical analysis. 
 
 2. The characters of crystals may be referred to two different 
 classes : 
 
of crystals, 3 
 
 I. Morphological characters, which result from the distribution 
 and geometrical relations of their plane faces ; 
 
 II. Physical characters, which result from their homogeneity and 
 the distribution of the physical properties. 
 
 3. I. Morphological Characters. 
 
 Conclusions drawn from observations on numerous crystals of 
 one and the same substance are the following : 
 
 On any individual crystal, faces presenting similar physical 
 characteristics, superficially exemplified in the lustre, striation, 
 hardness, c., of their surfaces, may usually be recognised as 
 recurring on different parts of the crystal, and, indeed, recurring in 
 a certain ordered and symmetrical arrangement. In general the 
 faces of a crystal are not all alike either in form or in the characters 
 of their surfaces ; they exhibit lustre that may vary from an ada- 
 mantine or a metallic brilliance, through a glassy or a nacreous 
 reflection, to a dulness that reflects no image : or, again, a striation, 
 that in some cases consists of a fine linear tracing, in others of 
 a coarse channelling of the surface ; the direction of striation being 
 generally parallel to certain edges of the crystal. 
 
 All the faces of the crystal fall into one or into several such 
 groups or forms ; each form comprising the symmetrically recurring 
 faces characterised by similar features and properties. 
 
 4. Crystallographic law. All the faces belonging to any one 
 crystal are connected by certain geometrical relations which obey 
 a simple law known as the Law of Rationality of Indices. 
 
 Different crystals of the same substance may present an indefinite 
 variety in their forms and combinations of forms : such crystals 
 may even have no form in common, and therefore no faces that are 
 directly comparable with each other. Yet, by virtue of the law just 
 mentioned, it is practicable to determine from any one crystal 
 of a substance all the faces that may possibly occur on the same 
 or different crystals of that particular substance, and to establish 
 a system of planes that shall be characteristic of it. By reference 
 to this system of planes, it is possible to establish the morphological 
 relationship of all crystals of the same substance, whatever faces 
 they may exhibit. 
 
 The faces of crystals of a given substance, even when they belong 
 
 B 2 
 
4 On the general properties 
 
 to the same form upon the same crystal, are usually found to differ 
 in size ; but while they obey no law as regards their relative mag- 
 nitude, the mutual inclination of every pair is the same as that of 
 every corresponding pair, whether of the same or different crystals 
 of the given substance, at the same temperature. 
 
 5. Crystallographic systems. Whereas the different forms upon 
 crystals of the same substance differ not only in the mutual inclina- 
 tions of their faces, but also in other characteristics, such as lustre, 
 striation, &c., they will all be found to conform to the same type 
 of symmetry. 
 
 Further, while in all crystals of the same substance the morpho- 
 logical features are distributed in accordance with one type of 
 orderly arrangement or symmetry, it will be hereafter shown that 
 there can be six and only six such types of symmetry to one or 
 other of which every crystal whatsoever can be referred. 
 
 These are termed the crystallographic systems and are distin- 
 guished as: 
 I. The Cubic system. 
 
 II. The Tetragonal system. 
 
 III. The Hexagonal system. 
 
 IV. The Ortho-rhombic system. 
 
 V. The Mono-symmetric (or C lino-rhombic) system. 
 
 VI. The Anorthic system. 
 
 6. Crystallographic elements. The geometrical relations connect- 
 ing the faces of a given substance are expressed by means of certain 
 constants termed the crystallographic elements, and, from what has 
 been said in Article 4, it follows that these are characteristic of the 
 particular substance. The crystallographic elements, and, as follow- 
 ing from them, the essential morphological characters of a crystal, 
 thus depend, not on the relative dimensions of its faces or lengths 
 of its edges, but on the relative directions of both, and therefore on 
 the dihedral angles of the latter. 
 
 While the crystallographic elements of an individual substance 
 are thus the same at a given temperature for all the crystals of that 
 substance, those of the different substances which crystallise in any 
 one of the systems present indefinite variety. Crystals of the Cubic 
 system are an exception to this statement ; for the crystallographic 
 
of crystals. 5 
 
 elements in this system are the same for all, so that analogous 
 forms have identical angles on such crystals. A like constancy 
 of angle attaches also to certain forms on crystals of the Tetra- 
 gonal and Hexagonal systems : the faces of such forms are parallel 
 to a particular direction, termed the morphological axis, peculiar to 
 such crystals, around which, either singly or in pairs, they are 
 symmetrically repeated in quadruple, sextuple, or triple recurrence. 
 
 7. II. Physical Characters. 
 
 Crystals are further distinguished from other matter in that 
 while the physical characters of ordinary uncrystallised matter are 
 generally the same, in crystals they are different, in different 
 directions. At the same time it is generally true that the properties 
 characteristic of any given direction in a crystal are found to 
 characterise also other directions in it. But the different directions 
 thus similarly endowed, which may be considered as if they were 
 repetitions of the first direction, will be found to be repeated sym- 
 metrically and in general accordance with the principle of symmetry 
 that controls the repetitions of the morphological features on the 
 crystal. 
 
 A brief review of some of the distinctive physical properties that 
 characterise crystals will serve to illustrate the above principle. 
 
 8. Elasticity and cohesion. 
 
 (a). The discussion of the conditions of elasticity in crystals 
 involves very complicated mathematical expressions. In the most 
 general case the determination of the stress requisite to produce 
 a given homogeneous strain involves as many as twenty-one inde- 
 pendent constants termed coefficients of elasticity : the constants, 
 however, become reduced as the symmetry of the crystal assumes 
 a higher type, and are reduced to three in the case of a cubic crystal. 
 In terms of these constants the coefficients of elasticity of volume 
 and the coefficients of elasticity of figure, or the rigidity, can be ex- 
 pressed. 
 
 It is found that directions which correspond in their morpho- 
 logical relations are endowed with the same characters as regards 
 elasticity. 
 
 (). When a solid body is acted upon by forces which produce 
 deformation it may behave in one of various ways. 
 
6 On the general properties 
 
 The deformation may be permanent or it may be only tem- 
 porary : if it be wholly or in part permanent, the original arrange- 
 ment of the particles is not restored when the stresses to which the 
 strain was due have ceased to operate ; a ductile body, for example, 
 undergoing a permanent elongation under a tensile stress ; a malle- 
 able body admitting of entire change of form by successive impacts 
 from a hammer or the continuous pressure of a roller, in both cases 
 without disruption. It is obvious that though a crystal may possess 
 such qualities, the exercise of them is incompatible with its particles 
 retaining their crystalline arrangement. 
 
 (c\ Glide-planes. Certain remarkable changes, however, involving 
 permanent deformation in a crystal without disruption have been 
 effected by pressure at the edges or quoins (solid angles). 
 
 Fig. i. 
 
 A simple mode of producing such a deformation in a cleavage- 
 rhombohedron of Iceland spar is one first suggested by Baumhauer. 
 Three edges of the rhombohedron meet on the morphological axis 
 at an apex where the three plane angles are each obtuse, being 
 109 8' 12". If a knife-blade represented by the plane bgg f be 
 applied at a point c' not very far from that apex, with its faces 
 perpendicular to one of the edges e, and be subjected to a steady 
 pressure or a slight blow, a wedge-shaped fissure c'gg'k is readily 
 formed, and the portion of the crystal between the blade and the 
 original face r of the rhombohedron is moved to a new position. 
 
of crystals. 7 
 
 A plane parallel to the edge <?, and equally inclined to the faces 
 forming that edge, is a not infrequent face of a calcite crystal. If 
 / be a plane parallel to such a possible face, meeting the two sides 
 of the fissure in the line gg' , the block included between the plane 
 / and the edge e has been sheared, the sheared part of the face r 
 becoming r'\ the particles of the line e between the blade and the 
 apex have glided until the displaced portion becomes kk' \ the 
 particles similarly aligned on all lines parallel to the edge e have 
 in the same way glided into new positions in the crystal-block, 
 rolling over, doubtless, through a half-turn during the deformation. 
 
 The glide-plane e' bisects the angle between the original face r 
 and the sheared face r* along the line mm'. The shear-plane 
 passing through the edge e cuts the glide-plane / and the plane 
 of the blade perpendicularly. The angles kk'm and kk'm' originally 
 obtuse are now acute, while the obtuse angle mk'm' has remained 
 unaltered in magnitude ; k' in the deformed part of the crystal is 
 thus no longer an apex situated on the morphological axis but 
 a lateral quoin similar to c. It will be easily seen that the 
 deformation has been effected without any change of volume of 
 the crystal. 
 
 (d). Cleavage. A characteristic property of most crystals is 
 that of their being more or less easily fissile along certain directions 
 which are termed cleavage-planes or cleavages. The cleavages (and 
 also the glide-planes) are usually parallel to prominent or crystallo- 
 graphically important faces of the crystal. 
 
 The facility with which the cleavages on the same crystal may 
 be effected will be found to be different where the faces to which 
 they are parallel have a different morphological significance: 
 where, on the other hand, they are crystallographically equivalent 
 (i. e. where they belong to the same form) the cleavages will be 
 equally facile. 
 
 The relative facility of two or more cleavages presented by 
 a crystal is frequently of assistance in determining the morpho- 
 logical symmetry. 
 
 The glide-planes, in the case of deformed crystals, are also 
 planes along which disruption can be easily effected. 
 
 (e). Impact-figures. When the cohesion is suddenly overcome as 
 
8 On the general properties 
 
 by the impact of a pointed instrument on a plane surface of 
 a crystal, linear cracks radiating from the point are produced, 
 giving rise to what is known as an impact- figure. These cracks 
 may be parallel to cleavages when such intersect the surface, but 
 they are often due to the production of glide-planes as described 
 in paragraph (c). 
 
 Since the directions of the cracks conform to the symmetry of the 
 crystal, the impact figures are often serviceable in determining the 
 character of that symmetry. 
 
 9. (/). Hardness. Attempts have been made to measure the 
 cohesion of bodies by determining their so-called hardness. 
 
 The hardness of crystals in different directions has been estimated 
 by means of an instrument termed a sclerometer. A pointed 
 fragment of hard material, one of diamond being employed in the 
 case of very refractory bodies, is loaded with a certain weight and 
 drawn over the surface a certain number of times along lines 
 parallel to a given direction. The loss of weight experienced by 
 the crystal during this process is taken as a measure of the hard- 
 ness in that direction. The operation is repeated for other direc- 
 tions in the same surface, and for other plane surfaces of the 
 crystal ; the values so obtained are always found to be symmetri- 
 cally repeated in accordance with the morphological symmetry of 
 the crystal. 
 
 Such a method is, however, obviously a rude one, and is subject 
 to mechanical sources of error that interfere with the accuracy of 
 the results obtained by it even as a means of determining hardness, 
 and it is therefore far from affording a true measure of cohesion in 
 a crystal. 
 
 The relations of hardness to cleavage have been carefully studied, 
 though by necessarily imperfect methods, with the results that 
 differences of hardness are found to be greater in crystals conspi- 
 cuous for cleavage ; that the hardness is more uniform in different 
 directions in the absence of distinct cleavages; that the planes 
 of cleavage are the surfaces of easiest abrasion ; and that hardness 
 increases as the direction is more nearly coincident with the normal 
 of a cleavage-plane. 
 
 (g). Results of erosion. Differences in cohesion may also be 
 
of crystals. 9 
 
 made apparent by other than purely mechanical agencies. Thus 
 a crystallised substance frequently manifests unequal solubility, or 
 different degrees of resistance to chemical action, along different 
 directions in the crystal, and this is probably universally true. 
 
 Lavizzari by the action of nitric acid reduced a sphere of calcite 
 into the form of a crystal with pyramidal faces. 
 
 The action of a solvent on the faces of a crystal depends both 
 upon the nature and strength of the solvent and the directions of 
 the faces, one solvent acting more energetically on some forms, 
 another solvent so acting on others ; but all faces belonging to any 
 one form are always affected in a similar way. 
 
 The action of an eroding agent on a single face of a crystal is 
 a valuable method of probing its symmetry. It generally com- 
 mences with the production of a number of minute hollows ; these 
 are sometimes small inverted or negative crystal forms, sometimes 
 grooves or mere lines ; but they infallibly indicate the symmetry of 
 the crystal face, and, where other morphological evidence has been 
 wanting, they have in important cases afforded the means of deter- 
 mining the merosymmetrical character of a crystal, that is to say 
 have revealed a partial development of symmetry in the crystal in 
 accordance with a law to be hereafter enunciated (in Art. 141, 
 p. 161). 
 
 10. Light. 
 
 The important subject of crystallographic optics will only be 
 treated here so far as to place the optical behaviour of a crystal 
 in general correlation with its other physical characters. In the 
 case of crystals belonging to the Cubic system, as in the case 
 of isotropic substances, along every direction a plane-polarised 
 ray may generally be transmitted having its plane of polarisation 
 in any direction whatever, and the velocity of the ray within the 
 crystal is the same whatever may be the direction of the ray or of 
 its plane of polarisation. In crystals belonging to other systems, 
 though a plane-polarised ray may be transmitted along every 
 direction, the plane of polarisation of that ray can in general have 
 only one or other of two mutually perpendicular positions, and 
 these are definitely related to other directions in the crystal. 
 
 The velocity of the ray within the crystal depends both upon the 
 
io On the general properties 
 
 direction of the ray and that of its plane of polarisation, as enforced 
 by the last-mentioned condition. 
 
 The direction, the velocity and the plane of polarisation of 
 all rays of a mono-chromatic light within a given crystal can 
 be geometrically represented by the aid of a single surface, the 
 optical indicatrix, whose form and position completely define 
 the refractive properties of the crystal for light of that particular 
 colour. In the Anorthic, Mono-symmetric and Ortho-rhombic 
 systems the indicatrix is an ellipsoid; in the Hexagonal and 
 Tetragonal systems a spheroid ; and in all of these the crystals 
 are doubly refracting with respect to light. In the Cubic system 
 the indicatrix is a sphere, and, by reason of the velocity being 
 the same whatever the direction and plane of polarisation of the 
 ray, the refraction on emergence cannot be other than single. 
 
 If the light be supposed to emanate in every direction from 
 a point anywhere within the crystal, it will after a small interval of 
 time reach a surface which is known as the ray-surface or wave- 
 surface. In the case of a cubic crystal it is evident from what has 
 been said that this is a sphere : but in doubly refracting crystals the 
 surface will consist of two sheets, since in any direction rays can be 
 transmitted with either of two velocities according to the position 
 of the plane of polarisation. Though the transmission of light 
 along any direction in a doubly refracting crystal may thus be 
 mathematically regarded as consisting in the separate and simul- 
 taneous transmission of two independent rays which arrive at 
 different points on the line after the lapse of the stated interval, 
 the physical process must be one of a more complex nature. 
 
 In crystals belonging to the Hexagonal and Tetragonal systems 
 the ray-surface consists of a sphere and a concentric spheroid 
 that generally touch each other at the ends of a common axis ; in 
 the remaining systems neither sheet is spheroidal or ellipsoidal 
 in form, but, for a given monochromatic light the ray-surface 
 possesses, in common with the ellipsoidal indicatrix from which 
 it may be geometrically derived, three perpendicular planes of 
 symmetry. 
 
 Although in form and position both the indicatrix and its 
 derived wave-surface may vary for monochromatic lights of 
 
of crystals. 1 1 
 
 different colours, they always do so in such a manner as to accord 
 with the morphological symmetry of the crystal a symmetry- plane 
 in the latter being invariably identical in direction with one or 
 other of the symmetry-planes of the ray-surface. 
 
 11. Thermal conductivity. 
 
 The degrees of facility with which heat is conducted in different 
 directions through the substance of a crystal have been investigated 
 by cutting thin sections with parallel plane surfaces along different 
 directions in the crystal, covering a face of such a section with 
 a thin layer of wax or paraffin, and observing the form ultimately 
 taken by the fused portion of the wax on the continued application 
 of a heated wire to a point on the surface. As the form is 
 invariably found to be either circular or elliptical, the continuous 
 isothermal surface which would result from the maintenance of 
 a given temperature at a point inside a crystal must be either 
 a sphere, a spheroid, or an ellipsoid. 
 
 In each case the isothermal surface is found to be symmetrical 
 to the planes of morphological symmetry, taking the form of 
 a spheroid when the crystal possesses a morphological axis, and 
 becoming a sphere when the symmetry is that of the Cubic 
 system. 
 
 12. Thermal dilatation. 
 
 The nature of the dilatational changes resulting from variation 
 of temperature in a crystal will be considered in a future chapter 
 in Part II, for they have an important bearing on the character of 
 crystal structure. 
 
 In general the amount of dilatation for a given change of tem- 
 perature varies with the direction in the crystal ; indeed, expansion 
 in one direction is sometimes simultaneous with contraction in 
 other directions. The mutual inclinations of the faces of the 
 crystal are thus in general dependent on the temperature: but 
 even for the largest attainable variations of the latter, the changes 
 of angle actually observed have never exceeded a few minutes 
 of arc. 
 
 If we conceive of a crystal-mass worked into a form which at 
 a particular temperature is a sphere, it may be asserted that at a 
 second temperature this will either remain a sphere, or will undergo 
 
12 On the general properties 
 
 a deformation whereby it becomes a spheroid or an ellipsoid ; and 
 it will have the one or the other form according to the symmetry 
 of the system ; planes of morphological symmetry being in every 
 case symmetry-planes to the dilatational changes wrought in the 
 crystal by changes of temperature. 
 
 13. Electricity. 
 
 While a crystal of tourmaline is being heated it becomes electri- 
 fied, and oppositely so at parts of the crystal situated at opposite 
 ends of the morphological axis {pyro-electricity} : during cooling the 
 polarity is reversed. This may be conveniently shown, as was 
 suggested by Kundt, by dusting the crystal during change of 
 temperature with a mixture of powdered red lead and sulphur 
 which become electrified during the process by mutual friction. 
 The positively electrified particles of red lead are attracted to 
 the negatively electrified end of the crystal and the negatively 
 electrified particles of yellow sulphur to the opposite end ; the 
 distribution of the electrification is thus indicated by the distribution 
 of colour. A crystal which exhibits this character is said to be 
 pyro-electric. 
 
 Compression of a crystal of tourmaline along its morphological 
 axis also produces electrification (piezo-electricity) ; the distribu- 
 tion of the electrification is the same as that caused by the 
 particular change of temperature, namely cooling, which, like 
 compression, produces a contraction of the crystal along the 
 axis. Conversely, it has been shown that electric induction causes 
 expansion or contraction of a crystal of tourmaline along its axis. 
 
 There are many other substances in which opposite electrifica- 
 tions are developed at opposite ends of the crystal by change of 
 temperature or pressure ; others, for example quartz, which possess 
 more than one axis of symmetry, may exhibit opposite electrifica- 
 tions at the extremities of more than one such axis. 
 
 In all these respects the electric properties of the crystal conform 
 to its morphological symmetry, since in the above cases the axis of 
 symmetry -along which electric polarity is manifested is also dis- 
 similar at its two ends as regards the disposition of the faces and 
 edges of the crystal. 
 
 Magnetic induction. Unmagnetised bodies if brought near a 
 
of crystals. 1 3 
 
 magnetic pole are either attracted or repelled by it, and are said 
 to be magnetised by induction ; being described in the former case 
 as paramagnetic and as diamagnetic in the latter case. When the 
 body is a crystal, whether it be of para- or of dia-magnetic 
 substance, the intensity of the induced magnetism varies, in general, 
 with the position of the crystal relatively to the lines of magnetic 
 force : and if the magnetic field be uniform the induced magnetism 
 has the same character. In the general case, for a given set of 
 physical conditions, there are only three positions of the crystal such 
 that the direction of the induced magnetisation is coincident with 
 that of the inducing magnetic force. These directions are mutually 
 perpendicular ; and the directions in the crystal which in these 
 several positions coincide with that of the inducing magnetic force 
 are called axes of magnetic induction ; for one of them the 
 intensity of the induced magnetism is a maximum, for another 
 a minimum. Their positions in the crystal and the corresponding 
 intensities of induced magnetisation accord with the symmetrical 
 requirements of the crystal. The properties of the crystal as regards 
 magnetic induction under the given conditions may be geometrically 
 represented by means of an ellipsoid known as the ellipsoid of 
 magnetic induction for a uniform magnetic field of a different 
 intensity, the ellipsoid may be different both in form and position 
 in the crystal, but the variation will always be such as to conform 
 to the morphological symmetry. 
 
 14. Summary. 
 
 On comparing the results obtained on the one hand by geo- 
 metrical investigation, and on the other by experimental observa- 
 tions belonging to the domain of physics, of which latter an outline 
 has been here given, one cardinal fact will hereafter be prominently 
 brought out in regard to the physical and morphological characters 
 of a crystal. It will be seen, namely, that these characters are not 
 localised in particular portions of the crystal, but are attributes 
 belonging to different directions in it, the properties of the crystal 
 in respect to any given direction being the properties exhibited by 
 every part of the crystal in respect to any line parallel to that 
 direction. In a word, the crystal though aeolotropic is a homo- 
 geneous body. 
 
1 4 On the general properties of crystals. 
 
 It will be found necessary for geometrical purposes to assume 
 lines and points fixed, not merely in direction but in position, 
 within and without the crystal, with a view to the comparison of 
 these its diverse characters in different directions, and in order to 
 establish the laws of crystal symmetry ; but it cannot be too clearly 
 understood at the outset that these form but the auxiliary scaffolding 
 by the aid of which the ideal structure of crystallography as 
 a science is reared, and that in Nature they have no reality or 
 significance. 
 
CHAPTER II. 
 
 MODES OF EXPRESSING AND REPRESENTING THE RELA- 
 TIONS BETWEEN THE PLANES OF A SYSTEM. 
 
 SECTION I. Elementary considerations as to the methods 
 
 of estimating the mutual inclinations of the Planes 
 
 of a System. 
 
 15. A System of Axes. The faces of a crystal have been 
 described in Article 4 as forming a system of planes whereof the 
 mutual inclinations or relative directions, but not the respective 
 magnitudes, nor therefore their relative distances from any arbi- 
 trarily chosen point within the system, have to be investigated. 
 The fundamental principle of the different methods by which 
 geometry enables us to determine and compare the directions of 
 any number of such planes consists in the employment of certain 
 lines or axes fixed in their directions as regards the system of 
 planes and intersecting in a point within the system, termed the 
 origin. 
 
 Any three or more, but usually only three such axes, which of 
 course must not all lie in the same plane, but in the general case 
 are otherwise arbitrarily taken as regards their directions, are 
 supposed to exist within the system of planes ; and to these axes 
 the planes are by various geometrical expedients referred. Where, 
 as will be the case throughout this treatise, the axes are three 
 in number, and intersect in a point, they are designated severally 
 by the letters X, Y, Z, magnitudes measured from the point of 
 intersection or origin along any axis being deemed positive or 
 negative according as they are reckoned in one arbitrarily chosen 
 
i6 
 
 A plane-system. 
 
 X 
 
 direction or its opposite. Each pair of axes, then, lies in a 
 separate plane ; and these planes, FZ, ZX, XF, divide the space 
 round the origin into eight hollow quoins or octants, which .may 
 be distinguished by the signs of the axes which contain them. To 
 each octant there will correspond three adjacent octants, which have 
 each one axial plane in common with the original octant ; three 
 
 attingent octants, in 
 
 +\Z contact with it only 
 
 along an axis; and 
 one opposite octant, 
 meeting it only at 
 the origin. Adja- 
 cent octants differ 
 in only one of their 
 signs, attingent oc- 
 tants differ in two, 
 and opposite octants 
 in all their signs. 
 
 It is often con- 
 venient to denote 
 the several octants 
 by numbers, as in- 
 dicated in Fig. 2. 
 Those adjacent and 
 opposite to the first 
 
 octant ( + X, + F, +^) are represented by even numbers; octants 
 attingent to it by odd numbers. 
 
 The eighth octant, X Y Z> is not lettered in the figure. 
 The angles at which the axes are inclined on each other will be 
 designated 
 
 the angle YOZ or YZ as 
 
 ,, ZOX or ZX as 77, 
 
 XOY or XY as ' 
 
 Each plane of the system must, if produced, intersect one and 
 may intersect two or three of the axes. The length along any 
 axis intercepted between the origin and the point in which a 
 
 Fig. 2. 
 
System of plane-normals. 1 7 
 
 plane of the system is met by that axis is the intercept of the 
 plane on that axis. 
 
 If several planes be supposed parallel to each other, their inter- 
 cepts on the several axes will only differ by a common factor : 
 where this factor is negative, the planes lie in opposite octants 
 or on opposite sides of the origin, being represented by opposite 
 signs. 
 
 Where two or more planes are not parallel to each other, they 
 must differ either in the relative magnitudes of their intercepts on 
 one or more of the axes, or in the signs of these, or in both 
 of these respects. 
 
 16. Normals to Planes. If a perpendicular from the origin 
 be drawn to any plane of the system which must be supposed to 
 be extended if necessary for the perpendicular to meet it, then the 
 direction in space of this plane and of any planes parallel to it is 
 known when the direction of this perpendicular is determined in 
 reference to the axes. A perpendicular so drawn through the 
 origin to a plane is called the centro-normal, or briefly the normal, 
 of that plane. 
 
 In this treatise the relative positions of planes will be represented 
 by expressions denoting the relative positions of their normals, 
 and normals will therefore be supposed to be drawn to all the 
 planes of the system. 
 
 We proceed to discuss the mode in which the relative positions 
 of planes can be thus simply represented. 
 
 17. If a plane P cut the axes X, Y, Z in the points A, B, C, 
 and for simplicity be supposed to lie in the first octant ; i. e. to 
 intersect with all three axes belonging to that octant ; and if the 
 intercepts of the plane be OA = a, OB = b, OC c, and if OP 
 (Fig. 3) be the normal of the plane ABC\ then 
 
 OP 
 
 cosPX, i.e. cosPOA ^\ 
 OA 
 
 cosPr=^, and C osPZ~. 
 
 Whence OP a cos PX = b cos PF = c cos PZ, expressions 
 which give the direction of OP and of the plane to which it is the 
 normal, whatever be the distance of that plane from the origin: 
 
 C 
 
i8 
 
 Position of a plane. 
 
 this direction being thus represented by the direction cosines of 
 the normal OP expressed in terms involving the intercepts of the 
 plane P. 
 
 Let now a second plane Q inclined to the first plane and lying 
 also for convenience in the first octant intersect the axes X, Y, Z 
 in the points H, K, L respectively ; then the intercepts OH, OK, 
 OL of this plane may be expressed by values involving those of 
 
 Fig. 3- 
 
 the intercepts of the plane P ; as for instance by taking factors 
 h, k, and /, such that 
 
 OH = \OA=^ OK=~OB = L OL = OC = j. 
 
 h h k k II 
 
 And, if OQ be the normal of the plane Q, 
 OQ = Otfcos QX, 
 
 = ^cosQX= -cosQr=- 7 
 Similarly, 
 
 QZ t 
 
 (A) 
 
 CL J) C 
 
 ~p^ ~ q C( ~r 
 
 would indicate the direction of a third plane R of the system, 
 
 the intercepts of which would be in the ratios : - : - on the 
 
 p q r 
 
 several axes taken in the order X, F, Z. 
 
Parameters Symbols. 19 
 
 18. The ratios a : b : c of the intercepts of some one plane 
 chosen as a standard or parametral plane are termed the para- 
 metral ratios or parameters of the system as referred to the axes 
 X, Y, Z, and these ratios are evidently two in number. 
 
 The literal symbols hkl, pqr, &c., or any numbers in the ratios 
 of hkl, &c., and generally the simplest numbers which represent 
 these ratios, are termed the indices of the planes Q, R, &c. 
 
 The indices of a plane placed between brackets, e.g. (hkl}, 
 (pqr], (321), &c., &c., form a compact symbol, which is the 
 symbol of the plane. 
 
 The parametral plane P evidently has for its symbol (i 1 1), since 
 
 a b c 
 
 its intercepts are - : - : - 
 iii 
 
 N. B. The brackets are in practice frequently omitted where 
 the use of the unbracketed symbol may involve no ambiguity. 
 
 In the cases so far considered the planes in question were 
 supposed to lie in the positive octant, i. e. to intersect, either 
 actually or if extended, the three positive axes. If however 
 a plane will intersect two only of the axes, it cannot but be 
 parallel to the third. Its intercept then on this last axis will 
 be indefinitely great, as its point of intersection with it is infinitely 
 remote. The index for the particular axis will thus become zero ; 
 
 AO a 
 
 since, for instance, an intercept , i.e. , is infinitely great. 
 
 Hence (hko), (p<?o) t (no) are the symbols of planes intersecting 
 with the axes X and Y but parallel to the axis Z\ (okt), (oqr\ 
 (on), (023) represent planes intersecting the axes Y and Z but 
 parallel to X ; and (hoi), (101) are symbols of planes intersecting 
 X and Z and parallel to Y. Furthermore, if a plane intersect only 
 one axis, it must have one of the symbols (100), (oio), or (ooi) 
 if it intersect the axis on the positive side of the origin ; any 
 other value than unity for the non-evanescent index being without 
 significance, since it is the direction, not the origin- distance, of 
 the plane that is to be expressed by the symbol. 
 
 Where a plane lies otherwise than in the first or positive 
 octant its position is at once defined by the signs of its several 
 indices. 
 
 C % 
 
2 o Designations for planes. 
 
 An index is taken as positive unless a negative sign is placed 
 over it; and the signs of the indices so inscribed will denote 
 the directions on the axes in which the intercepts of the plane are 
 taken. Thus (213) would be the symbol of a plane lying in 
 the sixth octant of Fig. 2. 
 
 Eight planes might therefore coexist having the same values 
 in their respective indices but differing in their signs ; one plane 
 lying in each octant. Where, in a rectangular axial system, the 
 intercepts of such a group of planes have the same magnitude 
 on each axis, the solid is the regular octahedron of Geometry; 
 and other solids more or less resembling the octahedron in form 
 result from the coexistence of planes intersecting with the axes 
 in each octant with the same intercepts on corresponding axes, 
 but with intercepts that differ on axes of different denomination. 
 A plane of such a group, whatever be the axial system, has been 
 designated as an Octaid, or, more correctly, an Octahedrid plane ; 
 it must have three indices that are not zero in its symbol. 
 
 For analogous reasons, planes parallel to a single axis but 
 meeting the other two axes, and having therefore one zero in 
 their symbol, will be termed Prismatoid planes ; and the term 
 Pinakoid planes (from 7riW|, 'a slab') will be given to such planes 
 as, intersecting with only one of the axes, have two zeros in their 
 symbol; namely the planes (100) and (loo), (oio) and (oTo), 
 (ooi) and (ool). 
 
 Each prismatoid plane may be said to lie in or belong to two 
 adjacent octants ; its zero index refers to that axis for which the 
 sign is different in the two octants. A pinakoid plane belongs 
 equally to four octants ; it is parallel to an axial plane, and inter- 
 sects with the axis which does not lie in that plane. 
 
 In the event of a polyhedron referred to a certain system of 
 axes not presenting octahedrid planes, parametral ratios may be 
 determined by the ratios of the intercepts of two prismatoid planes 
 which intersect with different pairs of axes ; their symbols be- 
 longing to two of the three groups, (on) or (oil), (101) or (Tor), 
 (no) or (Tio). 
 
 It is convenient to assume that the two planes thus chosen cut 
 that axis with which they both intersect in a common point. The 
 
Determining parameters. 
 
 21 
 
 intercept on that axis thus becomes the same for both planes, and 
 their other intercepts are directly comparable. 
 
 Let the plane LH parallel to the axis of Y cut the axes of 
 
 X and Z with intercepts OH |, OL -, and let the plane 
 
 BC parallel to the axis of X cut 
 those of Y and Z at distances 
 OB = b and OC = c. Then LH 
 and BC being taken for parametral 
 planes, let a plane parallel to the 
 plane LH cut the axial plane XZ 
 in CA. The intercepts of the new 
 plane will be OC on Z and OA on 
 
 OA Off , 
 X j and their ratio -^ - - that 
 
 of the intercepts of the original 
 plane LH ' and the two planes 
 AC, BC have their intercepts on 
 the Z axis in common, and their X 
 and ^intercepts are comparable. 
 
 The intercepts of the two planes are, on axes X and Z, 
 
 OA, OC', 
 
 and on axes Y and Z, OB, OC : 
 whence the parameters are 
 
 OA:OB'.OC=a:b : c ; 
 
 where a = OA = OC = - c 
 
 19. The angle contained between the normals to two planes 
 is the supplement of the internal angle 
 included between the planes them- 
 selves, i. e. that angle within which the 
 normals meet. 
 
 If OM, ON be the normals to two 
 planes ME and NE, and the plane 
 containing these normals is the plane 
 of the figure ; then will this plane be 
 
 Fig. 4. 
 
 Fig- 5- 
 
 perpendicular at E to the edge formed by the two planes; and 
 
22 
 
 The edge of two planes. 
 
 ME, NE are the edges formed by the normal plane with the 
 two planes, and the angle NEM measures the inclination of the 
 two planes ; t MENO therefore is a quadrilateral figure, whereof 
 OME and ONE are right angles. Therefore MEN+ MON 2 
 right angles, and each is the supplement of the other. 
 
 SECTION II. Expression for the direction of the Edge 
 formed by two Planes. 
 
 20. Let HKL, PQR be two planes assumed for simplicity to 
 lie in the same octant and having for their symbols (Ml), (pqr). 
 
 The most general case is that in which the two planes have no 
 intercept in common on either axis. 
 
 Let the two planes lie in the octant XYZ ^and intersect with 
 the axes in HKL, PQR respectively. In this case there will be 
 common points of intersection of the two planes with each other 
 
 -Y 
 
 and with each of the axial planes : let s be the common point 
 in the axial plane XZ, and u be that in the plane YZ. With 
 the third axial plane they will intersect also in a common point /, 
 in an adjacent octant. Then the points /, u y s will lie in one 
 and the same straight line, which will be the line of intersec- 
 tion or edge of the planes HKL, PQR. 
 
Expression for an edge. 
 
 The direction of this line will be parallel to that of all edges 
 formed by planes parallel to the original planes. Let then a 
 plane parallel to PQR be drawn through K intersecting with the 
 axes X and Z in E and G. The ratios of its intercepts will 
 thereby remain unaltered, and that on Y is common to the two 
 planes EKG, HKL. 
 
 By similar triangles, which need not be drawn, 
 
 OK 
 
 OE=OP. 
 
 OQ 
 
 and 
 
 The planes will now have only two points of intersection with 
 the axial planes ; namely, a 
 point K on the axis Y +\Z 
 
 common to the two planes 
 XY and YZ, and a point 
 M situate in the plane 
 ZX. 
 
 A line KM is therefore 
 the edge of the planes HKL 
 and EKG. Through M 
 draw MD parallel to the 
 axis Z and meeting the axis 
 X in D. And on the axis 
 ZtakeOM'=JDM: finally, 
 through the origin O draw 
 ON parallel and equal to 
 KM. 
 
 The direction of the line 
 ON passing through the 
 origin is therefore that of 
 the edge formed by the original planes or by any planes parallel to 
 them ; and the coordinates of N are 
 
 OD, -OK, and OM'=DM-, 
 
 ON being the diagonal of a parallelepiped with OK', OD, DM for 
 its sides. 
 
 In order to find the value of these coordinates, we have in the 
 similar triangles HDM and HOL, 
 
 Fig. 7. 
 
24 Expression for an edge. 
 
 OM'=DM=^.(OH-OD], (i) 
 
 DC 1 
 and OM'=DM=^jj,.(OE-OD). (ii) 
 
 Dividing (i) by OL and (ii) by OG, and subtracting, we obtain 
 OM' - = 
 
 Therefore 
 
 OH OE> 
 OM' OD 
 
 i i 
 
 UG~"OL 
 
 OM' OD 
 
 OQ i 6* (> 
 
 and 
 
 OHOKOP OKOROL 
 
 OM' OD 
 
 OH.OQ OK. OP OK. OR OL.OQ 
 and, by the symmetry of the problem, 
 
 OK 
 
 OL.OP OH. OR 
 Here ^ = \ , &c. ; 
 
 TST-f* 
 
 r r * J kq kp 
 
 Therefore mcrcraA-ap = ^' 
 
 &c., &c., = &c. 
 
 Substituting these values in the ratios and dividing by the com- 
 mon factor obey we obtain 
 
 OD OK OM' 
 
 a(kr-lq) ~ b(lp-hr) ~ c(hq-kpY 
 
 which expression gives the ratios of the coordinates of any point 
 in a line through the origin parallel to the edge of the planes 
 hkl and/^r, in terms of the indices of the two planes. 
 These ratios 
 
 a(kr-lq) t b(lp-hr\ c(hq-kp) 
 
A crystalloid plane-system. 25 
 
 may be written briefly as 
 
 an, dv, -w, 
 
 if u = kr lq, 
 
 v = Ip hr, 
 w = hq kp. 
 
 DEF. A plane parallel to any plane of a system but passing 
 through the origin will be termed an origin-plane, and any line 
 through the origin parallel to an edge of two planes of the system 
 will be termed an origin-edge, and hereafter a zone-line or zone- 
 axis ; and in speaking of the planes of a plane-system the term 
 'face' will be more particularly applied to them when considered 
 as the faces that bound a polyhedron. 
 
 SECTION III. A Crystalloid System. Principle of 
 Rationality of Indices. 
 
 21. In considering the general character of the expressions 
 for the relations of the planes in a system as referred to axes 
 of coordinates, the indices of a plane were not limited to any 
 special kind of values, integral or fractional, rational or irrational, 
 and the axial system might be arbitrarily chosen. If however 
 the nature of the system of planes be limited by the condition that 
 for a plane to belong to the system its indices must be rational, 
 that is to say, capable of being represented by integral numbers, 
 or one or two of them by zero, it will be obvious that some limita- 
 tion must also be imposed on the selection of the axes to which 
 the planes are referred. 
 
 22. It will be shown in Art. 75 that the axes must, in the case 
 supposed, be themselves possible zone-lines of the system. A 
 special case, confined to a single zone, will however serve here 
 to illustrate this important principle. 
 
 Let us suppose that OX, OZ are axes arbitrarily taken to which 
 two planes are referred, that may for convenience be prismatoid 
 planes, cutting only these two axes, and parallel to the third. Let 
 the lines AC, AL be the intersections of these planes with the 
 axial plane XOZ, which is that of the figure ; let OA = a, OC = c 
 be the intercepts on X and Z of one of the planes ; and let a and c 
 
26 A crystalloid axial system. 
 
 be taken as the parameters : and of the other plane let OA and 
 OL = - c be the intercepts. Then - is the index on Z of the 
 plane AL, and if the ratio of OC to OL be capable of being 
 expressed by whole numbers, is rational. 
 
 If we arbitrarily choose another axis OZ f elsewhere in the plane 
 XOZ, and take OC' and OL' as the intercepts on that axis of the 
 
 two planes j it is evident that the ratio 
 of these will vary with the direction of 
 OZ' and is not necessarily rational. 
 Wherefore the axial system may not be 
 arbitrarily chosen in the case of a sys- 
 tem of planes of which the indices are 
 rational. 
 
 If, again, we change the axial system 
 in such a manner that C becomes the 
 origin and CA, CO the new axes; and 
 if AO and AL be the lines of inter- 
 section of two planes with the axial 
 Fig. 8. plane CO A ; then CA and CO, the 
 
 intercepts of the plane OX, may be 
 
 taken as parameters for the system ; and the intercepts of the 
 other plane on the new axes are 
 
 CA = a', and CL = *^.<:, 
 
 so that its indices in respect to these axes are / h and /, and 
 are obviously rational. 
 
 Hence, in the special case supposed, any planes, the origin- 
 edges of which form axes to which it is possible to refer a system 
 of planes with rational indices, must themselves fulfil the con- 
 dition required for all planes of the system. 
 
 23. By the term a crystalloid system of planes we shall under- 
 stand an assemblage of planes parallel to the faces of a poly- 
 hedron finite in number and presenting such mutual inclinations 
 that if they be referred to an axial system formed by three different 
 origin-edges parallel to edges of the system, and with parametral 
 
Projection of poles on a sphere. 27 
 
 ratios determined by the intercepts of any plane of the system, 
 it shall be a necessary condition in order for a plane to belong to 
 the system that its indices be rational. 
 
 An axial system with axes and parameters so chosen will be 
 termed a crystallographic axial system. 
 
 The elements of such an axial system are five in number ; viz. 
 the three axial angles f, 77, and f, and the two parametral ratios 
 
 a c 
 
 T anc * T ' 
 b b 
 
 SECTION IV. On the Sphere of Projection, and the 
 principles of its Stereographic Representation. 
 
 24. A convenient means of representing and comparing the 
 relations of a system of planes forming a polyhedron is afforded 
 by treating their normals as radii of a sphere. A sphere of 
 arbitrary radius termed the Sphere of Projection is supposed to 
 be described round any point within the system taken as its 
 centre and as the origin of a system of centre-normals, or briefly 
 of normals, perpendicular to the faces of the polyhedron. The 
 point in which any such normal meets the sphere is termed the pole 
 of the plane to which the particular normal is perpendicular. A 
 pole may therefore also be defined as the point of contact of the 
 sphere and a tangent-plane parallel to a plane of the system on the 
 same side of the origin with the plane. 
 
 Where the system of planes is also referred to an axial system, 
 with the centre of the sphere for its origin, the point in which an 
 axis penetrates the surface of the sphere will be called its axial 
 point. 
 
 It is evident that if the poles of planes be connected by great 
 circles, their distances and relative positions, and therefore the 
 inclinations and relations of the planes themselves, may be measured 
 and investigated by the methods of spherical trigonometry : for in 
 the plane of the great circle, thus connecting the poles of any two 
 planes, the normals of these planes must lie ; and the arc between 
 their poles as measured on a great circle is that subtended by the 
 angle contained by the normals, and is therefore the supplement of 
 the angle of inclination of the planes themselves. 
 
28 
 
 Stereographic projection. 
 
 Another great advantage of this method of representing the 
 positions of all the planes of a system by points distributed on a 
 sphere is, that by a simple process of laying down such points in 
 
 a projection of the 
 sphere upon a plane 
 we do not need the 
 somewhat elaborate 
 process of drawing a 
 crystal by projecting 
 its edges, in order to 
 give a complete con- 
 spectus of all that 
 crystallography seeks 
 to represent; that is 
 to say, of the general 
 symmetry of the poly- 
 hedron and the dis- 
 tribution and relative 
 inclinations of all its 
 faces. Fig. 9 repre- 
 sents in orthographic 
 
 projection the faces and the poles of the cubo-octahedron, i.e. of.. 
 
 the two regular solids the cube 
 and octahedron united into a 
 single figure in which the faces 
 of the one figure truncate the 
 solid angles (or quoins) of the 
 other figure. Fig. 10 represents 
 the poles of the same faces and 
 the great circles passing through 
 those poles in what is termed 
 the Stereographic projection. 
 In the former case, which is 
 that usually employed for the 
 projection of the edges of crystals 
 in crystallographic figures, the eye 
 is at an indefinitely great distance (the crystal being seen as if 
 
 Fig. 9. 
 
Plane of projection. 
 
 29 
 
 from a considerable distance through a telescope) ; the visual rays 
 being parallel, and as a consequence all lines parallel in the ob- 
 ject remaining parallel in the figure that represents it. 
 
 The representation of the faces of a polyhedron by the 
 stereographic projection of its poles has the great advantage over 
 a drawing that it does not aim at representing the relative magni- 
 tudes of the faces ; while also by this method the poles of any 
 number of planes may be laid down at their correct angular 
 distances with speed and accuracy, whereas the number of faces 
 admitting of representation in a drawing is necessarily limited. 
 Furthermore, crystallographic problems may often be solved and 
 calculation greatly simplified by its means. 
 
 25. In the Stereographic Projection, which is the simplest 
 form of projection of a sphere with the above view (Fig. 10), 
 the eye is supposed to be at a point of the sphere's surface 
 and to see such poles or great circles as are distributed on the 
 hemisphere opposite to it projected (as on a screen) upon a plane 
 passing through the centre of the sphere and cutting the sphere 
 in the great circle at the pole of which the eye is situate. The 
 plane of projection thus bounded by a great circle of the sphere is 
 represented by the plane of the paper on which the circle is 
 drawn, which latter will be termed the circle of projection or 
 primitive circle. The advantage of this over other forms of pro- 
 jection of the sphere is that any great or small circles of the 
 sphere on the hemisphere opposite to the eye are by it projected 
 either as straight lines or as circular 
 and not as elliptical arcs, and thus, 
 by means of a protractor and com- 
 passes, all the great circles can be 
 laid down on which the poles of 
 planes are distributed. 
 
 That this is the case follows at 
 once from the properties of the ob- 
 lique cone ; but it may be otherwise 
 proved thus : suppose S, Fig. n, to 
 be the point of sight on a sphere of 
 which is the centre and OS therefore the radius. Let PP' 
 
30 Stereographic projection. 
 
 represent the plane of projection. The circle of projection will 
 then be a great circle of which S, the position of the eye, is 
 one of the poles. The apparent position on the plane of pro- 
 jection of any point A on the sphere as seen by the eye at 
 will evidently be that point in which a straight line drawn from 
 to A meets the plane PP f . Thus the centre O of the circle 
 of projection will be the point at which T, the pole of that 
 circle opposite to S, will be projected. So the arc FTP' on 
 the opposite hemisphere will be seen from S as a straight line 
 coincident with PP', and any great circle passing through T will 
 be projected as a diameter of the circle of projection ; such great 
 circles or parts of them are therefore projected as straight lines. 
 Thus a portion AB of the arc PTP' will be seen as db, a portion 
 of PP' limited by the points a and b, which are the projections of 
 the points A and B. It will further be seen that while all points 
 on the hemisphere opposite to the eye will be projected in points 
 within the circle of projection, the projections of points lying on 
 the same hemisphere with S will lie beyond the circumference 
 of that limiting circle. 
 
 26. We proceed to establish further that an arc of any circle on 
 the sphere not passing through the 
 point of sight is projected as a circular 
 arc. 
 
 Let ADB be a circular section 
 of the sphere, for convenience a 
 section by a small circle; and let 
 Fig. 12 represent a section through 
 S and also through any two points 
 A and B on this small circle. Let 
 PP' be the trace of the plane of 
 projection ; a, b the points in which 
 lines SA and SB cut PP' ; they are therefore the projections of 
 A and B. 
 
 Draw a tangent LS. Then 
 
 Therefore the four points A, B, b, a lie on a circle. 
 
 Now A and B in the section ABS being any points on the 
 
Projections of circles. 31 
 
 small circle, the locus of db must be the section by the plane of 
 projection of any surface containing all such circles as have been 
 proved to contain any two points and their projections. That 
 there is such a surface and that it is a sphere, and that consequently 
 the projection adb of the circle ADB is itself a circle, will be 
 evident from the following considerations. 
 
 Erect a perpendicular to the plane of the circle ADB from its 
 centre; then every point in the circle is equidistant from any 
 point in this perpendicular ; and a sphere described round such a 
 point C taken at equal distance from B and b will carry on its 
 surface all the four points A, B, D, b. 
 
 And d, the projection of any other point D on the original circle, 
 will also lie on the surface of this sphere. For if it do not, it will 
 lie somewhere else in the line SD, as at a point 8. 
 
 But, as in the case of the projections a, b of the points A and 
 B, we have proved baS = ABS] whence, by comparison of the 
 triangles baS and ABS, we obtain the reciprocal proportion 
 Sa . SA = Sb . SB. 
 
 Similarly, we should obtain for the points A, D, a, 8 
 Sa.SA = S&. SD. 
 
 But also, as SD must cut the described sphere somewhere, as in 
 a point*/, Sa.SA = Sd.SD 
 
 is also true, which can only be possible if d and 8 are one 
 and the same point. And this point is therefore at once the 
 projection of D and situate on the surface of the described 
 sphere. The locus of ab is therefore, as asserted, the section 
 by the plane of projection of the described sphere ; that is to 
 say, is a circle. 
 
 27. The proof applies equally to the section of the sphere by a 
 great circle : it may also be thus illustrated. 
 
 Let AB, Fig. 13, be a great circle intersecting with the plane of 
 projection, of which the trace is PP f t in a line perpendicular at 
 to the plane of the figure ; which plane passes in this case through 
 the centre of the sphere as well as through S. From S draw a 
 perpendicular to the plane AB meeting the plane of projection 
 in C, and draw SA through a the projection of A, and Sb through 
 
Stereographic projection. 
 
 B to meet the plane of projection produced in b which will be the 
 
 projection of B : then these lines will necessarily lie in the plane 
 
 of the figure. 
 
 As in the previous case, 
 
 Sab = SBA, 
 
 whence Aab = bBA, 
 
 and the four points A,a,B,b lie on a circle ; and it will follow, as 
 
 in the previous case, that 
 every point of the great 
 circle AB will lie in the 
 circular section in which 
 the plane of projection 
 will cut a sphere that 
 shall be so described as 
 to carry on its surface 
 the four points A, a, 
 B,b. 
 
 Fig. 13. COR. 1. The centre of 
 
 the circle in which the 
 
 original great circle is projected will be C, the point in which the 
 perpendicular from -S" on the plane of AB meets the plane of 
 projection ; for, since 
 
 SAB = SbC = CSb, 
 Ca=.CS=Cb; 
 
 and since aSbT lie on the same circle of which C is the centre, 
 two points in which the circle of projection is cut by the plane 
 through -S 1 and perpendicular to the plane of projection will 
 be the points of intersection of the circle of projection and the 
 circular arc in which that circle, whose trace is A and B, is 
 projected. 
 
 COR. 2. Since ASb is a right angle, b may be at once obtained 
 by drawing Sb perpendicular to AS and meeting the continuation 
 ofPP'ml,. 
 
 For the practical applications of the Stereographic Projection, 
 of which continual use will be made in this treatise, the following 
 propositions will be found necessary. 
 
To measure a projected arc. 
 
 33 
 
 28. PROBLEM I. To determine the magnitude of the arc of a 
 
 great circle which is represented by the projection of that arc. 
 
 Let V and C be two great circles, Fig. 14, which intersect in 
 the extremities of the diameter AA' of the sphere of projection, 
 and let their poles be P and S. 
 
 Now, every circle on the sphere, whether great or small, passing 
 through the points 
 P and S e.g. the 
 great circle U or the 
 small circle U' 
 will manifestly cut 
 in a similar manner 
 the two circles V 
 and C; and more- 
 over, any two circles 
 on the sphere thus 
 passing through the 
 poles of two great 
 circles on it must 
 intercept on these 
 two great circles 
 arcs of the same 
 magnitude; whence 
 it is evident that the 
 arcs intercepted on 
 
 the great circles V and C by the two circles U and U', or those 
 intercepted by U or U' ', and by a third plane, say U", passing 
 through A, must be equal ; so that 
 
 A Q = AR, A (/= AR', (?(/= RR', 
 
 if Q, R be the points of intersection of a circle U with the great 
 circles C and V, and (/, R' be those in which these great circles 
 are intersected by the circle U'. 
 
 If now the plane of the great circle C be taken for the plane 
 of projection, S being the position of the eye, every circle U 
 (U f , U", &c.), whether great or small, will be projected as a 
 straight line passing through p the projection of P. Indeed, 
 
 Fig. 14. 
 
34 
 
 Stereographic projection. 
 
 since every such circle U lies in a plane passing through P, S, 
 and a point R (or 7?') on the great circle V, straight lines drawn 
 from S to points on the circumference of the circle U can only 
 intersect with C in points upon a straight line passing through />, 
 which is at once the projection of the circle U upon, and the 
 intersection of its plane with, the plane C. 
 
 Consequently, while the great circle V will be projected in the 
 circular arc ArA', every one of the circles U t U', &c. passing 
 through the points R, R', &c. will be projected in a straight line 
 pQ) (pfft) & c -> which will further cut ArA', the projection of the 
 circle V, in a point r, (r f , &c.) which will be the projection of the 
 point R (or of R', &c.) ; r (or /, &c.) being in fact the point in 
 which the planes of three great circles V t U (or U', &c.) and C 
 intersect with each other. 
 
 It is thus that straight lines drawn from p, the projection of the 
 pole of the great circle V, through r and r' to the circumference 
 of the circle of projection (which straight lines are the projections 
 of two circles U, only one of which can be a great circle,) come 
 to intercept on this primitive circle an arc QQ' ; and this arc 
 QQ'= RR', the arc of which r/ is the projection, and QQ there- 
 fore measures the arc represented by r/. 
 
 Whence follows the rule : To determine the value of any arc 
 
 of a great circle as represented in 
 projection, find the projection of 
 the pole of this great circle (which 
 may be done by the succeeding 
 problem) ; then 
 
 i. The value of an arc of which 
 the projection is given may be 
 measured by determining on the 
 circle of projection the arc con- 
 tained between two straight lines 
 drawn from the projection of the 
 pole through the extremities of the 
 projected arc, Fig. 15; and, 
 
 2. An arc of a great circle of given magnitude is represented 
 in the projection of that circle by the portion of the projection 
 
Pole of a great circle projected. 
 
 35 
 
 determined by the intersection with it of two straight lines drawn 
 from the projected pole to the circle of projection and intercepting 
 on that circle an arc of the required magnitude. 
 
 If the point in which one of the straight lines intersects with the 
 projection of the great circle be fixed (i.e. is the projection of 
 a given point on the great circle), then the other extremity of 
 the projection of the required arc will alone have to be deter- 
 mined by the latter of the two methods. 
 
 COR. If the plane of a great circle, of which the projection is 
 given, be perpendicular to the plane of projection, the great circle 
 passes through S, the point of sight, and its projection becomes 
 (by Art. 26) a diameter of the circle of projection C ; and the pole 
 of the great circle will lie on the circle C, at the point of intersec- 
 tion with it of a diameter perpendicular to that in which the great 
 circle is projected. And this pole and its projection manifestly 
 coincide. 
 
 Hence arcs intercepted on the circle of projection by straight 
 lines drawn from this pole through the diametral line in which the 
 great circle is projected, will determine on that diameter the pro- 
 jection of an arc of any required angular magnitude. 
 
 PROBLEM II. Given the projection of a great circle, to find that 
 
 of its pole. 
 
 29. Let Diy, Fig. 16, be the diameter in which the given 
 projection, and consequently also 
 the original great circle, intersect 
 with the circle of projection. 
 
 Since the pole required will ne- 
 cessarily lie on the diameter per- 
 pendicular to that in which the 
 given projected circle intersects 
 with the circle of projection, and 
 must lie at a distance equivalent 
 to a quadrant on that diameter 
 from the point in which the given 
 circular projection intersects with 
 it ; let V be the point of this intersection, V that in which a 
 
 D 2 
 
Stereographic projection. 
 
 line D V produced meets the circle of projection. Then, by COR. 
 PROB. I, if V'P* be a quadrant, and a line DP' cut the diameter 
 through V and O in P, VP would be the projection of V'P* on 
 the line VP as seen from Z>; and since a similar construction 
 would hold good were the great circle AD'P' to be drawn perpen- 
 dicular to the plane of the figure, so as to be seen as A OP and 
 to pass through S the point of sight, and since the true pole of 
 V will lie on this great circle, the point P will be the projection of 
 that pole. 
 
 PROBLEM III. To draw the projection of a great circle, in which 
 projection two points are given, that do not loth lie on the 
 circle of projection. 
 
 30. Since only one great circle can pass through two points 
 on the sphere not extremities of a diameter, the centre of the 
 sphere and two such points suffice to determine the direction of 
 the plane, of such a great circle. Hence the projections of two 
 points and the centre of the sphere being given on the plane of 
 projection, it should be possible to describe the circle which is 
 the projection of the great circle on which the two points lie. 
 The most convenient way of doing this is to find the projection 
 
 of the point on the great 
 circle which is the op- 
 posite extremity of the 
 diameter on which one 
 of the points lies of which 
 the projection is given. 
 
 Thus, if two points on 
 a great circle V are pro- 
 jected in the points P 
 and Q, Fig. 17 both 
 of which do not lie on 
 the circle of projection 
 Fig. 17. DAI/ and if it be re- 
 
 quired to draw V t the 
 
 projection of V, it is necessary to find the projection of a third 
 point in the circle V'\ and the point diametrically opposite to either 
 
A great circle projected. 3 7 
 
 of the points P and Q is convenient for this purpose. If then we 
 would project in P', the point diametrically opposite to that point of 
 which P is the projection (and it is preferable for this purpose to 
 select the more remote from the centre of the two projected points 
 P and Q), we have to draw through the remoter point P a 
 diameter of the circle of projection ; and, where P does not lie 
 on the circle of projection, from 0, the centre of that circle, a per- 
 pendicular on PO is drawn to a point A in the circle of pro- 
 jection. A therefore is the projection of the pole of the great 
 circle which is projected in the diameter through P ; and P' will 
 lie on the line PO at a distance equivalent in the projection to 
 an arc TT from P. From A draw a perpendicular to AP, meet- 
 ing the prolongation of PO in P f then, by article 27, COR. 2, 
 P f and P are the projections of diametrically opposite points on 
 the original great circle, since the construction would equally 
 determine P f were the point A revolved round PP' till it became 
 coincident with S the point of sight. Perpendiculars from the 
 points of bisection of the straight lines joining Q, P, and P r 
 will now meet in a point C ; and from C as a centre a circle 
 drawn through P and Q will be the required projection V of the 
 original great circle V . It frequently happens that the points P 
 and P' coincide with the extremities of the diameter DD' in 
 which the two circles V and C intersect. 
 
 Otherwise; where the distance from the circle of projection 
 differs appreciably for the 
 two points P and <2, an 
 elegant mode of deter- 
 mining the position of the 
 diameter DD' is the fol- 
 lowing. Through P and 
 Q, Fig. 1 8, draw any circle 
 intersecting the circle of 
 projection in points p and 
 q ; (any circular disc laid on 
 the figure will determine 
 
 these points). Let the two straight lines PQ and pq be drawn 
 to meet in M. Then D&, the diameter of the circle of 
 
Stereograpkic projection. 
 
 projection which would traverse M if continued, is the diameter 
 required. 
 
 Through PQ and D describe an arc of a circle intersecting the 
 line MD in a second point A. Then 
 
 MP. MQ-Mp.Mq = MD . 
 
 and A and D' are the same point. Hence, DD' is the diameter 
 at the extremities of which the arc V (the projection of the arc V) 
 intersects the circle of projection. 
 
 If PQ and pq are parallel they are parallel to DD'. 
 COR. In order to draw the projection of a great circle where the 
 
 projection of its pole is given : 
 draw a diameter of the circle of 
 projection through the point M 
 in which the pole is projected; 
 then, as in COR. PROB. I, by aid 
 of the lines Pm, Pq, where the 
 
 fJT 
 
 arc mq = - , find the point Q on 
 
 this diameter equivalent to a 
 quadrant's distance from M, the 
 projected pole. A second dia- 
 meter PP f , perpendicular to the 
 first, will meet the circumference in the points of intersection with 
 it of the required great circle ; which may now be drawn by the 
 help of Problem III. 
 
 N.B. The centre K, from which this circle is to be drawn, 
 will be the point in which a perpendicular on PQ from B, the 
 point of bisection of PQ, will meet the diameter through QM. 
 And the case in which this centre K falls on the circle of pro- 
 jection is that in which OM and OQ represents each an arc of 
 
 - when K coincides with C, and 
 
 CQ = CP = chord 90 = r. 4/2 = 1.414^, 
 the radius of the required circle being CP. That in this case, 
 
 where OM and OQ each represent an arc of - , the radius of the 
 circle to be drawn is the chord CP is evident ; for then, 
 
To project a circle in position. 
 
 39 
 
 qPP^lqOr^y 
 
 and CQP = ---=-+-= QPC; 
 
 so that CPQ is isosceles, and CP = CQ. 
 
 PROBLEM IV. To draw the projection of a great circle which 
 intersects with another great circle at a given angle and in a 
 given point ; the projections of the latter great circle and of 
 the point being given. 
 
 31. Let a great circle V intersect a second great circle W 
 in a point r at an angle 0. Let V ', Fig. 20, be projected in the 
 arc V, and / in the point r. It is required to draw the projec- 
 tion of the great circle W. Project the pole of V in P, and 
 from r draw the line rPp to the circle of projection ; on which 
 circle let the points p and q inter- 
 cept an arc equivalent to : draw 
 U, the projection of the circle of 
 which r would be the projected pole. 
 U will therefore pass through P; 
 and let Q be the point in which U 
 intersects with qr. 
 
 Draw the projection W of the 
 great circle W', the pole of which 
 would be projected in Q: it will 
 be seen that the projection W will 
 pass through r\ and furthermore 
 that it will be inclined to the pro- 
 jected arc V at the required angle 6. For the distance of the 
 poles of two great circles, as measured on a great circle tra- 
 versing them on the sphere, is necessarily equal to the angle at 
 which the two circles intersect, and since r is the pole of the 
 projected circle through P and Q, the arc PQ represents the arc 
 on the great circle U measured by pq ; that is to say, represents 
 the angular magnitude 0. 
 
 Hence the circular arcs V and W are the projections of two 
 great circles V' and W' t which intersect in r' at an angle 0. 
 
 Fig. 20. 
 
Stereographic projection. 
 
 PROBLEM V. Given the projections T and P, Fig. 21, of two 
 poirits on the sphere ; to determine the position of a third point A 
 which shall be the projection of a point on the sphere distant 
 by an angular arc (j) from the point of which T is the pro- 
 jection, and by an arc from the point of which P is the 
 projection. 
 
 32. If we suppose the triangle APT to be drawn, and if the 
 angle at P = , and that at 7 = f, and the arc TP = a ; then 
 
 cos < cos cos a 
 
 cosf = 
 
 cos = 
 
 sin sin a 
 cos0 cos(/>cosa 
 
 sin 9 sin a 
 
 Draw, by Prob. Ill, Cor., UV, WV t the projections of two 
 great circles of which P and T are respectively the projected 
 poles ; and let them intersect in a point V. From P and T draw 
 
 straight lines P Vp, TVt meet- 
 ing the circumference in p 
 and t, and through a point x 
 at a circumferential distance 
 77 f from/) draw P.r, inter- 
 secting with the great circle 
 VU in U', and through 2 at 
 a distance TT f from / draw 
 Jz, intersecting with the great 
 circle VWm W. 
 
 Through U and W draw 
 the projection of a great circle 
 (Prob. Ill), and find A the 
 Fig. 21. projection of its pole (Prob. 
 
 II). Then will A be the point 
 
 required. For, by construction, APT and UVW are projections 
 of polar triangles. And since in UVW, UV=-n-, WV=v-, 
 and the angle at A-na, the angle 7=7r 0, and angle 
 
 Whence in the triangle APT, A T represents an arc 
 an arc 0. 
 
 and AP 
 
Position of a face-pole. 41 
 
 33. The manner in which the different varieties of crystals are 
 represented in stereographic projection will be considered in future 
 chapters. It is only necessary here to observe, that the plane of 
 projection is always so selected as to contain one of the most 
 important of the crystallographic planes, and is generally a plane 
 which divides the crystal symmetrically. 
 
 In the stereographic projections employed in this volume the 
 representation is such as would result from looking down on 
 a plane of projection on which the poles of the hemisphere 
 nearest the observer had been previously laid down, as seen by 
 an eye at the opposite pole of the circle of projection; that is 
 to say, at the nether pole of that great circle. It is often con- 
 venient to be able to represent the poles of both hemispheres 
 within the same great circle of projection. In such case the poles 
 of the upper hemisphere, as seen by an eye at the nether pole, will 
 be represented by black dots ; those belonging to the nether hemi- 
 sphere, as seen by an eye placed at its upper pole, will be repre- 
 sented by small eyelets. Where two poles fall on the same spot 
 the black dot is encircled by the eyelet. 
 
 Where literal or numerical symbols are employed to express 
 the character of the faces to which these poles belong, the signs, 
 or other values of these numerals, will usually serve to indicate 
 to which hemisphere they are to be applied. 
 
 When the symbols of both sets of poles are introduced, those 
 belonging to the nether poles will be the fainter and smaller in 
 type upon the projection. 
 
 SECTION V. Expressions for determining the position of 
 a pole on the sphere. 
 
 34. Let X, F, Z be the axial points, A,B,C the poles of the 
 axial planes YZ, ZX, XY to which a crystalloid system of planes 
 has been referred. 
 
 Hence ABC and XYZ, Fig. 22, are polar triangles, and A is 
 100, B is oio, C is ooi. Let P be the pole of a plane the indices 
 of which are + h, k, + 1. If however P lie on one of the 
 great circles passing through two of the poles A, B, C, it is the 
 
42 Position assigned to a face-pole. 
 
 pole of a face parallel to one of the axes, and one of its indices is 
 zero ; that is, the symbol is okl for any pole lying between two poles 
 B and C, hoi for a pole lying between two poles C and A, and 
 hko when lying between two poles A and B\ and in each case 
 the index in question must change its sign as it passes from one 
 to the other side of the respective great circles. 
 
 Whence the poles of all faces lying in any given octant will lie 
 within the corresponding one of the eight spherical triangles into 
 
 which the sphere is divided by the 
 great circles passing through two of 
 the poles +A, + B, +C and the poles 
 opposite to them A, B, C: 
 and the signs of the indices of the 
 face P will correspond with those de- 
 signating the particular spherical tri- 
 angles within which its pole lies: e.g. 
 the pole hkl lies within the spherical 
 Fig. 22. triangle +A B C. 
 
 35. Let P be the pole of a plane 
 
 of the system lying within the spherical triangle ABC. Through 
 P, Fig. 22, draw the quadrantal arcs XR, FS, ZT. 
 Then, 
 
 cos PX sin PR = sin CP sin BCP = sin BP sin CBP, 
 cos PY sin PS sin AP sin CAP = sin CP sin ACP, 
 cos PZ sin PT sin BP sin ABP sin AP sin BAP. 
 But the symbol of P is hkl, and by the fundamental equation A 
 in article 17, 
 
 ^-cosPX =|cosPJr= y cosPZ. 
 
 hkl 
 
 Substituting the above values for cos PX, cos PY, cos PZ we 
 obtain six equivalent expressions, 
 
 | sin CP sin BCP = | sin CP sin A CP, 
 h k 
 
 b - sin AP sin CAP = j sin AP sin BAP, 
 
 A? / 
 
 C - sin BP sin ABP = | sin BP sin CBP-, 
 I ti 
 
Expressions for position of a pole. 43 
 
 whence are obtained the equations 
 
 h asmBCP k b sin CAP I csinAP 
 
 
 k tsmACP I c s'mAP h a sin CBP* 
 which give the relations between the parameters and the indices 
 of the face P in terms of the angles which the arc joining its pole 
 hkl to any of the poles 100, oio, ooi forms with the adjacent 
 pairs of the arcs that unite these latter poles. 
 
CHAPTER III. 
 
 ON ZONES AND THEIR PROPERTIES. 
 
 SECTION I. Expressions for a Zone. 
 
 36. Definitions. If the centre of the sphere of projection coin- 
 cide with the origin of the axes to which a system of planes is 
 referred, the direction of any plane passing through this centre 
 (origin-plane) is determinable when the positions are known of 
 any two radii of the sphere not on the same diameter that lie 
 in the plane in question. And if these radii be the normals of 
 two planes of the system, the poles of these planes will be points 
 on the great circle in which the origin-plane intersects with the 
 sphere of projection. 
 
 Further, the two planes, and therefore also the edge in which 
 they mutually intersect, will be at once perpendicular to the plane 
 of the great circle containing their poles, and parallel to the dia- 
 meter of the sphere which is the normal of that plane : this is true 
 for all the planes the poles of which lie on the great circle, and for 
 the edges in which each pair of them may intersect. 
 
 Thus, the direction of the plane of this great circle and that 
 of its normal may be determined indifferently from any pair of 
 these planes. 
 
 The great circle which contains two or more of the poles be- 
 longing to a system of planes will be called a zone-circle, and 
 the plane containing a zone-circle its zone-plane : the planes or 
 faces perpendicular to the zone-plane are the planes or faces of the 
 zone. The diameter of the sphere normal to the zone-plane, and 
 to which therefore the edges formed by the intersections of faces 
 
Symbol of a zone. 
 
 45 
 
 of the zone are parallel, is the tone-axis. Two or more poles 
 (or their faces) are said to be tautozonal or heterozonal with a third, 
 according as they lie in the same or different zone-circles (or zones) 
 with it ; and when two zones have a face in common, that is to 
 say when their zone-circles intersect in a pole, they will be spoken 
 of as tautohedral in that face or pole. 
 
 37. Symbol for a zone. That two different centre-normals, 
 and therefore two faces not parallel, suffice for the determination 
 of the position of the 
 zone-plane to which 
 the faces are per- 
 pendicular is further 
 evident from this, 
 that two points, and 
 therefore two poles 
 on a sphere, not ex- 
 tremities of the same 
 diameter, can only 
 be traversed by one 
 great circle : and 
 the direction of the 
 zone-plane contain- 
 ing such a circle is Fig. 23. 
 
 known when that of 
 
 its zone-axis is known. In fact, if two planes in the zone be 
 PI or (h-i k^ /j), and P 2 or (h^ k z / 2 ), the coordinates for any point 
 on its zone-axis are (Art. 20) in the ratios, 
 
 A convenient symbol for representing a zone is formed by 
 placing the letters or symbols representing the faces by which 
 its direction is determined within square brackets, or so placing 
 the coefficients of the parameters a, b, c in the above expres- 
 sion. Thus, PP / . / 
 
 or 
 
 or 
 
46 Three tautozonal planes. 
 
 and, if P 3 be a third face of the zone, 
 
 [A ?*] or [u 2 v 2 w 2 ] , [P s P 2 ] or [14 v, wj , 
 are symbols that equally represent the zone [P l P z P 3 ~\ . 
 
 A. zone, its zone-axis and zone-plane or zone-circle, will be 
 represented by the same symbol. 
 
 It is however to be observed that the form in which this symbol 
 is presented has not the same signification as the symbol for a 
 plane of the system : moreover in the coordinates an, &v, <rw the 
 indices in the symbol are seen to be integral, not fractional, co- 
 efficients of the parameters. 
 
 But although the two kinds of symbol thus have, even where 
 their indices may be represented by the same numbers, entirely 
 different significance, there are cases in which the symbol for a 
 zone-plane or zone-axis comes to present identical indices with 
 those in the symbol for a plane of the system parallel to the zone- 
 plane. 
 
 These, and the conditions which have to be fulfilled by the 
 axial systems to which it is possible to refer the system of planes, 
 will be discussed in a future chapter. 
 
 SECTION II. Relations connecting three tautozonal 
 planes. 
 
 38. It results from the last article that if the different planes 
 belonging to a zone be taken in pairs, a fresh expression for the 
 zone-axis is obtained from each pair of the planes ; as, for in- 
 stance, in a zone containing the planes (h^ k^ /J, (/$ 2 2 / 2 ), (h 3 k z / 3 ), 
 the symbols 
 
 [u^wj, [u 2 v 2 wj, [u s v 3 w 3 ] 
 
 may be obtained, where [u^wj represents [h 2 k z l z -, ^ 3 / 3 / 3 ], 
 i.e. represents the symbol [ a / s / 2 s , l^h^l^ h 2 k z k^h^\, 
 
 [ U 2 v 2 wj being [> 3 V 3 ; ^i Vi]> 
 and [u 3 v 3 w 3 ] being [^^A; >M 2 / 2 ]. 
 
 And as these symbols equally represent the direction of an 
 identical zone-axis, it is clear that their indices can only differ 
 in their actual and not in their relative magnitudes; and these 
 
Different equivalent zone-symbols. 
 
 47 
 
 actual magnitudes can only differ, therefore, in the various symbols, 
 by a series of common factors. 
 
 It will thus be seen that it must be necessarily true that, for 
 instance, u k I k I 
 
 _ s _ 
 
 Ms- 
 
 <,1 . 
 
 T' 
 
 1 '*2 /V 3 ""3 ' V 2 
 
 where the value of A may in general be determined from either 
 of these ratios. It often happens, however, that from, for instance, 
 certain indices being zero, one or two of these expressions will 
 
 assume the indeterminate form - Since, however, all the indices 
 
 o 
 
 cannot equal zero, the value of A can usually be got by in- 
 spection and selection of the indices ; but it may always be found 
 if the entire symbols be compared, that is, by determining the 
 value of 
 
 which is A. 
 
 r i w i] K v i W J 
 
 39. Notation for Factor-ratios. This relation between the 
 symbols of a zone as derived from those of different pairs of the 
 planes lying in the zone will presently be seen to have much 
 significance. A convenient notation for the representation of such 
 a relation is afforded by confining the symbols of the planes in 
 question between square braces or simply between vertical lines, 
 the symbols or their equivalents being in the form of a fraction. 
 Thus, if p or fa ^ /,), Q or fa Z 2 / 2 ), R or fa k z / 3 ), 
 be three tautozonal planes, their zone may be equally expressed 
 as [PQ], [Q-R], or [RP}} but as these several symbols differ 
 generally by a common factor, we should have, for instance, 
 
 *.*, 
 
 A, 
 
 
 , h^l^ h^k*} ' 
 
 = A, 
 
Notation. 
 
 as various ways of representing the relation of the symbols for the 
 zone as deduced from two pairs of its planes ; and indicating 
 in this case that u 3 = Au 15 v 3 = Xv p and w 3 = Aw r Where 
 we would indicate that an expression for the zone is to be derived 
 by the cross multiplication of the symbols of a particular pair of 
 
 planes of the zone, the symbol may be written thus, 
 
 or where the ratio of two such expressions is to be indicated, 
 as in the above example, the expression will take the form of 
 double line therein represented. 
 
 Thus, if the planes P or (211), Q or (in), R or (100), S or 
 (oil), belong to the same zone, we may indicate this zone as 
 
 211 
 
 
 [PQ]; i.e. as [211, in], or 
 
 II I 
 
 which gives its symbol as 
 
 [on] ; or it may be designated from the planes PS, therefore, as 
 [211, oil] or [022] ; or from R and Q as [oil], or from R and 6" 
 as [oil], &c. 
 And the ratio 
 
 PQ 
 
 PS 
 
 211, III 
 
 211, Oil 
 
 Oil 
 
 02:2 
 
 and the value of A may be variously represented by A = J. 
 So the ratio 
 
 
 
 IOO 
 
 
 RQ 
 
 C\T 
 
 III 
 
 
 on 
 
 RS 
 
 UI 
 
 IOO 
 
 
 oil 
 
 
 
 oil 
 
 
 
 PQ 
 
 
 RQ 
 
 I 
 
 
 ~PS 
 
 * 
 
 RS 
 
 ~~ 2 
 
 And 
 
 represents the proportion of these ratios. 
 
 In expressions of the kind under consideration, the direction 
 in which an arc joining two poles on the sphere is considered to 
 be estimated, and therewith its sign, is indicated by the order of 
 the letters or symbols which represent the poles ; so that 
 QP=-PQ, 
 
 and 
 
 PQ_ 
 
 QS 
 
 QP 
 
 QP 
 
 &C. 
 
Condition for tautozonality . 49 
 
 40. Condition for a plane (hkl) to lie in a zone [uvw]. It 
 further results from the geometrical unity of a zone-axis and the 
 consequent invariability of the ratios of the indices in the various 
 symbols obtained for it as representing the edge formed by one 
 or other of the different pairs of planes belonging to the zone, that, 
 taking the symbols for any three tautozonal planes as before, 
 
 U 2 V 2 W 2 
 is evidently true. 
 
 Since the denominator of this last ratio is identically zero, as is 
 seen on substituting the values of u 2 v 2 w 2 in terms of h k l / x and 
 A s 3 / 3 , the numerator must also be so ; and consequently, 
 
 1 1 w 1 = o, 
 
 .e. ^(Vs-W 
 an equation establishing a relation between the symbols of any 
 three planes in a zone. 
 
 41. Symbol for a plane (hkl) in which two zones [u 1 v 1 w 1 ] 
 and [u 2 v 2 w 2 ] are lautohedral. Since (h k /) belongs to each of the 
 
 zones in question, 
 
 ,u 1 + /fcv 1 + /w 1 = o; 
 
 and /u 2 + /v 2 + /w 2 = o: 
 
 whence h (iij w 2 w x u 2 ) + k ( v x w 2 w x v 2 ) = o ; 
 and /^(U 1 v 2 -v 1 u 2 ) + /(w 1 v 2 -v 1 w 2 ) = o; 
 
 h _ VjWa-w^a h 
 
 d,lbU - 
 
 k wu 
 
 or, - = - - = - ; 
 
 V 1 w 2 -w 1 v 2 W 1 u 2 -u 1 w 2 U 1 v 2 -v 1 u 2 
 
 wherefore the indices of the plane (hkl) have the ratios repre- 
 sented by the symbol 
 
 KW.J w^, W 1 u 2 -u 1 w 2 , i^Va-VjUa). 
 
 And it will be seen that the form of the expression representing 
 the indices of a plane in terms of the symbols of two zones 
 tautohedral in it, is identical with that of the symbol of a zone as 
 derived from the symbols of two of the planes tautozonal in it. 
 
 E 
 
Derivation of determinants. 
 
 42. Expedient for deducing the determinant symbols. This 
 similarity in the form assumed by the symbol of a zone as de- 
 rived from a pair of its planes and by that of a plane common 
 to two zones (as obtained in Art. 40) leads to the adoption of a 
 similar process for obtaining the symbol in either case. An ex- 
 pedient for performing this operation, useful when dealing with 
 complicated symbols, is that of writing, one under the other, the 
 two symbols to be operated upon; each symbol being once 
 repeated on its own line and the first and last index on each of 
 the lines being struck out. The differences of the cross products 
 of each successive pair of indices on the two lines (these indices 
 being connected in the subjoined example by an X) form the new 
 indices, the product of the indices joined by the thin stroke being 
 deducted from the product of those joined by the thick stroke. 
 Thus if ^ 2 - 2 / 2 and ^ 3 / 3 / 3 be two planes in a zone, we have, 
 
 for the value of 
 
 k 2 / 2 h z V 
 
 
 XXX 
 
 
 k z ? 3 hs k z 
 
 
 Ik 
 
 
 I I h I 
 
 
 23 23 
 
 
 ii i? J? i 
 
 ! . 
 
 where 
 
 Ms - 
 
 or 
 
 2. 
 3- 
 
 [V s -/ a 8> 
 is the symbol of the zone sought. 
 
 Similarly, we get for the face (h k /) in which two zones [u v w] 
 and [u'v'w'J are tautohedral, the symbol 
 
 (vw' wv', wu' uw', uv' vu') or (h k I]. 
 The symbol for the zone containing the faces (342) and (324) 
 
 4234 II II 2432 
 
 or 
 2 43 2 4234 
 
 is 
 
 12 , 
 
 -6 
 
 ~fi) 
 
 or 12, 6, 6, 
 
 12, 6, 6; 
 
 which, reduced to their simplest forms, are [211] or [211], ac- 
 cording to the order in which the two planes are taken. 
 
Relations of edges and normals. 
 
 5 1 
 
 The same symbol [211] or [211] is similarly obtained from the 
 poles (ni) and (oil) which also belong to the above zone. 
 
 So again, the opposite poles in which two zone circles [in] 
 and [oTi] would intersect are (2!!) and (211). 
 
 43. Relations connecting the inclinations of the edges and of 
 the normals of three planes. 
 
 If XOY, FOZ, ZOX, Fig. 24, be three planes, OX, OF, OZ 
 the zone-lines parallel to their edges, the great circles joining the 
 axial points X, Y, Z of these zone-axes upon the sphere form a 
 spherical triangle the sides of which evidently measure the plane 
 angles at which each pair of edges meet in the solid angle 
 at ; the angles at X, Y, 
 and Z on the other hand 
 are those of the inclinations 
 of the planes. 
 
 If now OA, OB, OC be 
 the normals of the planes 
 YZ, ZX, XY- A, B, C 
 being the poles of these 
 origin-planes ; ABC and 
 XYZ are the angular points 
 of two polar triangles, and 
 the arcs AB, BC, CA are 
 
 P> 
 
 Fig. 24. 
 
 the supplements of the angles at which the planes YZ, ZX', 
 ZX, XY; XY, YZ are inclined to each other ; while the angles 
 at A t B, and C are the supplements of the angles YOZ, ZOX, 
 XOY at which the edges of the planes meet. 
 
 The arcs AB, BC, CA are generally capable of being deter- 
 mined by processes of measurement, so that from these the other 
 values can be calculated. 
 
 SECTION III. On the signs of the indices of a plane as 
 determined by the position of the plane in respect to 
 a given zone. 
 
 44. Two poles are said to be on the same or on opposite 
 sides of a given pole lying in a zone with them according as 
 
 E 2 
 
52 Signs of indices. 
 
 loth or as only one of them may lie within an arc-distance TT from 
 that pole as measured in the same direction on the zone-circle 
 that contains the two poles. 
 
 Let [uvw] be a zone-circle containing the poles p^ q^ r 1 , p 2 g 2 r 2 ; 
 then the equation u^ + vy+wg- = o gives the necessary condition 
 for a plane efg to lie in the zone [p 1 q^ r lf A ^2^2]' 
 
 Wherefore, for the plane hkl not belonging to the zone, 
 
 u/$ + v/ + w/> or <o; 
 
 so that, if the pole of this plane hkl lies on one of the hemi- 
 spheres into which the sphere of projection is divided by the zone, 
 this expression must have some definite value either positive or 
 negative. 
 
 If, now, for any particular pole this value be found to be positive, 
 it must be positive for all poles situate on the same hemisphere 
 with that pole, since the expression cannot change sign except by 
 the pole passing through zero, i.e. by its passing to the other side 
 of the zone-circle. 
 
 DEF. The hemispheres bounded by a zone-circle may be termed 
 positive or negative in respect to that zone-circle according as the 
 symbols of the poles lying in them give the above expression 
 positive or negative ; i. e. according as the determinant of the three 
 symbols two of which belong to planes in the zone may be positive 
 or negative. 
 
 45. The result of the process for determining the indices in 
 the symbol of a face by the rule of zones is ambiguous, as 
 regards the signs of the several indices, as is seen in the ex- 
 ample in Article 42; the ambiguity however can be removed by 
 determining the position of the pole on the sphere relatively to 
 two poles heterozonal to it, that is to say, relatively to the zone 
 containing those poles. And other conclusions determining the 
 distribution on the sphere of the different poles of a form, the 
 symbols of which differ in the relative order as well as in the signs 
 of their indices, can be arrived at by the aid of this principle when 
 put into the following form. 
 
 Determination of the signs of the indices in a symbol. If R, ft, Q 2 
 
Signs of indices. 
 
 53 
 
 be three planes in a zone, the ratio 
 
 will have a positive 
 
 value if Q l and Q 2 are on the same side of R, and will be negative 
 if they are on opposite sides of R. 
 
 If any pole K the symbol of which is \\LV be taken external to 
 the zone, while Q 1} Q z , R are e^g^ e 2 / 2 g 2 , pqr respectively, 
 then we have 
 
 and (^r-vg)e 1 + (vp-\r)/ 1 + (\g-fi,p)g 1 = C l ; 
 both positive or negative or one positive and the other negative 
 according as Q 1 and Q 2 are on the same or opposite sides of R, 
 that is to say, transposing the expressions, 
 
 ^(9gi-^A}+^(re 2 -pg^ + v(pf li -qe 2 } = C. 2 
 and \(qg l - r f l } + ^(re l -pg l } + v(pf l -qe l } = Q 
 present C 2 and C l with the same or with opposite signs according 
 
 .is 
 
 ~) and therefore 
 
 pqr 
 
 pqr 
 
 i.e. as 
 
 RQ* 
 
 is positive or negative; which thus follows from Q l and Q 2 lying 
 on the same or on opposite sides of R, that is, from their lying 
 on the same or on the opposite hemispheres divided by the great 
 circle [KR]. 
 
 Examples. 
 
 I. On a hexagonal crystal the zone-circle containing the poles 
 277 and 11:2, that is to say [in], intersects with the zone-circle 
 containing the poles oio, ooi, that is to say [100], in two poles 
 which are oTi and oil. 
 
 It is required to determine which of these poles is the nearer to 
 the pole 1 12, that is to say which will be on the same side with 
 iijs, of the pole 2!! on the zone-circle [in]. 
 277 
 
 Here 
 
 211 
 112 
 
 = has a positive value ; efg being the symbol 
 
54 
 
 Examples. 
 
 of the face in question, where e= o andy~= g, while/" and g are 
 unity. 
 
 The expression becomes 
 
 fgi ~( 2 g + e \ e+2/ 
 
 m 
 
 
 
 
 3 3 
 
 3 
 
 2J 
 
 n 
 
 333 
 
 333 
 
 and = \f\ f=-g\ so that if -- is to be positive, f must be 
 
 positive and^ must be negative, and the symbol is oil. 
 
 II. On a crystal of Calcite of which Fig. 25 is a projection the 
 forms r{ioo}, b {2!!}, v {20!}, /{SID} occur. 
 
 The poles v and v' are found by measurements of the angles 
 
 vb" and v' b" , ur and v 'r to 
 be symmetrically situated and to 
 belong to the same form. Simi- 
 larly, / 2 is found to belong to the 
 same form with f and so with /; 
 and / and v are the faces 310 
 and 2oT. 
 
 (i) Both faces / 2 and v' lie on 
 a zone with r and r" ', i.e. with 
 oio and ooi, and therefore on 
 [100]. 
 
 Let then / 2 be okl and v' be 
 
 /g- 
 
 Fig. 25. (2) Also both lie on the same 
 
 side with oio of the zone-plane 
 
 [in], which is the plane of projection of the figure, and therefore 
 k + l and/+- are both greater than zero, i. e. are positive. 
 
 (3) And they both lie on the same side with oio of the zone- 
 plane passing through 2!! and 100 (b and r}, that is to say, of 
 the zone [oil]. Hence/ g and k I are positive. 
 
 Hence from (i) k and / can only be 3 and i or 3 and I, or else 
 
 1 and 3 or Y and 3 ; and/" and g can only be 2 and Y or 2 and i . 
 From (2) k + l can only be 3+ i or i + 3, and 7"+^ can only be 
 
 2 i or 1 + 2. 
 
Four tautozonal planes. 
 
 55 
 
 But from (3) the second index must be greater than the third, or 
 k must be greater than / and f than g : wherefore the symbols are 
 of z>', 02!; and of / 2 , 031. 
 
 SECTION IV. Relations connecting four tautozonal 
 planes. 
 
 46. Let ^(Wi), &('!/!&), G a (' a /,^) and /> a (W 2 ) 
 be four poles lying on the same zone-circle, and let X, Y. Z be 
 
 * ,/A< 
 
 ,:& t; 
 
 cos = 
 
 cos XP 2 = cos 
 
 the axial points in which the axes to which the system of crystalloid 
 planes is referred meet the sphere of projection. Then 
 
 i cos P,Q, + sin XQ l sin P& cos P&X, 
 i cos P& + sin XQ 1 sin P& cos P&X\ 
 = cos XQ l sin />& cos P& 
 
 + sin ^ft sin P& sin^ft cos ^^ X, 
 sin P^i = cos XQ^ sin ^ft cos P& 
 
 -sin Z<2 X sin P& sinP^ cos 
 whence 
 
 cos XP^ sin P Z Q, + cos J^P 2 sin P& = cos ^Tft sin P 2 
 In the same way 
 
 sin P 2 Gi + cos FP 2 sin P 1 Q 1 = cos Y & sin P 2 
 
56 
 
 therefore 
 
 sin P, ft 
 
 Problem of four planes. 
 
 sin ftP 2 
 
 cos YQ^ - cos JT 2 cos XQ l 
 
 cos XQ l cos yP 1 - cos YQ l cos XP 1 
 
 cos 17^ -cos FP 2 cosXP^ 
 
 Similarly, by considering the poles P lt ft, ft, and again the 
 poles jP 15 ft, P z , we may shew that the above expressions are 
 equivalent to 
 
 sin ft ft sin ^ ft 
 
 cos XQ 2 cos J^ft-cos FQ 2 cos XQ l ~ cos XQ 2 cos FP 1 cos -Fft cos XP, 
 
 sin Q P 
 
 cos ^^ cos YQ Z - cos IT^a cos Zft 
 From the fundamental equations (A), 
 
 c 
 
 ! = f COS Z/*!, 
 
 I, 
 
 - cos ^ ft = 
 e i fi 
 
 cos XQ 2 = cos YQ, = - cos 
 
 - cos XP 2 = ~ cos FP 2 = - cos ZP 2 , 
 
 h, k z / 2 
 
 By substituting values for cosXP^ cosXQ l} &c. 
 sin^ft / t cos IT 3 ! k^-h,f 2 _ = 
 sin ft ft ^ cos^ft / VA/; " " 
 
 sn 
 
 sinft/> 2 
 Hence 
 
 _ /, cos ly, ^ a -^^ a _ _A 
 
 = &c. = 
 
 ft A 
 
 (1) sin/> i(? 2 . sin _ft^ 2 _ k * e *- h iA . /*%< 
 sin Pj ^ ' sin ft /* 2 ~~ k h^h^ k 2 ' f^ /i 2 < 
 
 ft?, 
 
 C 
 
Problem of four planes. 
 
 57 
 
 It may similarly be proved that 
 
 (2) 
 
 sin A ft t sin A ft _ 
 sin P l ft ' sin P z ft 
 
 Aft 
 
 Aft 
 
 If ft be external to A tne relation may be put into the form 
 
 V ' 
 
 sn 
 
 ft sinP 2 ft 
 
 sinAft' sin Aft" A" 
 In the expression (2) the ratio is positive, in (3) it is negative, 
 and these two relations, representing the cases where ft is internal 
 and where it is external to P z , may be comprised under the single 
 expression 
 
 sinAftsin(Aft-AA)_ Aft 
 
 sin (Aft- A A) sin A ft Aft 
 
 where A may be positive or negative. 
 
 = A. 
 
 By analogy, 
 
 sin Aft ^ A 
 sin^ft A 
 
 Aft 
 
 Aft 
 
 5 or 
 
 sn 
 
 ft _ 
 
 sinP 2 ft 
 
 Aft 
 
 Aft 
 
 *2/2 
 
 *2/2 
 
 ^2/2 
 
 ^2/2 
 
 c- M i r A Sm A ft 
 
 Similarly for - - =^-~ 1 ; and we have 
 A smP 2 ft' 
 
 sin A ft sin (/, Q, -/,/,) 
 
 sin (A ft- A A)" sin P, ft 
 
 _ ^2 Pi 
 
 = A 
 
 D' 
 
58 Anharmonic ratio, four planes. 
 
 where [p^rj, [p 2 q 2 rj are [^Vi A/xz;], [ 2 2 /2 A pv] re- 
 spectively, (A/xz/ 1 ) being the symbol of an arbitrarily taken pole 
 ^external to the zone-circle [P 1 Q Z ]- 
 
 By selecting for K a pole presenting one or two zeros in its 
 symbol, this symmetrical expression takes a very simple form. 
 
 47. It is seen from the form of the expressions leading to 
 C and D that a fourth plane in the zone is needed in order to 
 obtain a relation between the faces of a zone independent of the 
 axial distances of the several poles. 
 
 It will moreover be seen that the expressions thus obtained 
 for the relations of four faces belonging to a zone are in the form 
 of one or other of the anharmonic ratios of the normals of the 
 four faces ; which form a sheaf of lines lying in the plane of 
 the zone and meeting at the origin. And since these four normals 
 are homographic with the traces on the zone-plane of the origin- 
 planes parallel to the four tautozonal faces, it may be asserted alike 
 of the normals and of the faces, that in a crystalloid system the 
 anharmonic ratios of any four tautozonal planes must be rational ; 
 for it is obvious that the right-hand sides of the expressions C and 
 C' as well as of D and D' are rational. 
 
 The anharmonic function is in each case equivalent to the ratios 
 of the first minors of the . determinants formed by the symbols 
 of the faces taken in corresponding order. Since there are three 
 anharmonic ratios resulting from the division of an angle or of 
 a line, those of the four tautozonal planes P 1 Q I} Q^P 2 are 
 
 /I \ 
 
 Aft 
 
 
 ft ft 
 
 
 
 
 
 
 
 l/fg\ 
 
 V 1 ) 
 
 /o\ 
 
 AA 
 
 
 ftA 
 ftA 
 
 
 
 frr 
 
 
 
 #r 
 
 \A) 
 
 /q\ 
 
 Aft 
 Aft 
 
 
 ft ft 
 Aft 
 
 A 
 
 
 <XC., 
 Rrc 
 
 
 
 <xc. , 
 &C 
 
 \ 6 ) 
 
 Aft 
 
 
 Aft 
 
 1Y s 
 
 
 Ot^., 
 
 
 
 
 Solution of problems involving four tautozonal planes. 
 48. The expressions arrived at in the previous paragraphs 
 afford the means of determining the position on a zone-circle 
 of one of the four poles, or the symbol of one of them, when 
 
Problem to find an arc. 
 
 59 
 
 the remaining positions and symbols of the four poles are 
 known. 
 
 Thus, first; let the position of the pole Q 2 be sought where the 
 symbols of the four tautozonal poles P lt Q lt Q 2) P t1 that is, (^ k l /j), 
 ( e ifii)> ( e ifz^' (^2^24) respectively, and the arcs P 1 Q 1 and 
 PI.PV i.e. the positions of the poles P l Q 1 and P 2 , are known. 
 
 From the equation D we get 
 
 sin P, Q n sin P, 0, 
 
 tan 0, 
 
 sin(P 1 (2 2 ~P 1 / > a 
 where A = 
 
 = A 
 
 sin (P, Q,-P,P 
 
 and from this equation 6 can be determined, since the second limb 
 of the expression consists of known quantities. If the expression 
 D' be employed, the value of A is obtained from the right-hand 
 side of that equation. 
 
 In order to reduce the expression into a form adapted for the 
 use of logarithms, we have, 
 
 -tan 
 
 sin (P, Q,-P l P^-smP, Q 
 
 2 cos i (2 P l Q, - 
 = - tan (P l Qt-\P 
 
 sn - 
 
 cotan 
 
 and tan (/^ ft-l/^) --tan \P, 
 
 = taniP 1 P 2 tan(i 3 5-^) ..... E 
 
 49. Secondly, where the three angles between the four tauto- 
 zonal planes and three of the symbols of the planes are known, 
 and it is required to find the symbol of the fourth plane : 
 
 Let the angles P 1 Q 1J P l Q zt P^P 2 be given, and also the 
 symbols of P lf Q I} P 2 , viz. h^kj^ e^g^ h^kj^ respectively: 
 it is required to find the symbol of <2 2 , viz. 
 
 Putting the equation D into the form 
 
 A 
 
 A = 5 
 
6o 
 
 Problem to find a symbol. 
 
 where A is to be obtained from the left side of either of the 
 
 m 
 
 expressions D or D', --is rational and can be found from the 
 n 
 
 known quantities constituting the right-hand side of the equation. 
 Then 
 
 Aft 
 
 Aft 
 
 and 
 whence 
 
 m 
 
 ^ e 2 = m a / 2 
 
 A 
 
 by symmetry ; and we have for the indices in the required symbol 
 of ft> e = nh-mh \ 
 
 Examples. 
 
 I. On a crystal of Diopside there occur in a zone four faces, 
 their symbols, and the arcs between their poles, being 
 P l or (^ Vi) 01 ( IOO X 
 ft or fe/i^i) or ( I01 ). PI ft = 4939 / 
 P 2 or (h,kM or (ooi), ^P a = 73 59' ; 
 
 or 
 
 where the arc /\ Q 2 is to be determined. 
 From expression D, 
 
 or 
 
 IOO 
 
 301 
 
 OOI 
 
 301 
 
 OIO 
 
 030 
 
 Aft! 
 
 Aftl 
 
 IOO 
 IOI 
 
 OOI 
 IOI 
 
 In order to find the position of the pole Q 2 in the zone, the 
 
Examples. 
 
 61 
 
 symbols of and arcs between the other poles being given as above, 
 we have by equation E, 
 
 3 sin 24 20 
 tan (P, ft- ^ A A) = tan 36 59' 30" tan 103 20' 43", 
 
 = tan 107 28' 46"; 
 
 ' A <2 2 -36 59' 30" =107 28' 46", 
 and P,Q 2 = i4428'i6". 
 
 The value of A might have been equally obtained from the expres- 
 sion D'. Taking the pole (oio) for K, i.e. for (A, JK, v); we have 
 
 for [p^rj, 
 and 
 
 100 
 
 OIO 
 
 = [ooi], and for [p 2 q 2 r 2 ], 
 
 ooi 
 
 oio 
 o + o i o + o i 
 
 -3 + + ' 1+0 + 
 
 or A = 
 
 3 
 
 II. A zone on a crystal of Felspar presents the faces 
 
 p i or (203), 
 
 Q, or (In), 7^ = 31 1 2', 
 
 Q. 2 or (241), PiQ z is the arc required, 
 
 P\ or (130), Pfi = 86 12', and \P,P,= 43 6'; 
 
 _ _ 
 
 = = 
 
 Aft 
 
 
 203 
 241 
 
 
 12 4 8 
 
 
 Aft 
 
 3l2 
 
 
 I 3 o 
 
 241 
 
 
 3 i 2 
 
 
 Aft 
 
 312 
 
 and 
 
 A= 4; 
 
 =-4-- 5 and = -68 25' 47" 
 Whence tan (P, Q,- - P^ P 2 ) = tan 43 6' tan 23 25' 47", 
 
 = tan 2 2 4' 2 7", 
 
 and 
 
 In order to determine the value of A by the use of expression 
 r , we may take for k the pole ooi. 
 Then 
 
 wK 
 
 = [020], and p 2 q 2 r 2 = I *3 || = [ 3I0 j. 
 
 ooi 
 
62 Four tautohedral zones. 
 
 And by D', A = : - - = 4 ; 
 
 the same result as by the other process. 
 
 Conversely, if the symbols in the above zone for the poles P l , 
 Q! , P 2 are given and the arcs P l Q l and P^ P 2 as before, but the 
 arc P v Qi be observed as 65 20"; and it be required to find the 
 symbol of Q v 
 
 Then 
 
 203, in 
 130, In 
 
 sin 65 20' sin (3 1 12'- 86 1 2') 
 sin 31 12' sin (65 20'- 86 12') 
 
 _- _ _ s l n J^20 / sin55 _____ 
 sin 31 1 2' sin 20 52' 
 
 m 4 
 
 Assuming = > 
 
 we have by equation F for the symbol of Q 2 
 
 (-2-4, 0+12, 3-0), i.e. (6, 12, 3) or (241). 
 It will be seen, however, that on the assumption that the symbol 
 of Q 2 is ("241) and that the arcs P 1 Q 1 and P^P^ had been cor- 
 rectly determined, the true value of the arc P^Q t would not be 
 65 20', as measured, but is, approximately, as calculated in the 
 previous paragraph, 65 ic/27". 
 
 Problems relating to four tautohedral zones. 
 50. The method of introducing a fifth pole external to the 
 zone in which four poles lie, in order to represent the anharmonic 
 ratios of these four poles in terms involving two zone circles 
 intersecting with that of the original zone, may readily be extended 
 so as to involve four zone-circles passing through the poles of the 
 four tautozonal planes and also through an arbitrarily taken pole 
 external to their zone. 
 
 If k be this arbitrarily chosen pole and P lt Q lt Q 2 , P 2 be as 
 before four poles lying on a zone to which k is external, then k will 
 be the pole of an origin plane K or (Ajmv), and zone-circles passing 
 through k and through P I} Q lt Q^ P 2 will intersect with the great 
 circle of which k is the pole in the points/^, q^ q z , p% (Fig. 27). 
 
 Through p, the point in which the zone-circle P, P meets the 
 
Problems involving four zones. 
 
 great circle K, draw a straight line intersecting with radii of the 
 sphere Op lt Oft, Oq Ofa in a', &', /,/> 2 . 
 
 The zone-plane \P l P^\ will intersect the plane kpp^ in a line pd 
 which will meet in abed lines drawn from to P lt ft, P 2 , and Q 2 , 
 and from k to a V // 2 : thus the pencil OP l OQ 1 OQ Z OP 2 is homo- 
 graphic with the pencil Op l Oq^ Oq^ Op^ which pencils are the traces 
 of the four original zone-planes on the planes OP l OP 2 and Op^ Op 2 , 
 and their anharmonic ratios are, as in the expression D, 
 
 sin ft g a sin (A ft -A A) _ sinP 1 (? a sm(/ 1 (? 1 -/ 1 / a ) 
 
 sin (A? 2 -AA) sinftft sin (P^ Q^-P^ 
 
 = A, 
 
 where the arcs A?i>A 3i-> **& P\P* measure the angles at which 
 the zone-circles kP^ /fcft, ft, kP^ are inclined to each other. 
 
 It will be seen then from this construction that if there be 
 two poles ft, ft and the zone-circle denoted by them intersects 
 two other zone-cir- 
 cles the symbols of 
 which are [p : q t rj, 
 [p 2 <l2 r 2J which are 
 tautohedral in a 
 pole k\ then the 
 symbols (h^ k v l^) 
 and (h^kj^ of the 
 poles PI and P 2 
 in which the zone 
 
 <? 2 /2 ^?] intersects 
 
 the zone - circles 
 
 [PI <li r i] and 
 
 [P 2 q 2 rj are known ; 
 
 and equally the 
 
 symbol of the zone 
 
 [uj Vj w x ] passing 
 
 through (\ n v), i.e. k and 
 
 of the zone [/fcft], i.e. [A//i 
 
 i.e. ft and the symbol 
 , i.e. [u 2 v 2 w a ], are known. 
 
 There remain then the inclinations of the planes in the zone [ft ft] 
 
64 Problems involving four zones. 
 
 as measured by the arcs P^Q^ P^Q^, P\PV or tne mutual in 
 clinations of the four zone-planes 
 
 as measured by the arcs Aft A^a AA to be determined. And 
 where three of these are given, since the symbols are all known, 
 the fourth may be found. 
 
 51. If then the arcsAft? an ^AA ^ e gi yen the arc ft^ 2 may 
 be obtained by the method of Art. 48. In fact 
 
 sin A ft t sin (A ft -A A) _ sin/^C?., sin P^k ft 
 sin ( A ft A A) sm Aft sm ^ 2 ^(? 2 
 
 sin sin 
 
 whence 
 
 sin A 
 
 sin (A ^2 -A A) sin (A ft -A A) 
 and as in E 
 
 52. Or again, if the arcs be given between two pairs of poles. 
 the symbols of the zones [k P^\ and \_k P z ] and of the poles Q l , Q 2 
 being given as before, then the third arc may be found. Thus, 
 if P l Q z be the arc sought, we have from D', Art. 46, 
 sin P 1 Q a sin (P l Q 1 -P l P a ) = A sin P, Q, sin (P, Q 2 -P, P 2 ), 
 
 2 sin P, Q, sin (P, Q^-P l P 2 ) = cos (P, Q, -F Q, P z ) 
 
 -ooB( 
 
 whence 
 
 cos { 2 ^ (2 2 -(A ft- ft P 2 )} = (i - A) cos (P, ft- 
 
 an equation in which every quantity is known but PiQ 2) and 
 which therefore gives this required arc P 1 Q y 
 If P 1 Q l is a quadrant, the expression becomes 
 
A nalytical method. 6 5 
 
 SECTION V. Analytical investigation of the zone-law. 
 
 53. Expressions similar to those previously obtained may be 
 deduced more briefly and with greater elegance by the methods 
 of analytical geometry. Thus, if as before, OX, OJT, OZ be any 
 three axes, and % v z 
 
 -++ + -=! 
 
 a o c 
 
 be the equation to a plane of the system, then, if the system be 
 crystalloid in character, every other plane belonging to it will be 
 parallel to one of the planes represented by the equation 
 h k I 
 
 where, as before, and 
 
 o c 
 
 are the parametral ratios, and 
 
 the ratios of the intercepts of the plane in question ; h, k, I being 
 integers, or one or two of them zero. 
 
 It is frequently convenient to consider the planes of a system as 
 origin-planes : in which case the equation to the origin-plane, parallel 
 to a plane h k I 
 
 ~x + ~y + ~z = d, 
 a be 
 
 h k I 
 
 IS X + -ry H -- 2 = 0. 
 
 a be 
 
 54. If the equations to two origin-planes are 
 
 h k I 
 
 -x + -y + -2 = o, 
 
 a be 
 
 the line in which they will intersect will be the zone-line or 
 zone-axis x y z 
 
 ua v<5 w<:' 
 
 where u = kr lq, v = Iphr, w = hq- kp. 
 
 And where the system of planes is crystalloid in character these 
 values for u, v, and w are necessarily represented by whole num- 
 bers, or in the case of not more than two of them, by zero. 
 
 F 
 
66 Crystalloid zone-lines. 
 
 55. And if an origin-plane contain two lines passing through 
 the origin which are possible zone-axes, that is to say, which 
 have symbols involving only rational indices, this origin-plane is 
 parallel to a possible face of the system ; inasmuch as the equa- 
 tion to the plane containing the lines 
 
 x y z x y z 
 
 ?_ . cmn - 
 
 T" ZT i > 11U 7 
 
 ah <?k c\ ap <?q <rr 
 
 is (kr-lq)^ + (lp-hr)| + (hq-kp) * = o, 
 
 and the symbol for this plane, viz. (kr lq, Ip hr, hq kp). 
 contains indices which can only be integral, or equal to zero in 
 the case of one or of two of them; since the indices in the 
 symbols of the zones are only integers or zero. 
 
 56. Since the equations 
 
 par 
 x+ly+-z = o, 
 a o c 
 
 \p A<7 \r 
 
 and x + -~y H z = o, 
 
 a o c 
 
 are identical, representing one and the same origin-plane, the 
 symbol of which may be written indifferently as (/>, q, r) and 
 (A/, Xq, Ar), the indices in the symbol for an origin-plane of 
 a crystalloid system may be multiplied by any number positive 
 or negative, or be divided by a common divisor. And the same 
 is true for the symbol of a zone-axis, the equation to which is 
 x y z 
 ua v<$ w- 
 It may be written [uvw], or more generally [/xu/otvjuw]. 
 
 57. If (^! j /j), (^ 2 2 4)' (A ^s 4) t> e tnree origin-planes 
 intersecting in an identical zone-axis, their equations are 
 
 (1) ^ + ^+^ = 0, 
 
 (2) j.f + ^ + 4? = o 
 
 (3) ^f +H ! = 
 
The mle of zones. 
 
 Now x,y> and z have the same values in these three equations 
 and are not all equal to zero, consequently the determinant 
 
 V = 
 
 = o; 
 
 that is to say, 
 
 ^ ~~ MI \^2 3 ^2 ^3/ ~f~ ^1 (/2 ^3 ^2 ^3) ~f~ A (^2 ^3 2 3/ : ~ :=: ^* 
 
 58. Conversely, the three planes intersect in the same zone- 
 axis, if V = o. 
 
 For if the zone-axis in which, say, (2) and (3) intersect be the 
 
 line 
 
 or 
 
 that is to say, the line the symbol 
 
 for which is formed of the first minors of the determinant of the 
 above three equations, the line is 
 
 x y z 
 
 and the plane h^ k^ / x will contain this line if 
 
 h^ Uj + k^ v x + / x Wj = o, 
 that is, if V = o. 
 
 59. And furthermore it will be seen that the necessary and 
 geometrically sufficient condition for the plane hkl to contain a 
 possible zone-axis [u v w] is 
 
 /^u + ^v + /w = o ; 
 for the plane is 
 
 (2) 
 
 a o c 
 and the equation for the line must be 
 
 x y z 
 an <5v <:w ' 
 
 and any values for x, y> z satisfying (2) will satisfy (1); by sub- 
 stitution therefore in (1) of the values from (2) for the ratios 
 
 x y z 
 
 5 ^ 5 - j we obtain ^u + ^v + /w = o. 
 
 a o c 
 
 Since a fresh expression for a zone-axis is obtained from each 
 pair of planes in the zone, as, for instance, in the zone 
 
68 
 
 Analytical investigation. 
 
 the following symbols may be obtained for the zone-axis : 
 
 [u^wj, [u 2 v 2 w 2 ], [u 3 v 3 w 3 ]; 
 where Uj Vj Wj represent 
 h n k n 
 
 or koLi 
 
 ,-h,L 
 
 and similarly for the others : but as these can only differ in their 
 several actual, but not in their relative, magnitudes, the actual 
 magnitudes of the different symbols can only differ by a series 
 of common factors ; that is, adopting the previous notation, 
 
 u 2 v 2 w 2 
 
 or 
 
 2 2 2 
 
 h k / 
 
 J 
 
 w 
 
 results identical with those previously obtained. 
 
 60. The equation h z Ug-f-^g v 3 -}-/ 3 w 3 = o, which represents 
 the condition for a plane (7z 3 k 3 / 3 ) to belong to the zone [u 3 v 3 w 3 ] 
 or [^^/i, ^ 2 ^2 4] ^ s indeterminate ; the condition however which 
 has to be fulfilled by the symbol of the plane may be put into 
 another and more immediately applicable form. 
 
 Thus every origin-plane, the symbol of which has the form 
 
 and (h z / 2 7 2 ), 
 
 lies in the same zone with the two planes 
 and is a plane of the system if is rational. 
 For the determinant 
 
 = zero. 
 
 And, conversely, if (>$ 3 /^ 3 / 3 ) be the symbol of a plane belonging 
 to a zone determined by the planes (h^ ^ 4) and (/^ 2 2 / 2 ), its symbol 
 in general not its simplest, but some equivalent symbol can be 
 represented in the form 
 
 For if these two sets of indices represent the same plane, 
 
 4 =-"' 
 
Tautozonal symbols. 
 
 relations which may also be expressed thus, 
 
 ( 1 ) A h^ -f ft h^ + v A 3 = o, 
 
 (2) A^ +/u^ 2 -\-vk z = o, 
 
 By the reasoning in Article 57, we have 
 h, h n / 
 
 k k k 
 
 K K 
 
 = 0; 
 
 wherefore the above equations are consistent, any pair of them 
 giving the same values for A, //, v. 
 Thus (2) and (3) give 
 
 A v 
 
 V 8 -/ 2 
 
 A 
 
 -.4-4*1 
 
 or 
 
 ~ 
 
 = = ; 
 
 
 x 
 
 " 
 
 (3) and (1) give 
 
 
 = ~ = : 
 
 (1) and (2) give 
 
 A 
 
 -J^L - 
 
 W Q 1 
 
 Having then found values from any one of these systems for 
 A, jn, v, the ratios of which must clearly be rational, we may write 
 for (h z k z / 3 ) the equivalent symbol 
 
 or (A ^ 
 
 which was the expression to be arrived at. 
 
 Example. To find the symbols of a series of planes lying in 
 the same zone with the planes (m) and (320), i.e. in the 
 zone [231]. 
 
 Let A = 2, p. i, then the symbol is (542), 
 A =-2, /*= i, (102), 
 
 A= i, p= 2, (751), 
 
 A -i, fj.= i, (21!); 
 
 all of which symbols belong to planes of the zone [231] : thus from 
 the symbol (542) we have 
 
 5x2 4x3 + 2x1 =o, 
 and so for the others. 
 
7 o A nalytical investigation. 
 
 61. COR. Where A=/x= + i, the process is simply the addition 
 or subtraction of the indices in the symbols (h^ k^ /J and (h^ / 2 / 2 ). 
 
 Whence, whether we take the sums or the differences of the 
 corresponding indices in two symbols, we equally obtain a symbol 
 for a face belonging to the zone. 
 
 A face, the symbol of which is obtained by adding the indices 
 in the symbols of two other faces, will cut off or replace, and in 
 certain cases, where the faces are symmetrically disposed, will 
 truncate their edge; that is to say, will, in the latter case, be 
 equally inclined on the faces. 
 
 In cases where a face truncates an edge, symbols for faces 
 that bevil the edge (that is to say, would be inclined each at 
 the same angle on a truncating face) are obtained by adding the 
 indices of the symbol for the truncating face to those of the 
 symbols for any other pair of corresponding faces that meet in and 
 bevil the edge. [See Article 133.] 
 
 Thus, if (no) and (101) be the symbols of two faces the edge 
 of which is truncated by the face (211), the same edge would be 
 bevilled by the faces (321) and (312), as well as by the faces 532 
 and 523, &c. 
 
 The problems, of four tautozonal planes. 
 
 62. Let there be a system of tautozonal planes; the equations 
 to two of which are u = o and v = o ; where 
 
 - 
 
 a b c ' 
 
 Any plane passing through the same zone-axis [u v] with these 
 planes p^ g l r^ and / 2 g z r z must be represented by an equation of 
 the form u + /xv = o, or 
 
 t =o (2) 
 
 But if the plane be crystalloid and (hkt) its symbol, its equation 
 may also be written in the form 
 
Problems on tautozonality . 71 
 
 In order that these equations may be identical we must have 
 
 whence /x must be rational if the remaining letters represent 
 rational numbers; i.e. if the planes be crystalloid. 
 
 As in Article 46, let P 15 ft,P 2 , ft be four planes of a zone 
 [u v], of which u and v are any two planes ; and let 
 u + fa v = o be the plane P 1 or (^ ^ / a ), 
 u + ^v = o Q 1 or (tfj 
 u + M 3 v = o P 2 or (/$ 
 u + // 4 v = o ft or (<? 2 
 where /t^, /m 2 , ju 3 , ja 4 are, as above, rational. 
 Then the anharmonic ratio of these planes 
 
 Aft Aft _ sin ^ ft sinP.ft _ ^-^ ^-^ _ 
 Aft ' Aft ~ sin P 2 ft' sin ^ ft fS-^'ffi-Msi 
 
 and thus must be rational. 
 
 In order to determine the value of the anharmonic ratio in terms 
 of the indices of the planes ; since u and v are perfectly arbitrary 
 planes of the zone, we may take them such that each passes 
 through one of the crystallographic axes : and evidently such 
 origin-planes must be crystalloid, since each contains two zone-lines. 
 Take, then, u as passing through axis Y\ .'. q v = o : 
 v as passing through axis Z; .-. r z = o. 
 
 Hence, from equation (4), 
 
 k^ /j ^ j 
 
 =-- = 3 and ., = - 
 
 Similarly, 
 
 ft = lL/>, ft = ^A, y. t = !S* 
 
 Hence the anharmonic ratio is 
 
 - 2 f T? f 
 i / *i J i\ i K <i J i\ 
 
 A = " " " " 
 
 .. (8) 
 
 " (^2 ^2-4/2) (^1^1 
 
 whatever be the values represented by these letters. 
 
7 2 
 
 Analytical investigation. 
 
 In a similar way values may be obtained for A involving lh and 
 ge, or hk and ef\ and the anharmonic ratio may be written 
 
 as 
 
 _ sin P t ft sin^ft-PiP,) 
 
 sin (P a ft - Pj P 2 ) ' sin P x ft 
 
 = A (9) 
 
 the sign of an arc being indicated by the order of the letters. 
 
 The solution of problems involving the symbols and arc-distances 
 of four poles on a zone-circle follows from the last formula, as 
 in Articles 48 and 49. 
 
 Thus where the symbols of four poles and two arcs are given, 
 the position of the fourth pole ft is found from the formula 
 
 tan (P 1 ft - i A P 2 ) = tan J P 1 P 2 tan (135 - 0), 
 sin P 1 ft 
 
 where 
 
 tan 9 = A 
 
 and where the arcs are given and three of the symbols, the symbol 
 of the fourth pole may be otherwise obtained from the expressions 
 in equations (7). For 
 
 A = 
 
 sin^ft'sin^ft' 
 
 and by equations (7), 
 
 A = 
 
 A 
 
 As, 
 
 V, 
 
 4*. 
 
 **** 
 
 \ 
 
 where such pairs of symbols may be chosen by inspection as will 
 not result in an indeterminate value. 
 For the symbol e z / 2 g 2 of ft. 
 
 f % = nk^mk^ 
 
Problems on tautozonality . 
 
 73 
 
 Then 
 
 A = 
 
 m 
 
 V, 
 
 1 V, 
 
 I/I ft 
 
 In the same way, or by symmetry, 
 
 . A = 
 
 V, 
 
 it 
 
 m 
 
 k e\A 
 
 
 \ e \ 
 
 n 
 
 6 \f\ 
 
 
 /! h v 
 
 
 M, 
 
 
 . (10) 
 
 The multiplier of is thus any one of the fractions obtained 
 n 
 
 by taking as numerator any determinant formed by taking a pair 
 
 of columns from 
 
 determinant taken from 
 
 and dividing by the corresponding 
 
 The anharmonic ratio of the four planes 
 be written 
 
 , P z , Q 2 may 
 
 sin P, ft sin P, Q, [P, Q,] [P, 
 
 **! 
 
 = &c. = A. .. (11) 
 
 Since the zones [P t ft], [P z ft], [/\ ft], [P 2 ft] are identical 
 in position, corresponding indices in any pair of symbols must 
 bear the same ratio : let [AJ?C] be the simplest expression for 
 the zone. 
 
 Let [/> ft] = 
 
74 
 
 A nalytical investigation. 
 
 [ACJ = 
 
 [A<y = 
 
 Then from (11), 
 
 = A^At = ^B\ = 0~C 4 = y8 : 
 and, p t q, r being any three quantities whatever, 
 
 So, (C l 
 
 = [yAyByC} ; 
 
 <*f; (12) 
 
 (13) 
 
 By addition of equations (12) and (13) and resolution into factors, 
 
 Let [i^VjW-J be \^ifigi,pqr\ and [u 2 v 2 wj be 
 pqr being the symbol of any heterozonal pole. 
 
 & 
 
 Then 
 
 So 
 
 r=p 
 
 /i 
 
 q r 
 
 \A 
 
 v 2 + 7 a w 2 ) 
 
 ^ u 2 + ^ v 2 + 4 w 2 ) (^ 2 u x + X' 2 v x + / 2 w x 
 
 and [p 2 q 2 r 2 ] is [^^/ a> 
 
 and 
 If 
 
 by inverting the expression A, -r becomes A, and we have the form 
 
 A 
 
 of the ratio D', as in Art. 46 ; and similarly for any other form of 
 the anharmonic ratio. 
 
Zones with crystallometric angles* 
 
 SECTION VI. On Isogonal Zones. 
 
 63. Harmonic division of a zone. In investigating the relations 
 which connect the planes belonging to a zone, and establishing 
 the principle which in fact is but a more philosophical form of 
 enunciating the fundamental law of a crystalloid system that the 
 anharmonic ratio of four tautozonal planes is in such a system 
 rational, we are brought into a position from which we may 
 advance to discuss a problem which involves the whole principle 
 of crystalloid symmetry. 
 
 This problem deals with the conditions under which the angle 
 at which any two planes of a zone are 
 inclined on each other may be repeated in 
 the zone. 
 
 The simplest case to be considered will 
 be that in which two planes of the zone 
 are equally inclined on a third at an 
 angle <. 
 
 Thus, if P and P' be two origin planes 
 the angle between which is bisected by the 
 plane S, as in Fig. 28, where the lines OP, 
 OP', &c. are the traces of these planes on 
 the zone-plane which is that of the figure ; 
 it results from the principle of the harmonic division of an angle, 
 which is only a particular case of the anharmonic division, that if 
 2 be a plane in the zone perpendicular to S, 
 smPP' sinSS 
 
 (1) 
 
 sin .PS sinP / 
 
 = 2; 
 
 sin P2 smP'S_ 
 
 \6 ) . T-. ^. * ~ -r^tf-l A 
 
 sin/tf sinSP' i. 
 (d) snTPP > 'sinS 4 S'~2' 
 
 results easily verified by substituting the values for the angles in 
 the expressions ; thus expression (3) is 
 
 sin (j) cos </> i 
 
 2 sin </> . cos tf> sin 90 2 
 
76 
 
 Harmonic division of a zone. 
 
 Since the expressions are independent of the value of $ and 
 have rational values, we may assert : 
 
 1. That where two planes in a zone are equally inclined on 
 a third, a fourth plane perpendicular to the last plane is a possible 
 plane of the zone ; and 
 
 2. That, in a crystalloid zone containing two perpendicular 
 planes, for every possible plane of the zone another plane may be 
 found equally inclined with it on the two perpendicular planes, 
 which fulfils the condition necessary for its being a plane of the 
 system, namely that of its having rational indices. 
 
 In fact if S be (efg) and P and P' be (hkl) and (h f 
 then, 
 
 sinPP' sinSS 
 
 sin PS 
 
 and thus - is rational j p is determined from the relation 
 
 PP' 
 
 : 
 
 PS 
 
 o \* 
 
 = 
 
 kl 
 
 k'l' 
 
 
 
 fg 
 qr 
 
 SP' 
 
 kl 
 
 qr 
 
 
 f 
 k'l 
 
 where uvw are the indices of the zone 
 
 h kl 
 
 Hence the 
 
 indices p q r of 2 are necessarily rational that is to say, fulfil the 
 necessary condition for a possible plane of the system. This how- 
 ever, as will be hereafter seen, is not always a sufficient condition 
 in order that a plane may exist in a symmetrical system of crystal- 
 loid planes. 
 
 It is obvious that any relations that are established regarding 
 the anharmonic ratios of the planes in a zone apply equally to 
 those of their normals and to the sines of the angles of the arcs 
 joining their poles on the zone-circle. 
 
 It will further be evident that what particular values of </> may be 
 possible, or whether any two angles between three consecutive 
 planes in a zone can have the same value </>, will depend on the 
 nature of the zone, and ultimately on the axial elements to 
 which it is possible for the particular system of planes to be 
 referred. 
 
 64. We may next discuss the more general problem regarding 
 
The cry stallometric angles. 77 
 
 the repetition of the same angle of inclination between several 
 consecutive planes in a zone : where however this angle is limited, 
 by the condition that the number of planes in a crystalloid zone 
 cannot be infinite, to such angles as are commensurable with TT ; 
 so that the recurring angle 
 
 where p and q are integers. 
 
 And it is obvious that if the anharmonic ratios of four con- 
 secutive tautozonal planes inclined at the same angle $ are 
 rational, those of three of them and a fifth plane also inclined on 
 one of them at the angle < are rational, and that in fact the whole 
 zone-circle may be divided by n of its origin-planes into 2 n sectors, 
 
 the angles of which are each $, i. e. 2 TT. 
 
 DEF. A zone thus divided by n equally inclined consecutive 
 origin-planes will be termed an isogonal zone; and the angular 
 values which ( may be able to assume will be termed crystallo- 
 metric arcs or angles. 
 
 We shall proceed to prove that the number of possible values 
 for $ is extremely limited, and that in fact the only crystallometric 
 angles are 90, 60, 45, and 30. 
 
 It will be only necessary to establish this for the case of four 
 consecutive poles in a zone. 
 
 65. The only possible isogonal zones in which the 
 angle (</>) between consecutive origin-planes, and there- 
 fore between the normals of consecutive faces, is a sub- 
 
 multiple of TT are those in which <b is > > -> and o. 
 
 2346 
 
 Let P, Q, P f , Q be four poles in such a zone-circle, so that 
 
 PQ= Qp' = p>Q> = <l ) = 2 v. 
 
 sin PP f sin Qff . 
 
 Then the anharmonic ratio ^r-. ^-^ is rational, or 
 
 * 
 
 sin 2 </> . sin 2 <f> 
 
 = a rational quantity = A ; 
 sin 9 . sin 9 
 
 2 cos 20 = A 2 and is a rational quantity. 
 
Isogonal zones ; 
 
 By trigonometry* we have 
 
 2 COS ^2$ = (2 COS2<j)) q 
 
 COS 2 <f)) q ~ 2 + ... 
 
 r (2 
 
 where q zr is never negative and A 2 , A^, A 6) A 2r are all 
 integers. 
 
 But 2 cos ^2$ = 2 cos4/>7T = 2; 
 
 .'. (2COS24>) g -M 2 (2COS2(/>) 9 - 2 +... = 2. 
 
 Thus 2 cos 2 (/> must be a rational root of the Equation 
 
 x q + A. 2 x q -* + AtX-*+... = 2 . 
 
 t But if the coefficient of the highest power in a rational algebraical 
 equation be unity and the other coefficients integers, the rational 
 roots must be zero or integers. 
 
 Therefore 
 
 2 cos 2(> = 
 
 or = 
 
 or = + 2 
 
 for any other integer would give an impossible value for cos 
 
 and 
 
 , 7T 7T 277 n 
 
 2 = - 3 - 5 5 TT or o , 
 2 3 3 
 
 7T 7T 77 7T 
 
 <p = 5 - j -j -oro. 
 4632 
 
 66. If three tautozonal planes in a crystalloid system be in- 
 clined on each other at a common crystallometric angle (/> and 
 
 any fourth face belonging to the 
 zone be taken ; then any planes 
 in the zone inclined on this 
 fourth face at the same crys- 
 tallometric angle $, or at any 
 angle an integral multiple 
 of $, will fulfil the condition 
 necessary for being possible 
 planes of the system; i.e. 
 their indices will be rational. 
 
 Let /> Q, P 2 , R be four tau- 
 Fig. 20. 
 
 tozonal planes, the arcs between 
 
 the poles of which are P^Q = QP 2 = <, P 2 R = a ; then, if Tbe 
 a plane lying in the zone such that RT = </>, Tis a possible plane 
 of the system. 
 
 * Todhunter, Trigonometry, Article 286. 
 f Todhunter, Theory of Equations, 113. 
 
their properties. 79 
 
 For by the principle of rationality of anharmonic ratios, 
 sin 7\ P z sin QR __ sin 2 $ . sin ($ 4- a) _ 
 sin 7*! 7? sin (XP 2 ~" sin (2 </> + a) . sin$ 
 
 = 2 cos 6-. ~- ^ ......... (i) 
 
 r sm(2$ + a) 
 
 Also sin Pj Q sin 77*2 _ sin $ sin (</> + a) . 
 sin P l P 2 sin T^? sin 2 </> sin (2$ + a) 
 
 let 2 cos <h=. m, A 
 
 ' A = ^' 
 
 which is rational, since A is by hypothesis rational and m 2 is ra- 
 tional for all crystallometric values of <p. 
 
 Hence the symbol of T fulfils the condition of rationality in its 
 indices. And similarly another plane T 7 ' inclined also at the same 
 angle $ on T is a possible plane of the zone, and so on for any 
 more planes. 
 
 67. But in the same zone crystallometric angles of 
 30 and 45 cannot concur; for, 
 
 1. If P 1 Q = QP 2 = 4> = 30, and 7\S = = 45, then S is 
 not a plane of the system. 
 
 For the anharmonic ratio 
 
 sin 7*2 Q sin SP^ __ sin <> sin 2 
 
 sin P 2 P 1 sin SQ ~ sin 2 $ ' sin (</> + 0) ~ 3 _}_ ,/ ^ ' 
 which is irrational. And, 
 
 2. If R is, as in the last paragraph, a plane belonging to the 
 zone with P ly Q and P^ , then T 7 and T 7 ' are possible planes of the 
 zone forming with R three planes successively inclined at the same 
 angle $ at which the planes P 1 and P 2 are inclined on Q ; and 
 therefore a plane inclined on R at 45, if 4> be 30, or at 30 if $ 
 be 45, is not a plane of the system. 
 
 It is evident that in a zone containing three or more planes 
 inclined on each other successively at a crystallometric angle, each 
 of these planes will, together with planes perpendicular to them, 
 harmonically divide the zone ; and the latter planes therefore fulfil 
 the necessary condition for being possible planes of the zone. 
 
CHAPTER IV. 
 
 THEOREMS RELATING TO THE AXES AND PARAMETERS 
 OF A CRYSTALLOID SYSTEM. 
 
 SECTION I. On changing the Axial System to which a 
 Crystalloid Plane-System is referred. 
 
 68. IN crystallographic operations it occasionally becomes 
 necessary to change the parametral plane or to refer the crystal 
 to a new set of axes or to effect both operations simultaneously. 
 The expressions necessary for performing these transformations 
 may either be obtained directly by the methods of algebraic 
 geometry or may be deduced from the expression previously 
 obtained for the anharmonic ratios involved in four tautohedral 
 zones. 
 
 The latter, the more brief and elegant of these methods, is due 
 to Professor Miller (Tract on Crystallography, 21). 
 
 Both processes will however be given here as affording different 
 points of view of the operations performed. 
 
 (i) To change the parametral plane only. 
 
 69. a, b, c being the original parameters and a', b', c' the new 
 ones as determined by the intercepts of a plane (efg), i.e. by 
 the ratios a b c 
 
 ' /' ? 
 
 then a = a'e, b = b'f, c = eg, 
 
Transformation of axes. 81 
 
 and the intercepts of a plane (hkl) as expressed by the new 
 
 parameters are f e f & 
 
 #' ) b-ri c f -> 
 n k / 
 
 and the new indices of the plane are 
 
 j,' v /' h k I 
 
 ft . K , i or j T.J - 
 
 ' f 8 
 i.e. the symbol is (hfg, kge, lef). 
 
 If it should happen that any of the indices e, f, g be unity, 
 the corresponding parameter will remain unaltered. 
 
 ( ii ) To transform a crystalloid system of planes referred to one set 
 of axes to another set of axes formed by edges of the system. 
 
 7O. Let the system of planes be referred to axes OX, OF, 
 OZ with a parametral plane ABC, the equation to which is 
 
 - + i + -=r- 
 
 a o c 
 
 Let OX', OY', OZ' be zone-lines of the system which are to 
 form the new axes ; and let the equation to OX' be 
 
 i x _ i J / _i __ I 
 Uj a v x b w x c 0j 
 
 Then the plane ABC will cut the new axis OX' in a point A' 
 in which the coordinates of the plane and line are identical; 
 whence the coordinates of A' are 
 
 u, 7 v, w, 
 
 * = ' =*' =<' 
 
 W 
 
 l 
 
 ' 
 
 '- 
 
 and x = - -- .................. (1) 
 
 The new parameter for the axis ^' is Oyl', which is the 
 diagonal of the parallelepiped of which the sides are the old axial 
 planes, and the edges are the coordinates of A' on the original 
 axes. 
 
 If now HKL be a plane of the system the equation to which 
 referred to the original axes is 
 
 G 
 
82 Transformations 
 
 . . . 
 
 a o c 
 
 the coordinates of the point H' in which it cuts the new'axis of 
 X, that is, cuts 
 
 i oc _ i _r _ i z _ i 
 
 -a.! a Vi b \v l c 2 
 
 U l z V l W l 
 
 are # =0. , jy=. , 3 = f.-J. 
 
 A, SO 
 
 and ^ = 
 
 But, clearly 
 
 OH^ _ x^ _ 0, 
 ~OA' ~~ ~x ~ 6 
 
 similarly, 
 
 . W 
 
 where 0^ r and OL' are the intercepts of the plane HKL on 
 two zone-lines u 2 v 2 w 2 and u 3 v 3 w 3 , taken for the new axes of 
 Y and Z, and cut by the parametral plane ABC in ff and C'. 
 In fact the equation to the plane HKL referred to the new 
 axes 
 
 that is to say, is 
 
 U 3 v 3 w 3 
 x y 
 
 ~OH f 
 
 x y z 
 
 h' k' I' 
 
 in which the denominators are the intercepts, and the indices h' ', k', I' 
 of the plane HKL as referred to the new axes are the reciprocals 
 of the coefficients in these denominators of the parameters a', b', c, 
 that is to say of OA f , Off, OC'. 
 
of axes and parameters. 83 
 
 , , ,, OA' 
 
 wherefore / 
 
 Similarly 
 
 the zone-axes t^VjWj, u 2 v 2 w 2 , u 3 v 3 w 3 are the edges of three 
 origin planes p l ^ ^ , p 2 q^ r^ , p z q z r s , so that 
 
 KvjwJ is [ 
 [u 2 v 2 w 2 ] is [ 
 [u 3 v 3 w 3 ] is [Aftri* 
 
 values which may be substituted, if required, in the above ex- 
 pressions for >', ', /'. 
 
 If only one axis, say the axis of Z be changed, the plane A ? 3 r 3 > 
 common to the other two unchanged axial zone-lines, remains 
 unchanged and is ooi. 
 
 Whence 
 
 u i = 2v v i = A> w i = > 
 
 and 
 
 -A 
 
 These expressions give the values for the indices of a plane in 
 a crystalloid system when transformed to a new set of crystalloid 
 lines as axes, but referred to the same parametral plane. 
 
 (iii) T/" /#<? parameters are to be changed as well as the axes. 
 
 71. Let EFG be the new parametral plane and let efg be its 
 symbol as referred to the original axes. Then the coordinates of 
 
 G 2 
 
84 Transformations 
 
 E, the point in which the new axis of X intersects with the new 
 
 parametral planes, are 
 
 014 $Vj 
 
 v 0, 
 
 > ~~Z~ ' ^ ' 
 
 and 
 Hence 
 
 and the indices of the plane HKL as transformed to the new 
 axes, and also to the new parameters, become 
 ' ^ u i + ^ v i + ^ w i 
 
 Derivation of the expressions for transformation from those for 
 
 the anharmonic ratios. 
 
 72. Let the plane-system be referred to three new zone-lines 
 as axes, the symbols for which are deduced from any of the faces 
 belonging to their several zones. 
 
 Let [VW], [WU], [UV] be zones of which the symbols are 
 
 Fig. 30. 
 
 [ujViwJ, [u 2 v 2 w 2 ], [u s v 3 w 3 ] as referred to the original axes, and 
 let the new axis of 
 
of entire axial system. 8 5 
 
 X be the zone-line [u^Wj] which accordingly becomes [100], 
 Y [> 2 v 2 w 2 ] [oio], 
 
 z K^WS] [i"J; 
 
 UVW being poles in which the zone-circles intersect. 
 
 Let now P be the pole of a plane with the symbol (hkt) under 
 the old axial system, and Q be similarly the pole of a plane (efg) ; 
 then by the expression in article 51, which represents the form 
 adopted by Professor Miller for the anharmonic ratio, we have 
 
 sin V W/> sinUW/* u + v + w/ +v 2 _ + ^2 7 
 
 ~ ' ' 
 
 ' sinUW<2 ~ * + Vj/+ w x ' u 2 * + 
 sinWV/ > sinUV/ > 
 
 3 3t 
 
 sinWV<2 'sinUV<2 ~ ^^ + v 1 /+w 1 ^' u 3 ^-(-v 3 /+w 3 ,^' ' 
 and if the symbols of P and Q when referred to the new axial 
 system become (h'Kl'} and (e'f'g'), while those of 
 [u 2 v. 2 w 2 ], [u s v 3 w s ] become [100], [oioj, [ooi], 
 sin V WP sin U WP hk' 
 
 sinVW<2 sinUW<2 
 
 ''f'' 
 
 sin_WVP sinUV/ J _^ 
 slrTw V^ : sin U V (2 ~ / : /' 
 
 wnere by substitution and division, from (1) and (2), 
 
 Ul<? Vl Wl ^ '^ Vl Wl ' ( 5 ) 
 
 .. (6) 
 
 two equations which are satisfied if 
 h' = UA + V^ + W/ / = 
 
 which two sets of equations are those given by Professor Miller, 
 and thus comprehend all the processes of transformation. Besides 
 the symbols of the new axial planes in the two axial systems, it 
 is obvious that those of some fourth plane in each axial system 
 must be known. Such a fourth plane serves to give the parametral 
 ratios of the new axial system. 
 
86 Transformations. 
 
 The equations L then afford the means of determining either 
 the symbols of planes in the new system when their symbols as 
 referred to the old system are known ; or conversely. 
 
 Thus, if (e'f g') and (e fg) be the known symbols under the 
 new and the old systems respectively of any plane Q of the 
 system, and (h' tf F) be the symbol of the plane P as referred 
 to the new axial system, (h k /) being its symbol as referred to the 
 old system, then, from the equations L, we have 
 
 
 / = g ; 
 
 which will be equivalent to the equations previously obtained in 
 the last article in the case where //'/ is the face (in). 
 
 If then the symbols (efg\ (/ f g'} be given as well as [u^wj, 
 [u 2 v 2 wj, [u 3 v 3 w 3 ], the values of h'k'F can be found from those 
 of h k /, and conversely. 
 
 And where Q is the parametral plane ( 1 1 1 ) under the new 
 system, e '=f ' = g' = i, and where the parametral plane remains 
 the same for the two systems, only the axes being changed, 
 
 73. Let U, V, W, Fig. 27, be the poles (A?i r i)> ( A 2* r ^ 
 
 Fig. 31- 
 
 (p 3 (? 3 r 3 ); and let two zone-circles [hkl], [efg] intersect with the 
 zone-circles [UV], [VW], [WV], viz. 
 
Symbols of zones. 87 
 
 [hkl] with [UV] in H, with [VW] in K, with [WU] in L, 
 
 [efg] with [UV] in E, with [VW] in G, with [WTJ] in F; 
 then, by D', article 46, 
 
 sinUH sinVH h + k + r h + k + r 2 l 
 
 , x 
 
 -0 . . ( ^ ) 
 
 sinWF sinUF / 3 e + ^ 3 f-j-r 3 g A e + ?i f + r i 
 On referring the plane-system to new axes, let the poles U, V, W 
 become (100), (oio), (ooi) under the new axial system, and let the 
 new symbols of [hkl], [efg] at the same time become [h'k'l'j, 
 [e'fg']. 
 
 Then, by substituting and transposing in (1) and (2), we have 
 
 sinUE ' sinVE ~ 
 
 sinWL sinUL /> 3 h + ^ 3 k + r 3 l p, h + ^ k-l-r, 
 
 *~~" * 
 
 f' 
 
 Equations which are satisfied if 
 
 e'^e 
 
 f r = e + f + gr, > .. .. ,. N 
 
 from which expressions the symbols of any zones in the system 
 may be determined if the symbols of the three poles referred to 
 the old system are known. 
 
 The symbol of a zone when the parameters but not the axes 
 
 are changed. 
 
 74. [u v w] being the symbol of the zone \h k /, efg] as de- 
 termined by the original parameters, let [u'v'v/] or \h ' k' 'I 1 ', e f g'\ 
 be its symbol as determined by the new parameters, and let the 
 new parameters be a'b'c ; 
 
 u v w is kg If, I e h g, hf k e, 
 
 uVw' is /&y-/y, /v-^y, h'f-k'e' 
 
 and from article 69, 
 
 e'=eh'kl, f = fhk'l, g'^ghkl', 
 h'=he'f g) k'^kefg, l'=lefg'' } 
 
8 8 Transformations . 
 
 whence u' = K g ~t f = (kg -If) e f'g = u * 
 v ' = r / _y / =(le-Ag) c'fg' = v / 
 
 Hence the new symbol of the zone is 
 
 Examples. 
 
 I. To transform a crystal of Quartz from the customary axes to 
 a new set of axes such that the pole (41^) becomes the pole (100) 
 of the axial plane YZ, (241) the pole (oio) of the axial plane ZX, 
 and (i"24) the pole (ooi) of the axial plane XY. 
 Then [ui v i w J is [241, 124], i.e. [210], 
 
 [u 2 v 2 wj is [124, 412], i.e. [021], 
 [u 3 v 3 w 3 ] is [412,241], i.e. [102]. 
 
 The symmetry of quartz is of a type which renders it possible 
 and convenient to refer the crystal to an axial system in which the 
 angles at which the axes are inclined to each other are equal while 
 the parameters also are equal. It results from this that in trans- 
 forming a crystal of quartz from one axial system of this kind to 
 another of the same kind, the same face serves as the parametral 
 plane for both systems. Consequently, 
 
 / = / = /= i =e=/=g; 
 and the equations M become 
 
 _, tL^ + Vi/fc + Wj/ 2h+k 
 
 fl = - : - - = 3 
 
 ,,_ 
 
 u 2 + v 2 + w 2 3 
 
 , _ u s h + v 3 k + w s / _ 2 /+ h ^ 
 
 u 3 + v 3 + w 3 3 
 
 ' or (h'k'l') is (2h + k, 2k + l, 2l+h\ 
 
 Therefore, for the face 100 or r in the lettering of Miller 
 (Brooke and Miller's Mineralogy, p. 246), 
 
 a'=2, ^=o, r=i, 
 
 and the symbol under the new axial system becomes 201 ; for 2. 
 or Y22, the transformed symbol is 063, i.e. 021. 
 
Examples. 89 
 
 And the new symbols for the following faces,. 
 
 a, oil; b', 2li; b, 2?; k, n 4 7; II 4 7, 
 become 112"; ilo; oil; 651; 651; 
 
 those for AT, 4 I 2; ", 4^1; ^, 10 2 5; , 8 I 4, 
 become 740 ; 212 ; 2lo ; 5^0 ; 
 
 and those for w, 14 16 7; 14716; ^, 16178; 16817, 
 become 4 13 o; 7 10 6; 5 14 o; 8 n 6; 
 
 and 17 8 16; /, 10 14 5, become 14 o 5; and "2 n o. 
 
 II. A crystal of the (oblique) mineral Sphene as referred to the 
 axial system employed by M. Des Cloizeaux may present the faces 
 
 010, 001, 100, III, 110, 102, 102, 112, O2I. 
 
 It is required to transform these symbols into accordance with 
 the axial system employed by Professor Miller for this mineral ; in 
 which the zone-axis 
 loo 
 
 oio 
 
 , i.e. [ooi] is [u 1 v 1 wj and becomes [100]; 
 [oio] is [u 2 v 2 w 2 ] and remains [oio]; 
 , i.e. [201] is [u 3 v 3 w 3 ] and becomes [ooi]. 
 
 The equations M here become 
 
 7 / / I 7, f , k , , 
 h=e 5 /=/, l=g 
 
 -g / 
 
 And any plane of the system, the symbols for which according 
 to both axial systems are known, will serve to give the ratios of 
 the indices in the new symbol (h'tff) to which the original symbol 
 of the plane (h k I) has been transformed ; or, conversely, will give 
 those of (h k /) the original symbol from the indices of (h f k' /'). 
 
 The face In on the old system has for its symbol 123 on the 
 new : let (efg) then be In, (e'fff) be 123. 
 
 Then >&'=/, k f =2k, l'=l-2h. 
 
 Let now (h kl) be 1 10 ; then for the new symbol 
 
 h' k' I' 02^, i.e. = oil. 
 
 So if (hkt) be Io2, h f - 2, '= o, l'= 4, and (k'tfl') is 102 ; 
 i.e. the old symbol Io2 becomes the new symbol 102. 
 
9O Transformations. 
 
 Similarly, 102 becomes 100, ooi becomes 101, and 112 be- 
 comes no. 
 
 Conversely, for the face (h'k'l') or 112 of Miller the original 
 symbol would be (h k /), and we have for this 
 
 '=I=/, k'=T. = 2k, f=2=l2h; 
 
 .-. 2h = i, and (hkt) is Ii2. 
 
 Similarly, 163 on the new system corresponds to 131 on the 
 old. 
 
 III. A crystal of the (anorthic) mineral Axinite is referred under 
 the axial system adopted by Professor vom Rath * to axes which 
 would be the zone-lines of the zones [oio, ioT], [lol, 101], 
 and [101, oio] under the axial system to which Professor Miller 
 refers this mineral. Professor Miller's axes on the other hand 
 are the zone-lines which in vom Rath's system would have the 
 symbols 
 
 or To ill i.e. fujVjW/l for the axis of X, 
 01 1 
 
 oil I 
 
 or [loo], i.e. [u 2 v 2 w 2 ] Y, 
 
 Oil 
 
 100 
 
 or [on], i..e. [u 3 v 3 w 3 ] Z. 
 
 Whence by the equations M 
 
 f+g * f-g 
 
 The face 102 under vom Rath's system becomes the parametral 
 plane in under Miller's; and therefore efg is 102, and e f g 
 is in. 
 
 Hence, * = *L, K=h, /'= - 
 
 2 2 
 
 For the new symbols therefore of the face 101 (hkt) we have 
 
 '=4, '=!, /':=!, 
 
 or 1 2 1 is the symbol required ; and the symbols on the system of 
 vom Rath, 
 
 IOI, OOI, TO2, Y2O, I2O, III, ill, 211, III, Til, 
 
 * Pogg. Annal. cxxviii. 20 and 227. 
 
Axes are crystalloid lines. 91 
 
 become 
 
 121, 101, iTi, ili, nT, oil, on, 120, no, iTo 
 severally, on the system of Professor Miller. 
 So also for the symbol iY2 of Miller ; since 
 
 ,, k + l lk 
 
 h = i = - 5 k = i = h, and / = 2 = - > 
 
 2 2 
 
 the symbol under vom Rath's system is 113, and Miller's face 012 
 is 122 with vom Rath's axial system. 
 
 SECTION II. The axes of a crystalloid system are neces- 
 sarily origin-edges or face-normals. 
 
 75. In establishing the expressions in article 7O for the trans- 
 formation of an axial system the ratios of the values u^Wj &c. 
 were assumed as rational ; these formulae however are entirely 
 general and would be equally true were this restriction not intro- 
 duced. In that case however the component symbols resulting in 
 the symbol [u^wj would not be necessarily rational. It will 
 however be seen that in every case in which the symbols of the 
 planes as referred to the new axes are rational, these axes must 
 necessarily be zone-lines of the system and as such present rational 
 symbols. 
 
 In the article referred to it has been proved that if the equation 
 to the new axis X be 
 
 \x_\y_\z 
 U-! a Vj_ b Wj c 
 
 and hjiji be the transformed symbols of the plane hkl as referred 
 to the new axes, we have 
 
 _A 
 
 or 
 
 where h^ and h k I are by hypothesis rational. 
 
 In a similar way, if p^ ^ r be the new rational symbols for a 
 plane pqr, we have 
 
92 Plane- and zone-systems ; 
 
 and, solving these two simple equations, we get values for the 
 
 Tl "V 
 
 ratios and - - which must necessarily be rational. Hence 
 
 any axes for which the law of a crystalloid system of planes holds 
 good are themselves possible zone-lines of the system. 
 
 On the reciprocity of a zone-system and a plane-system. 
 
 76. It has been previously established that while the inter- 
 section of two origin-planes is a zone-axis, that of two zone-planes 
 is a normal to a face of the system ; or, which is the same thing, 
 that while two zone-axes lie in an origin-plane, two normals must 
 lie in a zone-plane. 
 
 It may further be proved, that whereas in a crystalloid system of 
 planes any set of origin-edges may be taken, together with a face 
 intersecting with them, for the axial system to which the system of 
 planes and of zone-axes parallel to the edges of these is referred ; 
 so, on the other hand, any set of normals of faces belonging to the 
 system may be taken together with a parametral zone-plane, for 
 the axial system to which may be referred the system of zone- 
 planes and of 'rays' (i.e. of radii of the sphere coincident with 
 normals to the planes of the system) in which these zone-planes 
 intersect. 
 
 And the expressions for a zone-axis as referred to three zone- 
 axes with a parametral plane belonging to the system are identical 
 in form with those for a 'ray' as referred to the axial system 
 formed by three normals and a zone-plane. In the one case, for 
 instance, the expression for the zone-axis is that already obtained 
 in art. 53, namely 
 
 x _ y z 
 \\.a vb we' 
 
 in the other case, that for the ray belonging to the pole hkl is 
 x _ y_^ z 
 
 where a, /3, y are the parameters on a normal system of axes, in 
 which the axes are the normals to the axial planes of the former 
 system. 
 
their reciprocity. 
 
 93 
 
 This may be thus proved. If OA, OB, OC be normals to the 
 axial planes YZ, ZX, XY to which a plane-system has been 
 referred, and (hkt) be the symbol 
 of P a pole referred to these 
 normals as axes, see Fig. 32; 
 then a zone-circle through A 
 and P will intersect the zone- 
 circle BC in a pole D which 
 will have for its symbol (p k /). 
 
 The position of a point P on 
 the normal of the plane (hkt) 
 is determined by the lines LM, 
 LN, LP parallel to the axes Fig. 32. 
 
 (and proportional to the inter- 
 cepts of the plane) ; and clearly the points 0, L, D are in a right 
 line. 
 
 y_ _ LN _ sin L ON _ smDC 
 
 ~z~Wd~~sm OLN ~~ sin DB ' 
 by the fundamental equation (A) 
 
 1 -cosZ)F = ^cosjDZ } 
 k L 
 
 or j sin DC sin BCA=. C - sin DB sin CBA, 
 
 K I 
 
 smDC 
 
 77 sin 
 to 
 
 Thus 
 
 and 
 
 k . _ " 
 
 - sin LA 
 b 
 
 - sin AB 
 c 
 
 X 
 
 a \jy oyu 
 
 
 7 ' 
 
 - sin BC 
 a 
 
 X 
 h^L 
 
 y z 
 
 ~kp- ly" 
 
 where 
 
 a 
 
 ft 
 
 y 
 
 smBC 
 a 
 
 sinCA 
 
 smAB 
 
 b 
 
 c 
 
94 Plane- and zone-systems. 
 
 Hence equation (1) represents the normal or 'ray' of the face 
 P as referred to normal axes ; and again, if 
 
 x y z 
 - = ^ = - and 
 
 x y z 
 
 be the equations to two such rays, these rays will lie in a plane of 
 which the equation is 
 
 x y z 
 u hv^+w = o ; 
 a /3 y 
 
 where u = /W- Ik', v = lh'-hl', w = hk'-kh', 
 and are rational. 
 
 Thus, if OU y OV,OWbe the intercepts of a plane parallel to 
 [uvw], then 
 
 OU _ OV_ _ OW 
 a : 1T ~ 
 h k / 
 
 and a, /3, y are the intercepts of the parametral zone-plane [i 1 1]. 
 
 The reciprocity in the expressions for a face-normal and for 
 a zone-plane as referred to an axial system of the kind adopted 
 in this treatise, and for a zone-plane and a face-normal referred to 
 such an axial system as has been discussed in this article, is com- 
 plete. And it will further be seen that the indices are the same 
 for the same face or for the same zone-plane as referred to the two 
 axial systems, though the symbols in which they are embodied 
 have a different significance in the different systems. 
 
 Some of the results of this reciprocal relation between the zone- 
 system and the plane-system of a crystalloid system of planes have 
 been examined by Professor Miller in his elegant tract ' On the 
 Crystallographic Method of Grassmann' (Part V of the Pro- 
 ceedings of the Cambridge Philosophical Society, Cambridge, 
 1868), wherein he has shewn that all the problems of crystal- 
 lography may be approached from the side of a system of rays 
 referred to normals as axes, and that this method yields expressions 
 identical in form with those which are obtained by the other 
 method. 
 
Zone-lines and normals. 95 
 
 Case of a zone-axis being coincident with a plane-normal. 
 
 77. An important question however arises as to the possibility 
 of a zone-axis being not only a normal belonging to a system of 
 zone-planes reciprocal to the system of actual planes, but being 
 also a normal to one of the faces of this actual system. 
 
 It is clear that in general it is not so; i.e. that a zone-axis and 
 a plane-normal, or that a zone-plane and an actual plane, can only 
 in particular cases be coincident. 
 
 Those cases will form a subject of enquiry in the Fourth Chapter, 
 in connection with the subject of Symmetry. 
 
 It will be well however to investigate here the general conditions 
 to which a plane-system is subject when presenting a plane or 
 planes the normals of which are zone-axes. 
 
 Taking the most general case, that namely in which the plane- 
 system is referred to oblique coordinates, let the origin-line Imn 
 perpendicular to the plane 
 
 Ax + By+Cz = o 
 
 be - = 4 = -> 
 
 ua vo \vc 
 
 then the condition for this line to be perpendicular to the plane 
 is that it shall be so to every origin-line I'mn lying in the plane. 
 But for the angle between the lines Imn and I'm'n, 
 cos 6 = II' + mm +nn' + (mri + m'n) cos f 
 
 + (nf+ n'T] cos r? + (lm f + I'm] cos f, 
 
 f, TI, f being the axial angles YZ, ZX, and XY\ and this ex- 
 pression equals zero, when the lines are perpendicular. 
 The condition for the line I'mn to lie in the plane 
 
 Ax + By + Cz = o 
 is At +Bm'+Cn'= o. 
 
 We have therefore 
 
 /+ ncosrj + m cos = kA, 
 m + I cos f + n cos f = kB, 
 cos f + / cos 77 = k C ; 
 
 where 
 
 Al+Bm+Cn 
 
9 6 Conditions of coincidence. 
 
 By substitution, the equation to the plane to which the line 
 x y z 
 ua v$ we 
 is perpendicular is 
 
 (/ + n cos if] -f m cos f ) AT -f (m + / cos f + cos f ) jy 
 
 H- (tt + w cos f + / cos ??) z o ; 
 that is to say, is 
 
 cos f +w<: cosf )jr 
 
 ;3 = o. 
 
 Comparing this expression for the plane in question with the 
 equation to an origin-plane parallel to the plane (efg), i.e. with 
 
 !*+/, + 0, 
 
 where 
 
 are the inverse ratios of the intercepts on the axes of the plane 
 (efg] ; we find for the case where the zone-axis [uvw] is normal 
 to the plane 
 
 cos v< + u<z cos 
 
 . 
 
 _ w c + v b cos f + u a cos T/ 
 
 = F I 
 
 Hence if we assume any rational values for efg, by these equa- 
 tions we may determine the symbol of a zone-plane [uvw] parallel 
 to the face (efg), but only in the case of u v w being also rational 
 will [uvw] be a zone-plane of the system. 
 
CHAPTER V. 
 
 ON THE VARIETIES OF SYMMETRY POSSIBLE IN A 
 CRYSTALLOID SYSTEM OF PLANES. 
 
 SECTION I. Application of the principles of Geometrical 
 Symmetry to crystals and crystalloid plane-systems. 
 
 78. Symmetry in Nature consists in the rhythmical recurrence 
 of a morphological element, that is to say, the repetition of the 
 element in accordance with a law of arrangement. 
 
 In animal and vegetable organisms the conditions of growth 
 appear never, or rarely, to be compatible with that geometrical 
 exactitude in the distribution of morphological features which is 
 requisite for their treatment by a geometrical science. 
 
 Crystallography, which treats of the morphology of inorganic 
 nature, offers, on the other hand, in the persistence of the angular 
 inclinations of the corresponding faces of crystals the means of 
 bringing the subject of their symmetry within the domain of 
 Geometry. 
 
 79. Geometrical symmetry of plane figures. In Geometry a 
 plane figure is said to be symme- 
 trically divided by a straight line (as 
 
 a 'line of symmetry') when a perpen- 
 dicular falling on this line from each -^ 
 point in the figure meets at an equal 
 distance beyond the line a point 
 corresponding to the first point. 
 
 If either of the halves into which 
 the line of symmetry .S divides the figure be reverted round that 
 line of symmetry, as an axis of revolution, through an angle of 
 1 80, it falls into congruence with the other half. 
 
 H 
 
 
 
Symmetry plane figures. 
 
 A plane figure may also be symmetrical to a point as a ' centre 
 of symmetry.' This point will then bisect all the lines traversing 
 it that join the several points of the figure with points corresponding 
 to them. Any one such line divides the figure into two halves 
 which become directly congruent by the 
 revolution of the figure in its own plane 
 through an angle of 180 round the centre 
 of symmetry. 
 
 A plane figure may furthermore be sym- 
 metrical with regard to a point within it 
 as a pole of symmetry when by successive 
 revolutions in its own plane through an 
 
 277 
 
 angular distance of round that point the 
 
 figure in each new position is congruent 
 with itself as seen in its original position. 
 
 If n = 2 the plane figure is symmetrical to a centre, and where 
 n = 2, or = 3, 4, or 6, the symmetry may be defined as being 
 diagonal, trigonal, tetragonal, or hexagonal. 
 
 And a plane figure may be simultaneously symmetrical to two 
 or more lines of symmetry and to a pole of symmetry. 
 
 DEF. A face of a crystal or any other plane surface or figure 
 symmetrical to one line will be said to be euthysymmetrically 
 divided by that line, as by S in Fig. 35 ; where it is symmetrical 
 
 Fig. 35- 
 
 to two lines perpendicular to each other it will be said to be 
 or tho symmetric ally divided by these lines. 
 
 An isosceles triangle, a deltoid, a symmetrical (as distinguished 
 from a regular) pentagon, Fig. 35, are euthy symmetrical figures ; 
 
Symmetry solid figures. 
 
 99 
 
 Fig. 36. 
 
 and a rhomb, Fig. 36, is orthosymmetrical to its diagonals, as a 
 rectangle is to diameters parallel to its sides ; and so the polygon 
 in Fig. 36 is also orthosymmetrical to the lines -S* and 2. 
 
 80. Symmetry of solid figures. In an analogous manner a 
 solid figure may be symmetrical to 
 one or to several planes of symmetry, 
 to an axis of symmetry, or to a 
 centre of symmetry ; or simultane- 
 ously to several of these. 
 
 It is so to a plane of symmetry 
 when corresponding points equi- 
 distant from the plane would lie on 
 any line drawn perpendicularly to 
 the plane. Where the solid figure 
 
 presents symmetry to only a single plane (and not to a centre 
 also) the corresponding portions of its surface cannot be brought 
 by reversion into congruence. They are to each other as either 
 would be to its own image if seen reflected by the plane of sym- 
 metry as by a mirror. 
 
 DEF. Such a correspondence of form will be termed antistrophe, 
 and such figures will be said to be antistrophic to each other. 
 
 A solid figure is symmetrical to an axis when every radius vector 
 moving in a plane perpendicular to the axis and meeting a point 
 of the figure would also meet corresponding points at the same 
 distances from the axis at each revolution through an arc-angle 
 
 277 
 
 of 
 
 n 
 
 The aspect of such a solid figure will not therefore be changed 
 by a revolution of the solid round this axis through the angle , 
 
 and any portion of its surface so revolving will move into a 
 position in which it will be congruent with another portion of the 
 surface entirely corresponding to it. 
 
 DEF. Congruence of this kind will be termed metastrophc, and 
 such corresponding parts will be said to be metastrophic to each 
 other. 
 
 DEF. Where n=2, or 3, or 4, or 6, the axis is one of diagonal, 
 of trigonal, of tetragonal, or hexagonal symmetry. 
 
 H 2, 
 
ioo Crystalloid Symmetry. 
 
 A solid figure may likewise be symmetrical to a point, or centre 
 of symmetry. Corresponding points will lie equidistant from this 
 centre on straight lines traversing it. 
 
 DEF. Such a figure will be termed centro symmetrical. 
 
 Other varieties of symmetry may be imagined : such, for in- 
 stance, as a spiral symmetry resulting from symmetry round an 
 axis along as well as around which the radius vector would be 
 supposed to move at a given relative rate of motion. 
 
 Application of principles of symmetry to crystals. 
 
 81. Equipoised polyhedra. It would obviously be futile to 
 attempt to apply these geometrical definitions directly to the 
 interpretation of such symmetry as crystals may exhibit. Crystals, 
 in fact, only in exceptional cases present any such complete 
 geometrical symmetry, since the magnitudes and the distances of 
 their faces from any point or planes within the crystal follow 
 no law. 
 
 But in the distribution on the Sphere of Projection of the Poles 
 (which represent the relative directions in space) of the faces of a 
 crystal, as also in the relative directions of origin-planes drawn 
 parallel to the faces, we have the means of establishing that a 
 crystal is, in a sense which is not the less real because somewhat 
 more elastic than the strictly geometrical sense* a symmetrical 
 polyhedron. The poles of its faces will in fact be found to be 
 symmetrically distributed on- the sphere, and inasmuch as any 
 planes parallel to the actual faces of the crystal would equally 
 represent those faces in a crystallographic sense, we may conceive 
 of a polyhedron so constructed that while its faces were all 
 parallel to those of the crystal, such of these as correspond sym- 
 metrically, that is to say which represent the same repeated face 
 of the crystal, should be taken at equal distances from the point 
 within the crystal which is the centre of the Sphere of Projection. 
 An imaginary polyhedron of such a nature is said to be in 
 equipoise. 
 
 The outline figures in which crystals are commonly represented 
 by the projection of their edges are drawn in equipoise; though 
 
Symmetry of a ''form! 101 
 
 faces of different kinds are generally projected with differently pro- 
 portioned magnitudes. 
 
 Before discussing the results established by observations made 
 upon actual crystals it will be well to determine what different 
 sorts of symmetry it is possible that a crystalloid polyhedron may 
 present, and the character of the limitations that the law of 
 Rationality of Indices must impose on the number of such 
 varieties of symmetry ; the expression ' symmetry ' being of course 
 understood in the sense just defined. 
 
 82. Crystalloid symmetry. Two planes, whether referred as 
 origin-planes to the centre or taken as faces of a polyhedron, 
 will be symmetrical in respect to or 'on' a third (which it will be 
 preferable to consider as an origin-plane) when they are tautozonal 
 with the third plane and the dihedral angle which forms their 
 edge is bisected by it. Their poles lying on the same side of 
 the plane of symmetry will be equidistant from either one of its 
 poles, those on opposite sides of it will be so from opposite poles 
 of that plane. 
 
 Evidently the two faces are antistrophically symmetrical: and 
 their edge lies in the plane of symmetry. 
 
 DEF. Two poles or planes thus symmetrically disposed in 
 regard to an origin-plane will be termed homologous to each other 
 in respect to that plane of symmetry ; and this term will be extended 
 to embrace all the planes or poles or other features which in a 
 system of planes correspond to each other as being symmetrically 
 repeated, whether in respect to one or more planes, or to an axis or 
 axes, or to a centre of symmetry. 
 
 DEF. A plane or its pole will be termed an independent plane or 
 pole when the pole is not the pole of a plane of symmetry, and 
 does not lie on the great circle in which a plane of symmetry 
 intersects the sphere of projection. 
 
 DEF. In a crystalloid system a group of homologous planes 
 will be comprised under the term ' a form' The general symbol 
 of a form will be represented by the indices of one of the planes of 
 the form enclosed in brackets, e.g. {hkl}, {no}, &c. 
 
 DEF. The great circle in which a plane of symmetry intersects 
 the Sphere of Projection will be occasionally termed a circle of 
 
IO2 
 
 Symmetral axes. 
 
 symmetry, and the axial point of an Axis of Symmetry a pole of 
 symmetry ; such an axis will be frequently designated by the 
 symbol of or by a letter indicating a face-pole in which it will be 
 found to meet the Sphere of Projection. 
 
 The adjective form symmetral will further be employed as a 
 convenient expression as applied to a plane, an axis, &c. of 
 symmetry. And we may speak of an axis or pole of symmetry 
 being diagonally symmetral, or diasymmetral, or orthosymmetral, 
 tri-, tetra-, or hexa-symmetral, according as the face or zone of 
 which it is the pole or the axis presents diagonal, orthosymmetrical, 
 trigonal, tetragonal, or hexagonal symmetry. 
 
 83. Di-n-gonal symmetry. But the most general and frequent 
 kind of symmetry that we have to deal with is that in which 
 an axis of a zone is an axis of symmetry by reason of the 
 symmetral character of a certain number of the planes belonging 
 to the zone. In such cases the recurrences of each feature take 
 place in pairs from repetition over each plane of symmetry; but 
 so that adjacent features are antistrophically, alternate features 
 metastrophically, repeated. It thus will happen that the zone- 
 axis of n tautozonal planes of symmetry will only be an axis of 
 ;2-gonal symmetry, notwithstanding that any single feature occurs 
 2 x n times. For instance, to take illustrations from plane-figures, 
 those known as the symmetrical hexagon, symmetrical octagon, or 
 symmetrical dodecagon (as distinguished from the regular figures), 
 and of which indeed all the sides but only the alternate angles are 
 similar, present as do the polygons in Fig. 37 only trigonal, tetra- 
 
 37- 
 
 gonal, and hexagonal symmetry when considered solely in the 
 aspect of symmetry round the centre of the figure as a pole of 
 symmetry, while the sides are repeated in three, four, or six pairs. 
 
Similarity of edges. 103 
 
 It will be in fact observed that the same distribution of features 
 in n pairs round an n symmetral axis, resulting from the influence 
 of n planes of symmetry, is virtually extant, even in such figures 
 as the equilateral triangle, regular hexagon, &c. ; since the ad- 
 jacent halves of their edges or angles as well as of the faces 
 themselves may be considered as due to repetition over three, 
 six, &c. planes of symmetry. 
 
 Where homologous elements of form are thus coupled or re- 
 peated in pairs, the symmetry may be designated as di-trigonal, 
 di-tetragonal, and di-hexagonal in character. 
 
 It is obvious that a polyhedron will equally, with a plane figure, 
 present such 2 -gonal symmetry in the distribution of its morpho- 
 logical features around an axis where that axis is the zone-line in 
 which n planes of symmetry intersect. 
 
 It is to be observed that, for reasons which will be developed 
 when the principles of crystalloid symmetry are further discussed, 
 n cannot exceed 6 and can never be 5, pentagonal symmetry 
 being precluded by those principles. The orthosymmetry which 
 is the result of two supplemental planes of symmetry perpendicular 
 to each other, to the exclusion of any other plane of symmetry in 
 the zone, is distinguished from merely diagonal symmetry by this 
 duplication of the recurring features. 
 
 84. Morphological features of crystals. The morphological 
 features of a crystalloid polyhedron are recognised in its edges, 
 its faces, and its solid angles or (its coigns or) quoins. Two 
 edges may be accounted in a geometrical sense as similar when 
 their dihedral angles are the same, and when further, for any 
 plane in the zone of or otherwise inclined on the one edge, a 
 plane is also possible in the zone of or inclined on the other 
 edge in such manner that the angular relations of this second 
 plane to the planes forming the latter edge are the same as those 
 of corresponding planes in the case of the former edge. 
 
 Since a face of a polyhedron is bounded by edges, two crystal- 
 loid faces are geometrically similar when they are bounded by the 
 same number of actual or possible edges severally similar and 
 inclined at the same plane angles, each to each. Such similar 
 planes distributed as they are on the surface of an equipoised 
 
IO4 Similarity of faces ; of quoins. 
 
 polyhedron, may be considered as plane figures irrespective of 
 their relative position in space, capable of being brought into 
 congruence either by direct superposition or by superposition 
 after retroversion : i.e. they are either metastrophic by the im- 
 agined congruence of the inner surface of the one with the outer 
 surface of the other ; or they are mutually antistrophically con- 
 gruent by the supposed contact of their outer or inner surfaces. 
 
 The faces of a form will, by the fact of their being mutually 
 homologous, fulfil these conditions of similarity. 
 
 In a crystalloid plane, edges that are symmetrically repeated will 
 be thus far similar, and such directions on the plane will be 
 similar as are equally inclined on similar edges. 
 
 85. Similar faces. In order that two faces on a crystal may 
 be crystallographically similar, their physical properties as well as 
 their geometrical characters will have to be identical in respect to 
 similar directions on the two faces. These must also be identical 
 along similar directions on the same face. And this will also be 
 true in respect of any sectional planes carried in similar directions 
 through a crystal; the normals of such arbitrary planes being 
 similarly inclined on similar edges. 
 
 N.B. We shall have hereafter to discriminate between this 
 geometrical similarity of features in the case of a crystalloid 
 polyhedron and the crystallographic similarity involving identity 
 of physical characters also, in that of a crystal. 
 
 86. Quoins : their symmetry. Similar quoins. Quoins are of 
 different kinds. Thus the faces which meet in a quoin may 
 have all their poles symmetrically distributed on the same small 
 circle of the sphere. In such a case the diameter of the sphere 
 passing through the pole of the small circle will, if continued, 
 meet the vertex of the quoin, and will be an axis of symmetry 
 to it. 
 
 And such a quoin may be formed of groups of faces of 
 which the poles are symmetrically disposed on different small 
 circles having a common pole. 
 
 Such quoins may evidently present the various kinds of crystal- 
 loid symmetry that are possible round an axis of symmetry. 
 
 And on the other hand there may be quoins that are sym- 
 
Forms with parallel faces. 105 
 
 metrical to one plane passing through their vertex, or again that 
 are devoid of any symmetry at all. 
 
 Two quoins of a crystalloid polyhedron will be similar if com- 
 posed of the same number of actual or possible edges severally 
 similar and inclined at the same angles, each on each. 
 
 87. Octants as quoins. An octant formed by three axial .planes 
 is a trihedral quoin transferred to the origin. Any two of its 
 edges, i.e. of the axes of the system, are therefore similar when 
 they are equally inclined on the third axis, and when the 
 intercepts of a plane, actual or possible, upon these axes are 
 the same. 
 
 Of the octants into which the axial planes divide space any two 
 will be similar when their respective elements are the same ; that 
 is to say when the corresponding axes are inclined at the same 
 angle in both octants. 
 
 88. Faces in parallel pairs. Since every origin-plane has 
 two poles lying at the opposite extremities of its normal on 
 the sphere of projection, while the indices in the symbol of 
 such a plane differ only in having opposite signs; we must 
 consider the two faces on either side of and parallel to the 
 origin-plane as equally representing that plane in the polyhedral 
 system in the absence of any special geometrical condition to 
 the contrary. 
 
 DEF. A polyhedron or a ' form ' of a polyhedron presenting 
 both the planes thus parallel to but on opposite sides of each of 
 its origin-planes will be termed a diplohedral form or polyhedron. 
 Where for each of its origin-planes the system or a form be- 
 longing to it has only one plane extant parallel to the origin-plane, 
 the system or form will be termed haplohedral. 
 
 It is evident that a centrosymmetrical system of planes must be 
 diplohedral) and vice versa. 
 
 89. Planes of symmetry in a zone presumably supplementary. 
 The consequence of the anharmonic ratios of four tautozonal 
 planes in a crystalloid zone becoming harmonic in the case 
 where two of the planes are symmetrical to a third, has been 
 alluded to in article 63. 
 
 It was there shown that a plane of symmetry in a zone implies a 
 
1 06 Condition for a plane of symmetry. 
 
 second or supplementary plane perpendicular to the first, which is 
 also a possible plane of the system, and like the first plane invests 
 with rationality the symbol of any plane symmetrical in respect 
 to it with some actual plane belonging to the zone. We have at 
 present only to deal with this general statement regarding the 
 symmetry of a zone to one of its planes, or to two perpendicular 
 planes in it. 
 
 If in a zone the planes are diplohedral, the plane perpendicular 
 to a plane of symmetry will clearly be actually symmetral : 
 in a haplohedral zone one of these planes is actually, while the 
 plane supplementary to it is only potentially symmetral, the zone 
 being as it were incomplete and its symmetry so far in abeyance. 
 
 SECTION II. Conditions for a crystalloid system of planes 
 to be symmetrical to one of its planes. 
 
 90. A plane of symmetry is parallel to a possible face, (i) If, in 
 a crystalloid system, an origin-plane S be at once a zone-plane and 
 a plane of the system, it is a possible plane of symmetry to the 
 system. 
 
 Let s, Fig. 38, be the pole of the plane S, p be the pole of any 
 plane P of the system. Then the zone-circles [/tf] and [S] will 
 intersect in m a pole of a possible plane 
 of the system. 
 
 If now mp'=. mp, the poles s and m 
 harmonically divide the zone-circle [//'], 
 and p' is a possible plane of the system. 
 Thus, for any pole existing in the system, 
 another pole symmetrical with it in re- 
 spect to the plane S is a possible pole of 
 the system. The entire plane-system 
 Fig. 38. must therefore be potentially symmetrical 
 
 to S. 
 
 (ii) Conversely, any symmetry plane of the system must be 
 at once a possible zone-plane and possible plane of the system. 
 
 Let//)' and q<f be two pairs of poles symmetrical to a plane S, 
 Fig. 38. Then the zone-circles [//'], [?/] will intersect in a 
 
Conditions for monosymmetry. i o 7 
 
 possible pole of the system. But they intersect in s the pole of 
 the great circle in which S intersects the sphere. is therefore 
 a possible plane of the system. And if the arcs />/>', qq are bisected 
 in m and n, the zone-circles [//'] an d \$f\ are harmonically divided 
 by s and m and s and n respectively : and therefore m and n are 
 possible poles of the system. Therefore \rtin\ is a possible zone 
 of the system. But m and n lie in the plane S which must there- 
 fore be at once, potentially, a zone-plane and a plane of the 
 system. 
 
 The necessary conditions therefore for a crystalloid system of 
 planes to be symmetrical to a plane are, that this plane shall be 
 at once a plane of the system and a zone-plane; or, they 
 may be expressed in the form, that two planes of the system 
 are perpendicular to a third plane of it. 
 
 91. Restrictions imposed on a crystalloid system by its being 
 symmetrical to one only of its planes. In the general case con- 
 sidered in the last article, if the arc mn = ^, the origin-planes 
 
 2i 
 
 M and N, of which m and n are the poles, will be perpendicular 
 to each other and will become potentially planes of symmetry 
 orthosymmetrically dividing the zone \S~\ ; but also each becomes 
 a zone-plane, M of a zone [PP'], N of [(?(/]. They are 
 therefore potentially planes of symmetry to the entire polyhedral 
 system : a condition inconsistent with the uniqueness of S as a 
 plane of symmetry to that system. 
 
 Evidently therefore, if S is to be the only plane of symmetry of 
 the system, no two planes belonging to the zone [S] can be 
 perpendicular to each other. 
 
 Nor, it may be added here, can any two of the planes belonging 
 to the "zone, the zone-plane of which is the primarily assumed 
 plane of symmetry, be inclined on a third plane at any other 
 crystallometric angle : for it will hereafter become apparent that if 
 such were the case, the zone lying in the plane of symmetry would 
 itself present trigonal, tetragonal, or hexagonal symmetry, ac- 
 cording to the value of the particular crystallometric angle between 
 the planes. And the symmetry of the whole plane- system will be 
 found to follow that of the zone in question, so that we should 
 
io8 
 
 A mono symmetrical system. 
 
 pass from the symmetry of the plane-system to a single plane on 
 to a kind of symmetry of a much higher order. 
 
 92. Axial-system where there is one plane of symmetry. Cha- 
 racter of a form. The two conditions that a crystalloid system 
 symmetrical to one of its planes must satisfy are most simply 
 embodied in an axial system wherein the plane of symmetry S, 
 and any pair of planes such as M and N y Fig. 39, perpendicular 
 to it but inclined on each other at an angle generally greater than 
 60 and less than 90, are taken as the axial planes, so that the axes 
 
 Zooi 
 
 g. 39- 
 
 are the normal of S taken for the axis Y and the intersections of 
 the planes M and N with the plane which are taken for the axes 
 Z and X. And, whereas these two planes are necessarily oblique in 
 their inclination on each other, the obtuse angle /3 which represents 
 their inclination (and cannot be of crystallometric value) is taken 
 for the positive angle ; that namely of X O Z and X Z ; the 
 supplemental angle /3'( TT 77) being taken for that of X O Z 
 and X OZ. The symbol of the plane of symmetry is then oio, 
 and its normal, the axis F, will coincide with the axis of the zone 
 containing the poles 100 and ooi that is to say, with the zone- 
 axis [oio]. 
 
Conditions for several planes of symmetry. 109 
 
 The planes M and N will have for their symbols H- 100 and 
 -hooi, and their poles will lie on the great circle of symmetry S, 
 distant from each other by an arc ft. 
 
 The parametral ratios may be provided either by a single plane 
 intersecting with all three axes, or by a plane in each of two out of 
 the three zones \MS\, \NS\, or [MN]. 
 
 The axial-system is thus represented by the expression 
 
 f = 90= C, j] > 9o< 120, a : b : c, 
 
 wherein two out of the five elements are fixed and the remaining 
 three are unfettered by conditions. 
 
 The first, fourth, fifth, and eighth octants, viz. 
 
 XFZ, xYz, x~r~z, ~XFZ~ 
 
 are similar : so are the remaining four octants adjacent to them. 
 
 The poles of an independent form {hkl} will lie on a great 
 circle passing through the pole oio and the diplohedral form 
 will have four faces. If the pole in which this great circle con- 
 taining the poles of the form intersects with the zone-circle [o i o] 
 lie between ooi and 100, the symbol is {hkl}, and the four faces 
 are h k /, h k /, h k /, li k T; if it intersects with the zone-circle [oio] 
 in_ a pole lying between ooi and Too, the symbol of the form is 
 {h k 1} and its faces are h k /, h ~k /, h k 7, h k I. A pole lying on the 
 zone-circle of symmetry \S\ will belong to a form {hoi} or 
 to a form {h o /} which will comprise only two parallel faces. 
 
 A form {hko} will have four faces the poles of which lie on the 
 great circle [100, oio]. The four poles of a form {o//} will lie 
 in the great circle [oio, ooi]. The normal to the plane of sym- 
 metry oio is an axis of diagonal symmetry when the system of 
 planes is diplohedral. 
 
 SECTION III.- Conditions involved in a crystalloid poly- 
 hedron being symmetrical to more than one of 
 its planes. 
 
 93. Where a plane-system is symmetrical to more than one 
 plane it is obvious that not only must each pole or plane or other 
 actual morphological feature, but that also each zone-circle and 
 each plane of symmetry, must be virtually as such repeated over 
 
no Definitions and Nomenclature. 
 
 each and every plane of symmetry of the system. Where one 
 plane of symmetry S^ is so repeated over a second plane of 
 symmetry 2 the third plane of symmetry S z is originated ; and the 
 distribution of all the features of the system, as illustrated for 
 instance by the poles belonging to a form, will necessarily 
 be identical when viewed as circumjacent to the two planes of 
 symmetry S. 
 
 But the distribution of these features of the system in respect to 
 either of the groups of planes will not be the same, nor will it 
 ever be so, in respect to two adjacent planes of symmetry. 
 
 DEF. Where the situation of a pole or where the distribution of 
 the poles of a form is different when considered as circumjacent to 
 one or to another of two planes of symmetry, these two planes are 
 said to present unconformable symmetry ; where the situations of 
 the poles fall into congruence by such a revolution of the system 
 round the zone-axis of the two planes as brings one of the planes 
 of symmetry into coincidence with the position previously occupied 
 by the other, the two planes of symmetry will be termed similar, 
 or of conformable, or also of congruent symmetry. 
 
 Further, it may happen in certain cases that the distribution of 
 the features of a crystal may be unconformable in respect to the 
 two halves into which each of three planes of symmetry is divided 
 by the common zone-axis ; so that for instance three similar zone- 
 circles, as in Fig. 45, article 115, may present conformability in 
 the symmetry due to alternate hemizones. Such zone-circles or 
 planes of symmetry will be- termed hemicy die ally conformable in 
 the symmetry they govern. 
 
 94. Nomenclature for planes of symmetry. DEF. The plane or 
 planes to which a plane-system is symmetrical will hereafter be 
 called its systematic planes; where it is symmetrical to different 
 planes or groups of planes not conformable in their symmetry, 
 these several planes or groups of planes are designated as proto- 
 systematic, deutero-systematic, and trito-systcmatic planes or groups 
 of planes : and in this treatise these designations will correspond 
 to the letters S, 2, and C by which the different planes and groups 
 of planes of symmetry are denoted. 
 
 95. Planes of symmetry are inclined at crystallomelric angles. 
 
Symmetral planes are isogonal. 1 1 1 
 
 It follows from what has been said regarding the mutually re- 
 petitive character of planes of symmetry, that two planes S 1 and 
 2 1? reflected each by the other in planes S 2 and 2 2 , &c., form a 
 zone of planes of symmetry in which the planes alternate with 
 the planes 2, and the inclinations of the planes on each other are 
 equal. 
 
 This equality in the angles at which successive planes of sym- 
 metry are inclined to each other, leads directly to the necessity 
 of these angles having only crystallometric values; and indeed 
 having only a single crystallometric value in the case of planes 
 of symmetry lying in any particular zone. And it is further evident 
 that a zone cannot be symmetrical simultaneously to two inde- 
 pendent sets of planes. 
 
 Were the angles in question not commensurate with TT, these 
 angles would continue to recur in the zone through each successive 
 revolution round the zone-axis, while also in each such revolution 
 new series of planes of symmetry inclined on each other at new 
 angles of inclination would present themselves until the number of 
 such planes would become indefinitely great, and the symmetrical 
 character of the zone would entirely disappear. 
 
 The necessity will thus become apparent for the limitation 
 which was imposed in article 65, on the character of crystallo- 
 metric angles, whereby they were confined to such as were 
 
 commensurate with TT and not greater than - 
 
 96. And this necessity for the angles between planes of 
 symmetry being crystallometric is no less imperative in the case of 
 a plane system than it is in the case of the symmetry of a zone ; 
 so that we have to recognise that the condition, necessary and 
 sufficient for a single plane to divide the plane system sym- 
 metrically namely, that it shall be simultaneously a zone-plane 
 and parallel to a face is no longer sufficient in the case where a 
 second plane is, simultaneously with the former plane, a plane of 
 symmetry ; but that this has to be supplemented by the condition 
 that when a certain plane is established as a plane of symmetry to 
 a zone or plane-system, any other plane or planes of symmetry 
 can only be inclined on it at crystallometric angles. 
 
H2 Symmetry in the case of crystals. 
 
 97. Relations between the degree of symmetry of a crystalloid 
 plane-system and the axial systems to which it may be referred. 
 The equations P, obtained in article 77, implicitly contain the 
 conditions under which a crystalloid plane-system may present 
 symmetry to one or more of its planes. 
 
 It has been proved that a plane of symmetry must be parallel to 
 a face of the system, and that when there are more planes of 
 symmetry than one these must be inclined on each other at 
 crystallometric angles. The equations P give the conditions ne- 
 cessary for a zone-axis to be perpendicular to a face, and it is 
 obvious that these equations will become greatly simplified in 
 cases where the plane-system can be referred to axes that are 
 rectangular and where two or all three of the parameters are 
 equal. 
 
 On the other hand it is evident that where there is a plane of 
 symmetry the plane system is capable of being referred, as in art. 
 92, to axes whereof one is perpendicular to the other two. 
 
 We might proceed to enquire what would be the conditions 
 under the different varieties of axial systems to which a crystalloid 
 polyhedron might be referred, in order that the normal of a face 
 and a zone-axis may be coincident in direction. 
 
 But hereafter, when the changes of volume accompanying changes 
 of temperature in a crystal are discussed, it will be shewn that, 
 in a crystal as distinguished from a crystalloid polyhedron, the 
 parametral ratios can never be permanently rational except where 
 one or both of them is unity; and that the cosines of the axial 
 angles are equally only capable of being momentarily rational 
 where the axes containing them are not coincident with the 
 normals of actual or possible planes of the system, or, which is 
 an equivalent statement, do not lie in an isogonal zone. 
 
 98. It will be sufficient, then, for our purpose, to consider here 
 the cases arising under each kind of axial system in which zone- 
 planes will be parallel to faces of the system ; i. e. the conditions 
 under which the substitution of the designated values for the axial 
 elements in the equation P, article 77, gives rational indices for 
 the coincident normals and zone-axes. And in fact it will be 
 seen that with the selected axial elements this resolves itself into 
 
Crystals with rectangular axes. 113 
 
 determining all the cases in which the equations P will reduce to 
 the form u v w 
 
 i.e. to the cases in which the zone and the face to which it is 
 parallel have the same indices in their symbols. 
 
 I. Clearly one series of values by which this condition is satisiied 
 will be that in which the plane-system is capable of being referred 
 to an axial-system in which the axial-angles are 
 
 = ,= = 90 
 and the parameters are equal, i. e. 
 
 a = b = c. 
 
 Then 5 = v = HL 
 
 ' f g 
 
 is true_/0r the symbol of every plane and of a possible zone-line normal 
 to if, in the system. 
 
 II. If, the axes being as before rectangular, two only of the 
 parameters are equal, for instance 
 
 a = b $ c, 
 
 the equations P become 
 
 u _ v _ r 2 w 
 
 ~7"" ; 
 
 where the third expression must be taken as irrational (since ^ 
 
 can only be temporarily rational), unless, 
 
 (a) w = o and g o, and therefore the condition holds good 
 for all the normals in the zone [ooi] for which efo is the symbol 
 so long as e andy are integers or one only of them o. It holds 
 therefore for the normals of the faces 100 and oio : and 
 
 (b] it is true also for the normal of the face ooi, since it is true 
 for the case where 
 
 u = v = o = <? = /5 
 
 simultaneously with w and g being finite ; as then the equations 
 become ^! u _ f! v _ ^ 
 
 <**-<* f~ g 
 and are satisfied if w = g = o. 
 
 I 
 
H4 Case of two oblique axes. 
 
 III. If again, with rectangular axes, the parameters be all 
 unequal, the equations P would become 
 
 u 70 v a w 
 
 a? = o* = <r a : 
 
 writing the equations as 
 
 w 
 
 (#) the second and third expressions are irrational (since ^ 
 
 c 2 
 and must be treated as if they were permanently so), if v,/", w 
 
 and g are not all zero, while u and t remain finite. The condition 
 
 holds, then, for the zone-axis [100], which is also the axis X, and 
 
 coincides with the normal of the axial plane. 
 
 And similarly (3) for the zone-axis [oio] or axis F, 
 
 and (c) for the zone-axis ooi which is the axis Z. 
 
 Hence in this case the intersections of the axial planes are the 
 
 only cases of coincidence in zone-line and normal, and in the 
 
 symbols of these. 
 
 IV. Assuming the parameters still to be unequal but only two 
 of the axial-angles to be right-angles, i. e. 
 
 a ~ b ^ c and f = = 90, 
 
 77 being an angle greater than 90, the cosine of which may be 
 treated as irrational, the equations become 
 
 #U <TWCOS?7 <5v <TW <2UCOS7/ 
 
 _ - = 7.=- 
 
 a b c 
 
 and can only be satisfied by the first and third ratios which contain 
 the irrational cosine becoming indeterminate, in which case 
 u = w = o simultaneously with e = g = o and v and f remain 
 finite. 
 
 Under the assumed conditions therefore the only zone-axis that 
 is also a face-normal is that in which the indices for the X and Z 
 axes are zero ; namely, where the symbol is [oio]. 
 
 99. V. In the case where only a single axial angle can be a 
 right angle and the parameters are unequal it will be found that no 
 zone-axis can be a face-normal of the system. It remains then to 
 
Cases of three oblique axes. 115 
 
 consider the cases in which all the axial angles are oblique. And 
 here we have first that in which these angles are equal as well as 
 the parameters ; i. e. 
 
 a = b = c ; cos f = cos rj = cos = ^ 
 
 Here, as in case IV, the angle ?] not being a permanently per- 
 sistent angle the value of -^ may be treated as irrational. Then the 
 equations become 
 
 and 
 
 and u + v + w = (e +f+g). 
 
 Subtracting the last from each of the three previous equations in 
 turn, 
 
 where 
 
 TT U V W 
 
 Hence 
 
 2)-^ 7(8+2)-^ ^(6+2)-- S 1 
 
 Since - is rational, let = n, and = n, 
 
 v v w 
 
 Similarly, (8 + 2) (>-/) = (n-i)S. 
 
 So long as 6 is irrational this can only be true provided either 
 that 
 
 nfe = o, ng-f= o, and 6" = o = t+/+g t 
 
 I 2 
 
1 1 6 Res^d^lary cases. 
 
 U V "WT 
 
 that is, -== , where e +/+ g = o ; 
 
 * / g 
 
 or that / * = o, ngf o, and n i = o = w' i ; 
 that is, u = v = w and * ==./'= " 
 
 There are therefore two cases that here arise in which the 
 coincidence of face-normals and zone-axes is possible ; the one is 
 the case of all normals of faces the indices of which fulfil the con- 
 dition e +f+g = o, i. e. for the normal of every face in the zone 
 [111]; and the other case is that of this zone itself, the axis of 
 which is the normal of the face (in). 
 
 100. VI. It will have been noticed that the cases in which 
 the coincidences under consideration are the more numerous are 
 those in which the greater number of the axial elements are fixed 
 in their values, and that in case IV, where only a single zone-plane 
 was parallel to a face of the system, two of the axial elements were 
 fixed. 
 
 And all remaining cases that may be conceived will be found, 
 where they present any zone-line coincident with a plane-normal, 
 to resolve themselves into one or other of those already discussed. 
 
 Thus, for example, if we take 
 
 we have to eliminate the cosines of f and f from the equations P, 
 
 and we get 
 
 u wcosrj &* v w u cos 77 
 
 _(U + W)(I-COST?) 
 
 u + w e+g , a* , i x 
 
 or M = , where M = -jol* *) 
 
 v / ^ 5 
 
 and is irrational ; and the equation is satisfied if 
 
 v = o, /=o, 
 
 and if u + w -o, *-(- = o, 
 
 i.e. if w= u, g e. 
 
 In the case supposed then there will occur at least one case of 
 a coincident symbol for a zone-line and a normal, that, namely, of 
 the normal to the plane (*o^) or lol. But this normal must of 
 
The kinds of crystalloid symmetry. 1 1 7 
 
 necessity bisect the arc Too,ooi, since the parameters a and c are 
 equal; and the normal 101 will bisect the supplementary arc 
 100,001. And the zone [oio] is thus symmetrical to the two 
 perpendicular planes 101 and lol : and, since the axis of that 
 zone is also perpendicular to the normals 100 and ooi, and 
 therefore to 101 and roT, the case in question resolves itself into 
 one presenting identical conditions with those discussed under 
 case III of this article. 
 
 The assumption of irrationality of the parameters and of the 
 cosines of the axial angles, on which the reasoning in articles 
 97, 98 and 99 is founded, involves statements regarding physical 
 characters which, while true of a crystal, have no place in a plane- 
 system that is simply crystalloid. 
 
 For the discussion of the principles involved in the parallelism 
 of zone-axes and normals in a system of the latter kind, see a 
 memoir by H. J. S. Smith, Savilian Professor of Geometry, in 
 the Proceedings of the London Mathematical Society, vol. viii. 
 Nos. 109 and no, an abstract of which is given in the Proceedings 
 of the Crystallogical Society, Phil. Mag. y Ser. V. vol. iv. p. 18. 
 
 101. Conditions for more than one plane of symmetry. It has 
 been seen that the conditions necessary for a single plane to 
 be a plane of symmetry to a crystalloid system, namely, that it 
 be at once a zone-plane and parallel to a face, are not sufficient 
 to impart to a second plane the character of a plane of symmetry. 
 Such a second plane, in fact, must furthermore be inclined on 
 the first plane, and therefore also on every other plane of symmetry, 
 at one of the crystallometric angles. 
 
 The method of reasoning adopted in article 90 suffices to 
 prove that every plane of symmetry to a crystalloid plane-system 
 must be also, by that fact, the plane of a zone : and it is shown in 
 article 96 that two planes of symmetry must always belong to an 
 isogonal group in the zone that contains them. 
 
 In fact, if S l be a plane of symmetry and Sj be a plane of the 
 system, Sj will be symmetrically repeated over S 1 ; and where the 
 two planes are inclined at a crystallometric angle, the zone [*S*2] 
 will be isogonal as regards the repeated planes and symmetrical to 
 each of them. And by article 63 the zone will further be sym- 
 
1 1 8 Crystals planes of abortive symmetry. 
 
 metrical to a series of planes perpendicular to the planes 6" and 2, 
 and therefore also included with them in an isogonal group 
 harmonically dividing the zone : indeed in all cases except where 
 
 < = - these supplementary planes will each fall into coincidence 
 
 o 
 with one or other of the planes *S" or 2. 
 
 It is evident that in this case each of the planes of the isogonal 
 group is at once parallel to a face and to a zone-plane. And 
 each will, in accordance with the reasoning in article 90, be 
 potentially a plane of symmetry to the entire plane-system. 
 
 That a plane inclined at other than a crystallometric angle on a 
 plane of symmetry cannot be a true plane of symmetry to a plane - 
 system may be seen by considering the class of cases that have been 
 already alluded to in articles 98. I and II a, and 99. V; in which 
 a series of tautozonal planes fulfil the primd facie conditions for 
 being planes of symmetry to the zone they lie in, while, from the 
 plane of that zone being parallel to a possible face, it might be 
 assumed that all or some of them would be planes of symmetry 
 to the entire plane-system. And yet they are not true planes of 
 symmetry either to the system or even to the zone ; for the reason 
 that they are not inclined at crystallometric angles on certain 
 planes belonging to the zone, which, being at once (like the planes 
 in question) zone-planes and parallel to faces, are on the other 
 hand unlike those planes in belonging also to an isogonal group, 
 and so being established in the position of true planes of symmetry 
 alike to the zone and to the entire plane-system. 
 
 102. Planes of abortive symmetry. The cases just alluded to, 
 in which all the planes lying in a zone may in unison with other 
 planes perpendicular to them (in accordance with article 63) 
 ortho symmetrically divide the zone, need some further consider- 
 ation. It is clear that for any plane belonging to such a zone 
 the symbol of another plane can be calculated that shall be 
 rational, provided this second plane is equally inclined with the 
 first on some third plane belonging to the zone. In fact, the 
 symbols of the three planes will, with the symbols of a plane 
 perpendicular to the last, form a harmonic ratio. Every plane of 
 the zone would in short fulfil the first condition of a possible plane 
 
Abortive symmetry. Case I. 119 
 
 of symmetry to the zone ; and by virtue of a plane perpendicular 
 to it, and of the plane of the zone itself being both planes of the 
 system, each plane of the zone would seem presumably to be a 
 plane of symmetry to the entire plane-system. 
 
 But such presumptive planes of symmetry are evidently pre- 
 cluded from really possessing symmetral characteristics by the 
 principle that if they were actual planes of symmetry they would 
 of necessity be recurrent ; whereas if they were so they would 
 mutually repeat each other in numbers without limit, since their 
 angles of mutual inclination cannot be commensurate with n. For 
 it is the essence of a plane of symmetry, actual or potential, that all 
 the morphological features of a zone or of any form be not 
 capriciously or partially, but systematically, or not at all, repeated 
 in respect to it. Where a plane may hereafter be spoken of as 
 only potentially a plane of symmetry, it will be implied that its 
 influence on a form is as it were suspended or in abeyance in 
 respect to the entire form. In the cases under consideration 
 however a zone would contain an indefinite number of planes; 
 and the faces of a form would be repeated at an ever-increasing 
 variety of angles : from a plane-system we should pass to a solid 
 with a curved surface. Symmetral characters in planes of the kind 
 under discussion cannot therefore be spoken of as being merely 
 in abeyance : they are impossible, and such planes are con- 
 sequently abortive as planes of symmetry, and that for the reason 
 that they are not inclined at crystallometric angles on certain 
 particular planes which in a crystal are naturally selected from 
 among the possible planes of the zone as true planes of morpho- 
 logical symmetry. 
 
 103. The several cases of abortive symmetry. In articles 98 
 and 99 the whole of the cases have been recounted in which 
 planes of abortive symmetry occur. Case I in that article pre- 
 sents an entire plane-system the faces of which are all such that 
 they are parallel to possible zone-planes ; and it will be seen that 
 these faces, when a system of this kind falls under consideration, 
 will divide themselves into such as are parallel to true systematic 
 planes (of symmetry) and such as cannot be parallel to such planes 
 nor be inclined on them at crystallometric angles. 
 
1 20 One zone of Case II, one of Case V. 
 
 The other cases are those of case II, article 98, and case V, 
 article 99, which however are each confined to the faces of a single 
 zone. In either case we shall find that the true planes of sym- 
 metry in the zone have a crystallometric angle less than - : such, 
 
 in fact, are the only cases not included under case I in which 
 
 we shall meet with zones of this kind. 
 
 The point under discussion is sufficiently important to receive 
 
 illustration from an example, which may be taken from the second 
 
 case in article 98. In that case 
 the plane oo i is parallel to the 
 zone-plane containing the axes 
 X and Y and also the poles 
 100 and oio which lie on the 
 normals of the planes YZ and 
 ZX ; the parameters on these 
 axes being equal. Conse- 
 quently, two planes 2 1? 2 2 , Fig. 
 40, perpendicular to each other 
 bisect the right angles between 
 the planes ZX and YZ which 
 may be designated S l and S 2 , 
 
 Fig. 40. 
 
 and these four planes are successively inclined at 45, and will be 
 recognised in a future article as the planes of symmetry for the 
 zone [ooi] and also for the whole plane-system. A pole p 1 on 
 the zone-circle [ooi] will be repeated over the plane 2 X in a pole 
 //, and this over the plane S 3 in a pole p z ; and the angle 100 
 on pi equals the angle oio on / 2 , and the arc p 1 p z is a quadrant. 
 So /j fulfils the primary condition for being a plane of symmetry 
 to the zone. In fact, if we assign a rational symbol, say 320, to 
 the pole^, then a pole q^ equidistant with 100 from p l must have 
 a rational symbol. In fact, the symbol of />/ will be 230, and of 
 p 2 will be 1230, and the ratio 
 
 is harmonic, and 
 
 100 
 
 100 
 
Types of crystalloid symmetry. 
 
 121 
 
 
 230 
 
 hko 
 
 230 
 
 IOO 
 
 320 
 
 hko 
 
 320 
 
 IOO 
 
 whence 5/6 = i2h, and the symbol of ^ is 5 12 o. 
 
 Now taking the poles 100, p lt 110,010, by the problem of four 
 planes, the arc 100,^ = 334i' = A?i and the arc^, oio = 2237 / . 
 So that if^ were the pole of P a plane of symmetry, this plane P 
 and the plane S l would be mutually repeated at angles of 
 3 3 41', while with ^ and S 2 , P would form a series of planes 
 inclined at n 19" and 56 19', while Q the plane of which q is the 
 pole would furnish repetitions with these planes at again new 
 angles. 
 
 It would be futile to pursue such repetitions into their results. 
 In a word, the zone can exhibit no symmetry other than to the 
 planes and 2. 
 
 104. It has been established in the preceding articles that in 
 order for a plane-system to be symmetrical to more than one of 
 its planes, each such plane must be parallel to a possible zone- 
 plane and inclined on each other plane of symmetry at a crystal- 
 lometric angle. It is however to be observed that where this 
 condition is fulfilled by two planes of an isogonal group in a zone 
 that are not supplementary, it is necessarily true of the remaining 
 planes belonging to that group. 
 
 We may now proceed to discuss the various kinds or types of 
 symmetry which result, in the first place from assuming different 
 crystallometric angles between planes in a zone fulfilling the above 
 conditions, and in the next place from the discussion of the 
 problem of the possible modes in which such isogonal zones are 
 capable of intersecting with each other. 
 
 CASE I. The type of symmetry where (b = - 
 
 2 
 
 105. The first case to be considered will be the simplest, that 
 namely in which two planes of symmetry are perpendicular to each 
 other. 
 
 If there be two and only two planes of symmetry and 2, 
 
122 Two Symmetral planes perpendicular. 
 
 Fig. 41, in a zone, these must be perpendicular to each other and 
 divide the zone-circle or tho symmetrically: and the zone-plane C 
 containing the normals of the two planes will be parallel to the face 
 in which their two zone-circles intersect and be itself potentially a 
 plane of symmetry. And the three zone-lines j/, o-o-', cc, which 
 are the normals as well as origin-edges of the three planes S, 2, C, 
 will, where the plane C is an actual plane of symmetry, be 
 
 axes of orthosymmetry to the 
 system. And it will further be 
 seen that the system will, as a 
 consequence of its symmetry to the 
 plane C as well as to S and 2, 
 be centrosymmetrical. 
 
 The conditions of symmetry 
 assumed in the particular type 
 under discussion preclude the 
 possibility of other than two per- 
 pendicular symmetral planes in 
 either of the zones [S2], [2C], 
 or [CS]. 
 
 And they therefore preclude the possibility of a pole of any form 
 lying on a great circle that should bisect either of the right angles 
 formed by the intersection of the planes (or zone-circles) S, 2, C, 
 and therefore also of a pole bisecting any of the quadrantal arcs 
 connecting the poles of the .planes S, 2, C. For if such a pole 
 existed, e.g. if a pole m 1 were to bisect one of the quadrants on the 
 great circle [C] between the poles of the planes S and 2, a great 
 circle [m 1 c~\ would bisect the angle 2 and would be perpendicular 
 to a second zone- circle \cm\ bisecting the supplementary right 
 angle of 2, and these two zone-circles thus intermediate to, and 
 equally inclined at 45 on 6" and 2 would each be potentially a 
 plane of symmetry to the system ; and this would superpose a fresh 
 condition to those assumed for the type of symmetry under 
 discussion; so that the plane-system would in fact belong to 
 another and a more complex type of symmetry than that to only 
 three perpendicular planes. 
 
 In the type under consideration, therefore, any independent 
 
Systematic triangle. 1 2 3 
 
 pole p must be situated asymmetrically, that is to say, at a point 
 on the sphere of projection the distances of which from the great 
 circles S2 and C must all be different ; wherefore, every plane of 
 an independent form (to which such a point p would be the pole) 
 must be inclined at different angles on the three planes of sym- 
 metry and will meet their zone-axes j/, cro-', cc at different distances 
 from the origin. 
 
 106. The systematic triangle. The necessity for this unsym- 
 metrical or eccentric position of the pole in a case where three 
 adjacent great circles of symmetry so intersect to form a spherical 
 triangle as that other great circles of symmetry are precluded 
 from invading the triangle, is not confined to the case in which 
 the triangle is quadrantal. It in fact is true for every spherical 
 triangle formed by the intersections of adjacent great circles of 
 symmetry. And it will be well therefore to make a brief digression, 
 in order to give form to this principle. 
 
 Evidently a spherical triangle of the kind in question will have 
 for each of its angles an axial-point in which a zone-axis of a zone 
 of symmetry-planes and is consequently an axis of symmetry 
 meets the sphere of projection ; while, further, each such point is 
 the pole of a possible face. 
 
 DEF. A spherical triangle formed by the intersection of ad- 
 jacent planes of symmetry with the sphere of projection, and which 
 therefore may not be intersected by any other circle of symmetry, 
 will be termed the systematic triangle for the particular type of 
 symmetry to which it belongs and which moreover it charac- 
 terises; since it is not conceivable that the surface of the sphere 
 should be symmetrically subdivided into systematic triangles of 
 more than one kind. 
 
 107. The general independent form is a scalenohedron. It is 
 obvious that one and only one pole of a form can occur in each 
 systematic triangle, and that the position of this pole in every such 
 triangle will be the same relatively to corresponding sides and 
 angles. It is also evident that the angles of a systematic triangle 
 can have only crystallometric values. 
 
 Furthermore, since from the nature of a systematic triangle no 
 two of its sides can be in a crystallographic sense homologous, 
 
1 24 General Scalenohedron. Type ^ -. 
 
 and since the edges of adjacent faces of an independent form 
 drawn in equipoise will lie in the several planes to which those 
 faces are symmetrical, the edges of each face will necessarily be 
 three in number and will all be dissimilar ; so that every face of 
 such a general independent form will have the character of a 
 scalene triangle. Such a general form therefore, in every case 
 where the type of symmetry admits of a systematic triangle, may 
 be termed a general scalenohedron of the system. 
 
 108. Systematic triangle and axial system where <\> = -. The 
 
 systematic triangle in the case last under consideration, that namely 
 of three perpendicular planes of symmetry, will be formed by three 
 quadrantal arcs of the great circles S, 2, and C, and be represented 
 
 by the expressions TT 
 
 o = 2j = L = 5 
 2 
 
 s = o- = c = 90 ; 
 
 S, 2, C being the arcs that form the sides; s, o; c the angles 
 opposite to them, of this quadrantal triangle. 
 
 Taking the three planes of symmetry for the axial planes, their 
 normals become the axes for the polyhedral system and the axial 
 octants coincide with the eight systematic triangles. The octants 
 and the systematic triangles adjacent to each other, and the faces of 
 any form of which the poles lie in them, will evidently be anti- 
 strophic in the character of their symmetry, those belonging to 
 attingent octants will be metastrophic. 
 
 The axes being the zone-lines or origin-edges [SO], [2], [2S] 
 are dissimilar, since from 105 it is seen that no plane is possible 
 in the system which intersects any pair of these axes with equal 
 intercepts ; so that equal, and therefore commensurate, parameters 
 are equally precluded. 
 
 For the purpose of uniformity in the representation of crystals 
 that accord with the same type of symmetry, it will be desirable to 
 assign an ' orientation ' of a definite kind to the different axes ; and 
 for this purpose in the type under consideration the order of the 
 magnitudes of the parameters will be adopted. Thus the longest 
 parameter will be that assigned to the X axis, the mean to the 
 Y axis, and the least to the vertical or Z axis. 
 
Forms are or tho symmetrical. 125 
 
 The expression representing the axial system will therefore be 
 
 in which the parametral ratios a\b\c are unfettered by conditions 
 and may be different for each distinct system of planes conforming 
 to this type of symmetry. The remaining three permanently fixed 
 elements of the axial system are therefore engaged in satisfying 
 those conditions which all plane-systems must fulfil in order to be- 
 long to this type of symmetry ; the conditions namely that a plane 
 of the system is the plane of a zone symmetrical to one of its planes, 
 or, as a consequence, that the plane-system is symmetrical 
 to three perpendicular planes which are also zone-planes. 
 
 109. The character of a form. As each octant is conterminous 
 with a systematic triangle, eight 
 faces will be comprised under 
 the symbol of an independent 
 form {hkl} ; see Fig. 42. And 
 as the intercepts on any one of 
 the several axes will have the 
 same value for every face of 
 the form, the position of the 
 indices in their symbols will 
 not admit of permutation ; the 
 signs of these will however 
 change, following those that 
 designate the octant in which 
 a particular pole may lie ; and 
 therefore undergoing every possible interchange, 
 the eight planes are therefore 
 
 hkl, Jikl, hkl, hkl, Tkl, Jikl, hkl, JiTl. 
 
 The position of the axes in this system being so taken that the 
 zone-lines [SC], [2CJ, and [S2] are the axes X, Y, Z respectively, 
 the plane S has for its symbol (oio), 2 is (100), and C is (ooi). 
 
 The prismatic forms {hko}, {hoi}, and {okl} are constituted 
 each of four planes, the first form being technically termed a 
 prism; the other two, domes (from the Greek fioyia). Thus the 
 dome-form {101} comprises the planes (101), (loi), (iol), (i~oT). 
 
 The parametral form { 1 1 1 } is a scalene octahedron. 
 
 The symbols of 
 
126 Type < = - ; five symmetry planes. 
 
 CASE II. The type of symmetry in which < = - 
 
 4 
 
 110. The case next to be considered involves the condition 
 precluded by the conditions of the last case, which required two 
 adjacent planes of symmetry to be perpendicular to each other. 
 If a pole be recognised as possible on a great circle that would 
 bisect an angle and the quadrantal side opposite to it of the 
 systematic triangle in the case last considered, we admit the con- 
 dition that a face of the system may be equally inclined on and 
 intersect with equal intercepts two of the axes, for instance, the 
 axes X and Y ; for this would evidently be true of every plane the 
 pole of which lies on such a great circle. 
 
 Then the pole in question will with the pole C (or ooi) serve 
 to designate a zone-circle intersecting with the great circle C in 
 
 another pole equidistant, that is to say, separated by an arc of - 
 
 4 
 from the poles of the axial-planes ZX and FZ; see Fig. 42. 
 
 The plane corresponding to this last pole would thus be inclined 
 
 at the crystallometric angle of - on each of the two perpendicular 
 
 4 
 planes of symmetry S and 2 characteristic of the last considered 
 
 type. It, and obviously also a plane perpendicular to it and also 
 lying in the zone [S2], would therefore each fulfil the necessary 
 conditions for a plane potentially symmetral. 
 
 Adjusting the letters to indicate conformable planes of sym- 
 metry, it will be seen that we have in the case under consideration 
 two proto-systematic perpendicular planes S lt S 2 alternating with 
 two perpendicular deutero-systematic planes 2 A , 2 2 , tautozonal 
 with the planes S and forming with them an isogonal zone with 
 the crystallometric angle of 45. 
 
 Each pair of planes S and of planes 2 will be conformable 
 inter se, but unconformable each with the other. And the plane 
 C of the zone-circle [S2] will also be potentially a trito-sys- 
 tematic plane of symmetry to the system unconformable with 
 both the pairs of planes and 2. Moreover, it will be evident 
 that the poles of a form on either of the hemispheres divided by 
 the equatorial plane C will be repeated in the same manner of 
 
The tetragonal Scalenohedron. 
 
 127 
 
 distribution on the opposite hemisphere, whether the repetition be 
 due to the system being centrosymmetrical or to the influence of 
 the plane C as an actual plane of symmetry. 
 
 The sphere of projection will be divided by the five planes A^ , 
 2 X , S t , 2 2 , and C into sixteen systematic triangles, each having 
 
 two of its sides S and 2 quadrants and the third side C = - 
 
 4 
 
 The systematic triangle will therefore be represented by the 
 expression 
 
 e v r ^ 
 
 o=2/ = ? C = j 
 
 2 4 
 
 s = o- = 90, c = 45. 
 
 111. The distribution of the poles of the tetragonal scalenohedron. 
 If under any type of symmetry the letters designating the sys- 
 tematic triangles be read in one order of rotation (e. g. that of the 
 hand of a clock) it will be 
 seen that these triangles fall 
 into two groups indicated by 
 letters taking the order, in the 
 one group ccrs, in the other 
 group csv. 
 
 The triangles of the one 
 group and the faces whose 
 poles lie within them are mu- 
 tually metastrophic, but are 
 antistrophic to those of the 
 group designated by letters in 
 inverse order to theirs. Thus 
 if in the type under consider- 
 ation the poles in the group of triangles cscr be designated as 
 p lt /> 2 , &c., and those in the triangles ccrs as //, //, &c., the 
 scalenohedron {p} will consist in the assemblage of the faces 
 
 AAAA> P\ / 2 /s /*> 
 
 AAAA> P\ p\ P\P\\ 
 
 wherein the minus sign is used to indicate poles on the lower 
 of the hemispheres divided by the plane C, and the faces be- 
 
 43- 
 
128 A morphological axis. 
 
 longing to the poles p are antistrophic to those belonging to 
 the poles /'. 
 
 112. The axial system for this type. The necessary conditions 
 for the type of symmetry under discussion are four in number, 
 and may be embodied in the statement that a plane of the system 
 is a zone-plane, and that a second plane of the system also 
 parallel to a zone-plane is inclined on the former plane at 45. 
 Of the five elements constituting the axial system four will be 
 constants occupied in satisfying these conditions, a single element 
 only remaining to vary with the particular system of planes. As 
 in the previous case, three of the perpendicular planes of sym- 
 metry, namely the two planes S and the equatorial plane C, are 
 taken as axial-planes, the zone-axis [S2] becoming the Z axis, 
 and the zone-axes \Sf\ [S 2 C] the axes X and F, respectively. 
 
 While any plane having its pole situate on one of the zone- 
 circles [2] will meet the X and Y axes with equal intercepts, such 
 a plane will, under the conditions assumed for this type of sym- 
 metry, meet the Z axes at a distance incommensurable with the 
 intercepts on the axes X and Y\ that is to say, the parameters 
 a and b are equal but are unequal to and generally incommensurate 
 with the parameter c. Hence also there can be no pole on a 
 great circle bisecting the quadrantal arcs S or 2 of the systematic 
 triangle or the right angles which they subtend. 
 
 Taking then for a parametral plane a face belonging to the 
 zone S l , the axial system is represented by the expression 
 
 f = ?? = C=9> 
 a = b ^ c. 
 
 The symbols of the axial-planes will be for S 1} S^ the poles of 
 which lie on the Y and X axes respectively, oio and 100; the 
 symbol for the pole of the plane C which will lie on the Z axis 
 being ooi. 
 
 DEF. Where two or more planes of symmetry lying in a zone 
 are conformable, their zone-axis will be termed a morphological 
 axis or axis of form for the system ; and where there is only one 
 such axis to the plane-system, the plane (potentially symmetral) to 
 which this axis is the normal will conveniently be termed the 
 equatorial plane. 
 
Symmetry of forms ditetragonal. 
 
 129 
 
 In the type of symmetry under discussion the morphological 
 axis is an axis of tetragonal symmetry for all forms of the system, 
 since two congruent planes of symmetry perpendicular to each other 
 intersect in this axis; and when the symmetry is complete it is 
 ditetragonal. 
 
 Of each zone-circle S and 2 the two hemizones on either side 
 of the morphological axis are similar. 
 
 113. Symbols for forms of tetragonal type. Each octant of the 
 axial system is composed of two systematic triangles t antistrophic 
 to each other, in which the order of the indices, like that of 
 the letters indicating the systematic triangle, will be direct for 
 metastrophic, inverse for antistrophic planes. Hence the indices 
 
 h and k relating to the similar axes X and Y will be interchanged 
 in position in the symbols for the two faces of a form {hkl} that 
 lie in the same octant ; the position and value of the index / re- 
 maining unchanged, except that its sign is + or according as 
 the face intersects with the axis Z above or below the origin ; see 
 Fig. 44. 
 
 K 
 
130 Symmetry-type, cf) = - 
 
 ) 
 
 The symbols of the faces of the general scalenohedral form will 
 therefore be 
 
 hkl khl, hkl 'khl, Jikl Yhl, hll khl, 
 hTl khl, hk~l khJ, hkl khl, hkl ~khl. 
 
 If the poles lie on the zone-circles 2, h and k do not differ in 
 value, and the symbol of the form is {hhl}, which is an isosceles 
 octahedrid form and, for the parametral plane, becomes { 1 1 1 } ; 
 while the zones C and 2 are tautohedral in the faces of a four- 
 faced form {no}. 
 
 If the poles of the form lie on the zone-circles S, the symbol is 
 {hoi}, and the form has the eight faces 
 
 h o I, oh I, hoi, oh I, hoi, oh I, hoi, o hi. 
 
 And the symbol for a form consisting of eight poles distributed 
 on the zone-circle C will be {hko}, all the faces being parallel to 
 the axis Z. The angles between planes belonging to this zone 
 will be constant for all plane-systems presenting this type of sym- 
 metry. 
 
 The zone-axes [SC] and [2C] are, as in the type of symmetry 
 previously discussed, axes of orthosymmetry, since they are the 
 intersections of two perpendicular planes of symmetry. 
 
 The abortive character of the symmetry which every plane in 
 the zone-circle C simulates has already been exposed in article 103. 
 It is only in the case of the systematic planes that this symmetry 
 is real. 
 
 CASE III. The type of symmetry in which </> = - 
 
 114. We have next to enter on the consideration of the case 
 in which a plane of symmetry is inclined on another plane of the 
 
 system at the angle </> = Here therefore there will be two 
 
 planes of symmetry S^ and S. 2 inclined to each other at 60, which 
 will be mutually repeated in a third plane S 3 symmetrical to each 
 in respect of the other and inclined on both at 60. Three 
 simultaneous tautozonal planes S lt S 2 , S 3 result; their common 
 zone-line being a morphological axis for the system of planes. 
 This axis divides the zone-circles into hemicyclically conformable 
 
Potential symmetry-planes. 131 
 
 hemizones. And the great circles [S] divide the sphere of pro- 
 jection into six lunes, the alternate lunes becoming congruent by a 
 revolution through 120 round their zone-axis as an axis of 
 trigonal symmetry. 
 
 115. Distribution of the poles of a form of trigonal type. A 
 form of a system symmetrical to three such planes will present,, 
 if/ be an independent pole, six poles / ditrigonally grouped round 
 the axis of form : viz. p lt p\, p 2 , p' z , / 3 , />' 3 , as in Fig. 45, in which 
 the poles on the nether hemisphere are indicated by eyelets, those 
 on the hitherward hemisphere by dots. 
 
 If the pole lie in a zone-circle S, there will be but three such 
 poles, viz. r lt r 2 , r s , or f lt / 2 , /,. 
 
 Since however the form is to be generally assumed to be centro- 
 symmetrical, and therefore diplohedral, there will be six addi- 
 tional poles p in the former case, or three new planes r or / where 
 they lie on the zone-circles S. There are, thus, the poles 
 
 A> /i, A, / 2 > A> / 3 ; A> /i> A /2>_ A / 3 ; 
 
 and r lt r^ r 3 , r^ r 2 , r 3 ; or / t , /, / 3 , 7 15 / 2 , / 3 . 
 
 Since the zone to which the planes S belong is symmetrically 
 divided by these three planes, it 
 may therefore, by article 101, also 
 be symmetrical to three new planes 
 2, each perpendicular to one and 
 inclined at 30 on the other two 
 planes S; and these planes 2 
 will thus be also potentially planes 
 of symmetry for the entire plane- 
 system : as is also the equatorial 
 zone-plane C. 
 
 Hence, that these planes 2 or 
 the plane C may become actual 
 planes of symmetry entails no condition that is not involved in 
 those originally assumed in this case ; a point in which the type 
 of symmetry now being considered differs from the otherwise 
 somewhat analogous case where two intermediate planes of sym- 
 metry are equally inclined on two perpendicular symmetral 
 planes. 
 
 K 2 
 
132 Type $ - merged in Type <p = -. 
 
 116. The quasi-independent nature of this trigonal type of 
 symmetry. Another and very characteristic point of dissimilarity 
 between the rectangular systems already discussed and the system 
 now under discussion is that diplohedral and therefore centro- 
 symmetrical symmetry is independent in the present case of the 
 influence of the equatorial plane as a plane of symmetry. In the 
 former systems centro- symmetry involved symmetry to the plane 
 C and vice versd. Here these are independent of one another, and, 
 according as we may suppose one or the other principle of sym- 
 metry to be in abeyance, the system will assume distinct characters. 
 To assume a form to be haplohedral is however inconsistent with 
 the view hitherto taken of the developement of a plane-system as a 
 system of faces parallel in pairs to a system of origin-planes, though 
 hereafter we shall have often to deal with such forms. But the 
 supposition that the influence of a group or groups of planes of 
 symmetry may be in abeyance puts no such strain upon the 
 conceptions of symmetry that we have formed. And in fact, 
 though it involves no new geometrical condition for a system 
 
 symmetrical to three planes in a zone inclined at - to pass into 
 
 o 
 a system symmetrical at once to two triads of tautozonal planes 
 
 inclined at the crystallometric angle of -, and to a plane 
 
 equatorial to these, yet the distinct and quasi-independent 
 characters of the symmetry to a trigonal axis as in the former 
 case, and of symmetry to a hexagonal axis as in the latter case, 
 would alone justify a separate treatment of the two resulting 
 varieties of symmetry. But this treatment will be further borne 
 out when we find ourselves hereafter dealing with a veritable 
 trigonal system of symmetry, such that the circumstances of the 
 particular system (the tesseral system) in which it occurs are in- 
 compatible either with the three planes intermediate to the original 
 three planes or with their equatorial zone-plane being recognised 
 as actual planes of symmetry. 
 
 But where the zone-circles 2 are only potentially symmetral, 
 i. e. where they are not actual planes of symmetry, a pole may lie 
 on one and be repeated on each of these great circles simply as 
 
Symmetry of forms ditrigonal. 
 
 133 
 
 the result of the symmetral character of the -S-triad of planes : 
 since the existence of such poles does not invalidate the conditions 
 assumed for this type of symmetry, as it did in the case where 
 
 = - \ inasmuch as in the present case no new condition has to 
 
 be assumed as necessary to be fulfilled by the system of planes in 
 order to render the presence of the faces corresponding to these 
 poles possible. 
 
 Such poles, which we may designate by the letter u, will be 
 repeated according to the same law as the poles /, namely in 
 six poles which, through hexasymmetrically grouped round the 
 morphological axis, are in the case in question to be viewed 
 (see Fig. 46) as grouped in a ditrigonal manner round that axis. 
 And if the poles u fall on the 
 zone-circle containing the poles 
 of the planes S they coincide 
 with the poles s of the planes 
 of symmetry. 
 
 In the same way, if the 
 three poles r on the great 
 circles S fall on the equatorial 
 zone- circle C they coincide 
 with three alternate poles a- of 
 the great circles 2; and if 
 this form be diplohedral, the 
 poles opposite to r falling on 
 the zone-circle C would coin- 
 cide with the remaining poles a : and in this case also the result 
 is a form with six poles grouped round the axis [S], that may be 
 viewed as an axis of ditrigonal symmetry. 
 
 It will be seen then that some forms presenting even the aspect 
 of regular hexagonal symmetry may be conceived as resulting from 
 a law of trigonal symmetry in which, as it were, there lurks though 
 concealed the potentiality of a hexagonal type of symmetry. But 
 such a system is evidently incapable of being represented by a 
 systematic triangle, since the surface of the sphere is partitioned 
 into six similar lunes alternately congruent. 
 
 Fig. 46. 
 
134 Symmetry Type, - 
 
 The only real axis of symmetry is the zone-axis of the planes S, 
 ditrigonal in its character. 
 
 CASE IV.- The type of symmetry where (/> = - 
 
 117. The discussion of the last case shows that no new 
 fundamental conditions will have to be assumed in order to pass 
 to the consideration of a plane system in which a plane of sym- 
 metry is inclined at the crystallometric angle of on another 
 
 plane of the system which is at the same time a zone-plane. It is 
 in fact the case in which the symmetry resulting from a plane 
 
 parallel to a zone-plane and to a face being inclined at the angle - 
 
 \j 
 
 on a plane of symmetry has received its complete developement : 
 it is the case, namely, in which the latter kind of plane system is 
 symmetrical simultaneously to its centre and to its equatorial 
 plane ; and in which as a consequence the planes 2 influence the 
 system as a triad of planes of actual symmetry (of deutero- 
 symmetry). The polyhedron thus becomes symmetrical to seven 
 planes, namely, to one triad of proto-systematic planes S, a triad 
 of deutero-systematic planes 2 unconformable with the planes S, 
 and to the equatorial plane C as a trito-systematic plane. These 
 planes 2 lf 2 a , 2 3 are then perpendicular each respectively to one 
 of the planes S lf S 2 , S s and inclined at 30 to the other two, 
 their mutual inclination being 60 : the systematic triangle where 
 the symmetry is thus complete being represented by the symbol 
 
 S = S = -, C = l, 
 
 2 6 
 
 s = o- = 90, c 30. 
 
 It will be seen that the sphere of projection is partitioned into 
 twenty-four such triangles. The number of fa.ces presented by an 
 independent form is therefore twenty-four. And the morphological 
 axis [-5*2] becomes an axis of hexagonal or generally of dihexa- 
 gonal symmetry. 
 
 The conditions which the crystalloid axial system and therefore 
 also the particular polyhedral system that is to accord with such 
 
A. rectangular axial system. 135 
 
 an axial system has to satisfy are embodied in the statements that 
 a plane is to be at once a plane of the system and a zone-plane, 
 
 and that it is to be inclined either at - or at - on a second plane 
 
 which must be parallel to a face of the system. In satisfying 
 these conditions four out of the five axial elements will be em- 
 ployed, one only being left, as in the case where the morpho- 
 logical axis was tetragonal, to vary with the different varieties of 
 the plane systems. 
 
 118. Axial Systems. Various axial systems may serve for the 
 geometrical treatment of such a polyhedral system. Thus : 
 
 (1) Three of the planes of symmetry, perpendicular to each 
 other, e.g. S l} 2 1 , and C, may be taken as the axial-planes; and 
 if C be taken as parallel to the plane ooi and therefore the 
 morphological axis be the axis Z, a plane the pole of which lies 
 on the zone-circle S 3 may be taken for the parametral plane ; or 
 the plane S 2 may be taken to determine the parametral ratio for 
 the axes X and Y\ this ratio having therefore the constant value 
 
 j = A/3 ; while another plane the pole of which lies in a great 
 
 circle S 1 or 2j will serve to determine the parametral ratios - or - 
 
 for the axis Z. 
 
 The axial elements would thus be represented by the ex- 
 pression 
 
 f=i? = C=- a : 6 : c, 4/3:1 :c; 
 
 the paramental ratio for the axis Z being the single varying 
 element. 
 
 Such an axial system, the details of which have been elaborated 
 by Schrauf (Sitzb. d. k. Acad. Wien, 1863), presents the insu- 
 perable disadvantage, that while the three planes of symmetry S 
 divide the sphere symmetrically with respect to a trigonal axis, the 
 axial-planes so divide it that congruent lunes are not similarly 
 situated in regard to the axes, so that even the simplest trigonal 
 forms have to be represented by double symbols. 
 
 (2) This difficulty of representing by a single symbolical ex- 
 
1 36 Isoparametral oblique axes. 
 
 pression even the simplest and most frequent forms of a crystal 
 presenting trigonal or hexagonal symmetry, when its forms are 
 referred to rectangular axes, led to the adoption by some crystal- 
 lographers of an axial system itself according with this symmetry, 
 the three proto-symmetral planes being taken with the equatorial 
 plane C as axial-planes. Thus, their intersections form four 
 axes to which the system of planes is referred, three of these 
 presenting an axial-angle of 60 with each other and being 
 perpendicular to the fourth which is the morphological axis. 
 Since however three points are sufficient to determine a plane, 
 an axial system by which a plane has in fact or virtually to be 
 represented by four points involves an element in excess of what 
 is needed. The further discussion of such an axial system will be 
 entered on in a future chapter. 
 
 119. The axial system of Hauy and Miller. On the other 
 hand, an axial system more in accord with geometrical method is 
 provided by the selection for axial-planes of three origin-planes 
 parallel to faces of the system, symmetrical in regard to and 
 therefore equally inclined on the morphological axis ; the poles of 
 which planes lie on the great circles of proto-symmetry S. 
 
 Thus, r x being the pole of a plane R^ and lying on the zone-circle 
 [SJ, two other poles r 2 and r 3 , poles of planes R z and R z , will lie 
 on the alternate hemizones of the zone-circles [S 2 ] and [S s ~\, equi- 
 distant with r x from c the pole of the equatorial plane C. 
 
 Accordingly the zone-axes [r 2 r a ], [rjrj, [i^rj which are the 
 edges of the planes R Z R Z , & 3 R l} and R^R^ become the axes 
 X, Y) Z, and the axial-points A x , A 2 , A 3 in which they meet the 
 sphere will also be equidistant from <:, on the same side with r lf r 2 , r, 
 respectively on the circles [S^], [S z ], [S 3 ]; and they will only 
 coincide with the poles TJ, r 2 , r 3 in the case in which the planes R 
 and therefore their edges also are perpendicular to each other. 
 
 This however involves a condition that would remove the 
 system of planes into another type of symmetry, in which not only 
 would the parameters be equal, but the axial-angles would, besides 
 being equal, become right angles. 
 
 In fact it will be seen hereafter that 
 
 tan arc r^ = 2 cotan arc Aj c % 
 
Twenty-four systematic triangles. 137 
 
 so that where r^ and Aj coincide 
 
 r^c \c = 54 44.14' and r^ 2 = \ 1 \ 2 = 90. 
 120. That in the axial system under consideration the 
 parameters will be equal is involved in the statement that the 
 pole c is equidistant alike from the poles r and the axial-points A.. 
 
 For then, evidently, a face parallel to C, the equatorial plane, 
 meets the axes with equal intercepts and serves as the parametral 
 plane in or III; the former symbol being taken for that lying 
 in the upper hemisphere on C. 
 
 Such an axial system as that here defined will be represented 
 by the expression 
 
 =i) = f f a = & = c; 
 
 where the required four conditions are satisfied by an equal 
 number of the axial elements, and only the particular angle TJ at 
 which the axes are inclined remains a variable element character- 
 istic of the particular system of planes. 
 
 Great circles passing through each pair of the axial-points XYZ, 
 X Y Z will divide the sphere into octants, two of which, XYZ and 
 X Y Z, are similar, while the remaining six, viz. 
 
 ZXY, YXZ, XYZ, 
 
 are also similar to each other but in ' zig-zag/ that is, alternately 
 inverted in their position on the sphere. 
 
 It will thus be seen that the systematic triangles are not co- 
 terminous with the octants formed by the axial system : and, as a 
 consequence, the symbols of a complete hexagonal form will in 
 certain cases have a double character. Thus of the general 
 scalenohedron, one pole will lie in each of the twenty-four 
 systematic triangles : but the poles lying in those adjacent pairs of 
 triangles that have in common for one of their sides an arc S 
 containing one of the axial-points X, Y, Z or of the poles 100, oio, 
 ooi, will have different intercepts on these axes from those of the 
 planes the poles of which lie in the triangles which in pairs 
 alternate with them. Hence the symbol for the general form will 
 be of a double kind; the poles corresponding to those of a 
 simple trigonal form retaining in their symbol indices which will 
 be distinct from those in the symbols for the poles of the other 
 
138 
 
 General 2 /[.-faced Scalenohedron. 
 
 trigonal form correlative to the former, and which united with the 
 former complete the form in its hexagonal type. 
 
 Retaining the letter p for the general trigonal form and desig- 
 nating the trigonal form correlative to it by the letter q, the 
 general symbol for the completed form will be {pq}. The dis- 
 tribution of the poles, including certain of those on the negative 
 hemisphere, is seen in Fig. 47. 
 
 They will be AAA, P\P\P'*> 
 
 the faces in the two left-hand blocks being, as in article 109, 
 
 antistrophic to those in the right- 
 hand blocks, while the faces the 
 symbols of which lie in either 
 block are mutually metastrophic. 
 The six poles r and six poles 
 / in Fig. 47 unite to form a 
 double form {rt}, the poles of 
 which lie on the proto- systematic 
 great circles S ; while the poles 
 u, alike twelve in number in the 
 trigonal (article 115) and in the 
 hexagonal type, constitute a form 
 of which the poles lie on the 
 great circles 2. The poles of a 
 
 form m on the great circle C will also be twelve in number 
 
 in the hexagonal as in the trigonal type of symmetry. 
 
 The poles of the proto-systematic plane S are six, lying in the 
 
 zone-axes [2 C] on the great circle C, namely, 
 
 ^li S 1> S 2> S 2> S 3> ^3 J 
 
 those of the planes 2 being 
 
 "i> o^'i, " 2 > a 'v "s> "'s 
 
 on the axes [SC]-, while the zone-circle C has two poles c 
 and /. 
 
 121. Symbols of the different forms. The poles r lf r a , r s of the 
 
 Fig. 47. 
 
Symbols of planes of symmetry. 
 
 139 
 
 axial-planes R^R^ R^ must have for their symbols 100, 010,001; 
 and since (in accordance with the principle established in article 
 99) the planes of symmetry [S] which pass through these poles 
 and the pole c or 1 1 1 have the same indices in their symbols as the 
 zone-axes have which are their normals, their symbols are, severally, 
 
 in 
 
 for ,, , i.e. on or on ; 
 
 100 
 
 for 6*2, loT or Toi ; and for S 3 , ilo or Tio. 
 
 Of these symbols, by article 45 the former of the two will in 
 each case be that of a pole lying on the same side with 100 of the 
 great circle passing through the poles in and oio, the latter will 
 be the poles lying on the same side of that zone-circle with the 
 pole ooi. 
 
 The symbols of the planes 2 lf 2 2 , 2 3 , the poles of which lie in 
 the intersection of the zone-circles S and C, are, of 2 X , "21 1 or 2!!; 
 
 Fig. 48. 
 
 of 2 2 , 121 or 121 ; and of 2 8 , 112 or 112. And of these the 
 former will be the symbol of the poles lying on the same side with 
 ooi of the zone-circle passing through in and ilo. The symbols 
 will then be distributed as in Figs. 48 and 49. 
 
140 
 
 Zone of abortive symmetry -planes. 
 
 The symbols of all the axial-points forming the angles of the 
 systematic triangles having been determined, we may proceed to 
 consider those of a pole lying on one or other of the sides of the 
 systematic triangle. 
 
 With respect to the symbol of any form of which the poles lie 
 on the zone-circle [m], it is evident that the sum of the three 
 indices is zero ; i. e. the symbol has to fulfil the condition 
 
 p + q + r = Q or p=q r. 
 
 And this is precisely the zone which corresponds to one (namely to 
 Case V) of the two cases in which it was shown in articles 98 and 99 
 
 mm. 
 
 that a plane-system might present a zone which could be con- 
 ceived as being symmetrical to each of its planes, all these planes 
 being parallel to possible zone-planes. In respect to these planes 
 however it was shewn that this symmetrical character can be only 
 abortive where the planes in question are not inclined at crystallo- 
 metric angles upon any of the systematic planes S or 2, that is 
 to say, must be so for all planes other than these. 
 
 In considering the arcs S forming the sides of the systematic 
 triangle we shall have to distinguish between those which contain 
 the poles of the form {100} and those which do not; since the 
 poles of this form on either of the hemispheres that stand on C lie 
 
Symmetry of a form di-hexagonal. 141 
 
 only on the quadrantal arcs S of alternate systematic triangles. 
 Now whereas the symbol of any pole lying on the great circle [-SJ 
 i.e. [oil] which traverses the pole r lt must fulfil the condition 
 kl=o, it is clear that the symbol must be of the form mnn\ 
 and if it lie on the positive side of the great circle [in], m + 2n 
 must be greater than zero ; if, again, it lie on the same side of the 
 great circle [2!!] with the pole 100, 
 
 2m2n>o and m>n; 
 
 and if it lie on the other side of [2 IT], m < n. Forms fulfilling 
 the former condition will be termed direct, those fulfilling the latter 
 will be termed inverse. 
 
 If the pole r 1 for which m > n have for its symbol (hkK), and 
 the so-called ' transverse ' pole / equidistant with r from in on 
 the other side of in, and therefore also homologous with r, 
 be (eff\ then / 1? / 2 , / 3 will be three poles homologous with and 
 severally correlative to r lt r 2 , r 3 , that is, to (hkk\ (khk], and (kkti). 
 Now /j will be symmetrical to r 2 in respect to the plane 2 3 and 
 to r s in respect to the plane 2 2 , and a great circle passing through 
 /, and r z will also pass through i il ; / x therefore is the pole in 
 which the two great circles [112^ oio] and [oil] are tautohedral 
 and the indices in the symbol (eff) are 
 
 ^kh, k+zh, k+2h. 
 
 Thus, for example, the poles correlative to 100, oio, ooi are 
 722, 212, and 221". 
 
 Hence the six homologous poles r and / lying on the great 
 circles S on one side of the zone-plane C will have their two 
 correlative triads of symbols connected by the relation just estab- 
 lished between the symbols hkk, khk, kkh and the symbols 
 eff, fef, ffe ; and on the opposite hemisphere the symbols of 
 planes parallel to these will only differ from them in having 
 opposite signs. 
 
 The symbol for a pole (min) lying on the side 2 of a systematic 
 triangle is characterised by one of its indices being the arithmetic 
 mean of the other two. Thus, a pole hkl lying on the great 
 circle [2J, i.e. on [2!!] or [211], must fulfil in its indices the 
 condition 
 
 2hklo or 
 
142 Composite symbol for scalenohedr on. 
 
 k + / 
 where h - For a pole situated on the positive side of 
 
 the zone-plane C, i.e. of [in], 
 
 h + k + l > o or k + l > o; 
 
 and if it be also situated on the same side with oio of the zone 
 [oil], k I > o, that is k > /. Whence if min be the symbol of a 
 
 pole u where m > n and z'= - , the poles on the upper or 
 
 positive hemizones of the great circle 
 
 2 2 are mtn, m'm, 
 
 2 X imn, tnm, 
 
 and on 2 3 mni, nmi; 
 
 the poles on the nether or negative hemisphere of faces parallel to 
 these having their signs reversed. 
 
 122. Composite symbol of the general di-scalenohedron. Of the 
 general independent form hkl a face lies eccentrically in each 
 of the twenty-four systematic triangles of the hexagonal system. 
 If the pole hkl\)Q that lying in the triangle cv^'^ Fig. 47, the pole 
 symmetrical with it in respect to the plane S l will lie in the triangle 
 ca l s 3 on a great circle passing through the poles oil and oTi : 
 and since the positive sides of the great circles [aTI] and [in] 
 are those containing the pole 100, the symbols for the poles in 
 the two systematic triangles in question must satisfy the conditions 
 
 zhkl>Q and h + k + l > o, 
 or 2h>(k + l)' and h>(k + t). 
 
 So that the first index must be greater than the other two ; and as 
 h > k > / is the assumed order of magnitudes of the indices, the 
 first index is h in the symbols for both the poles. So again the 
 pole hkl is on the same side with oio of the great circle S 1 or 
 [oil], and therefore the second index is greater than the third, 
 while for the pole lying in adjoining systematic triangles the third 
 index is greater than the second. 
 
 Hence the symbols of the poles in the triangles 
 
 cv^s'i and ro-jjg are hkl and hlk, 
 and in C(T z s \ an d cv^'^ are khl and Ihk, 
 
 C(T Z S\ and co- s s t are klh and Ikh. 
 
Symbols of correlative semiforms. 143 
 
 The symbol of the pole q transverse to and correlative with a 
 pole/ or (hkl) may be readily found by the problem of four planes. 
 
 It lies on the zone \J>c\ t i.e. or [k /, / h, hK\, at a 
 
 distance qc=ipc=0 from c. And as this zone will intersect with the 
 great circle [in] in a pole d, or (pqr), the symbol of which will be 
 2 h-k-l, 2 k I h, 2l-h-k, 
 
 m 
 
 n 
 
 cd 
 
 ^~ (article 49), 
 
 k} sin 6 cos 
 
 / /A h k 2sin0cos0 2' 
 
 and the indices in the symbol efg of the pole q are obtained by 
 equations P, article 49. They are, substituting the values obtained 
 for/^r, 
 
 e = npmh = 2(-{- /)//, 
 
 g = nrml= 
 
 where t <f < g for ef= 3^ 3^, and, ^ being greater than /, 
 f>c, also/" = 3/ 3/^, where, / being greater than l,f<g- 
 
 The same ratios for */ are obtained directly from the symbols 
 of the zones [^^/, in] and [1^1, lkh\ which are tautohedral in 
 \efg\ since [efg~\ and \lkh~] are symmetrical on the plane 2 2 . 
 They may also be written 
 
 123. Symbols of the prisms. Reverting to the poles lying on 
 the equatorial zone-circle, we have from the last article 
 
 p=2h-k-l t q=2k-l-h, r=2l-h-k, 
 whence q + r = k + l, q r= ^(k /) ; and, since k > /, also q > r, 
 So that the absolute relative magnitudes of the indices are / > q > r 
 for all poles not lying at the intersections with C of the great 
 circles S or 2 ; and these for the axial points at the intersections 
 of the zone-circles give q = r = i, while those at the inter- 
 sections of the zone-circles 2 gives p = r = i, the values of p q r 
 being taken absolutely. 
 
1 44 Congruent heterozonal symmetry-planes. 
 
 Planes of congruent symmetry not in one zone. 
 
 124. The four cases have now been considered in which it 
 is possible for two or more tautozonal planes of symmetry to 
 be inclined on each other at one of the crystallometric angles ; 
 and they have been seen to involve three distinct types of sym- 
 metry. In fact the symmetry of the whole system of planes is 
 controlled in these cases by the symmetry of the zone to which 
 the proto- and deutero-systematic planes belong. 
 
 The question however remains as to what other types of sym- 
 metry may be possible that are not included under these three and 
 that of symmetry to a single plane. 
 
 Thus of the polyhedral systems hitherto considered some were 
 symmetrical to planes at once tautozonal and conformable in their 
 symmetry; but it remains to be determined whether it may not be 
 possible for three or more heterozonal planes of symmetry to 
 present conformability; and after these have been considered, there 
 will remain the question whether there may not be yet other types 
 of crystalloid symmetry. 
 
 Since it follows that an axis of symmetry potentially tetragonal will 
 result from the existence of an independent pole lying on a circle 
 that bisects one of the right-angles formed by three perpendicular 
 symmetry-zones and bisects therefore also the quadrant of the 
 systematic triangle which subtends that right-angle ; we may en- 
 quire what will result from the further condition that any pole or 
 poles may lie on a great circle bisecting the right-angle formed by 
 another pair of the three perpendicular planes. Or, which is the 
 same thing, it may be asked what will be the nature of the sym- 
 metry resulting from a third plane of symmetry being conformable 
 with as well as perpendicular to either and therefore to both of two 
 perpendicular and conformable planes of symmetry. 
 
 CASE V. Three heterozonal planes of congruent 
 symmetry. 
 
 125. Case of three conformable planes of symmetry that are hetero- 
 zonal. In the case suggested in the last article each of the three 
 right-angles s, a; and c of the systematic triangle, Fig. 40, will be 
 
Three symmetry -planes conformable. 145 
 
 bisected by a zone-circle lying in what will be potentially a plane 
 of symmetry but not conformable with the two planes with which 
 it intersects. Hence also each of the three zone-lines in which 
 the proto-symmetral planes intersect becomes an axis of tetra- 
 gonal symmetry, since in these same zone-lines two perpendicular 
 deutero-symmetral planes will intersect with the two former planes 
 at 45- 
 
 Assigning similar letters to conformable planes of symmetry, 
 namely, S to the three planes of proto-symmetry and 2 to the 
 planes of deutero-symmetry, we have three perpendicular planes -S" 
 intersected by six planes 2 (see Fig. 50) that are in pairs tautozonal 
 with two and perpendicular to the third of the planes S. 
 
 It will be seen that one zone-circle 2 of each of the three 
 deutero-symmetral pairs must intersect with other two belonging to 
 
 *'ig- 50. 
 
 the other pairs, in a point o that will be equidistant from the 
 poles of the planes of proto-symmetry, and this will be the axial 
 point of an axis of trigonal symmetry. 
 
 For the three zone-planes 2 X , 2 2 , 2 3 (Fig. 50) will obviously 
 intersect with each other at the crystallometric angle of 60; since 
 
146 Cu bo-octahcdra I Symmetry . 
 
 in the triangles hdo (Fig. 50) we have the angles h 45, d 90. 
 
 and the side/id or = -; whence also ho or D = 54 44-14', and 
 4 
 
 <&? or # = 35 15-86', and D + ff= 90, and of the eight points 
 the mutual distance of any two adjacent poles as measured 
 on that arc of the zone-circle 2 which traverses a pole d is 
 70 3 1-7', while the distance of two poles o measured on the arc 
 traversing a pole h is 109 28-3'. 
 
 On comparing the relative distances of the points o and the 
 points h with the angles between faces of the polyhedron known as 
 the cubo-octahedron, it will be seen that they are supplementary to 
 each other, and that in fact the points o and h correspond to the 
 poles of the regular octahedron and cube respectively which 
 combine to build up that figure. The conditions supposed will 
 thus give rise to a type of crystalloid symmetry different from 
 those which have been hitherto considered. 
 
 126. The systematic triangle, and axial system. In it the sphere 
 of projection will be divided by the nine intersecting circles -5" and 
 2 into forty-eight systematic triangles, of which the sides 0, D, II 
 opposite respectively to the angles o, d, h have the values 
 
 = ^, = 54 44- 14', ^=35 15-86', 
 
 while the angles are o = 60, d= 90, h 45; 
 and there will be 
 
 three pairs of tetra-symmetral poles h, 
 
 four pairs of tri-symmetral poles o, 
 
 six pairs of ortho-symmetral poles d. 
 
 The general scalenohedron under such a type of symmetry will 
 therefore have forty-eight faces. 
 
 Taking the proto-symmetral planes S for the axial planes and 
 consequently their perpendicular intersections for the axes, and 
 taking a face of the octahedron for the parametral plane, it is 
 clear that since the latter truncates a quoin of the cube it will cut 
 the axes with equal intercepts, so that all the parameters are equal : 
 and the conditions for this type of symmetry as embodied in the 
 axial system thus chosen are expressed by the symbol 
 
Forty -eight systematic triangles. 147 
 
 It will be seen that all the five axial elements are required to 
 have special values in order to fulfil the general conditions of the 
 system .; no element remaining variable to characterise any different 
 systems of planes that may conform to this type of symmetry. 
 
 The six poles h are those of three symmetral planes S, parallel 
 to the faces of the cube ; and they are also the axial points in 
 which the axes of the quoins of the octahedron meet the sphere ; 
 the eight poles o are those of the faces of the octahedron and are 
 the axial points of the quoins of the cube ; and the twelve poles d 
 are those of the six symmetral planes 2, parallel to twelve faces 
 which, truncating the edges alike of cube and octahedron, con- 
 stitute the form {no}, termed the rhombic dodecahedron (the 
 dodecahedral rhombohedron), a figure the quoins of which have 
 axial points in common with those of the cube and of the octa- 
 hedron. 
 
 127. Symbols for the forms in such a system. Since the axes 
 and the edges of the proto-symmetral planes S are coincident, 
 it is evident that the faces of the cube will have the six symbols 
 arising from the various permutations of the indices 100 and the 
 interchanges of the sign of the unit index ; and the different faces 
 of the octahedron will be represented by the eight interchanges 
 of + and sign of which the general symbol { 1 1 1 } for this 
 form is susceptible. And since the axial octants are formed by 
 the intersections of the proto-systematic planes S, each of them 
 will be conterminous with six of the systematic triangles ; so that 
 six poles of the general scalenohedron will lie in every octant. 
 
 In fact, since the edges S 2 S 3t S S S 13 S 1 S 2 , and therefore also 
 the axes, are necessarily similar, a face presenting three different 
 indices in its symbol will be so often repeated in the octant as is 
 necessary to interchange each pair of the indices for every two 
 axes ; so that the different symbols for the faces in an octant will 
 be six in number, corresponding to the six interchanges of position 
 in their indices of which three different numbers are susceptible, 
 and falling each into one of the six systematic triangles in the 
 octant. Further, if we consider the symbols thus indicating the 
 six poles of the form {hkl} in an octant, it will be seen (see 
 Fig. 49) that those in which the indices follow the same order as 
 
 L 3 
 
148 Cyclic permutations in symbols. 
 
 hklhk or as Ikhlk are symbols of poles lying in alternate 
 systematic triangles, which, as in article 125, may be designated 
 as hdo or hod, and their faces are metastrophic; while those in 
 which the order of the indices is reversed belong to antistrophic 
 faces lying in adjacent triangles. 
 
 In passing from one octant to another, it is obvious that the six 
 faces belonging to any octant will have their signs in the same 
 position in their symbols, the position namely of the signs of the 
 XYZ which designate the octant. 
 
 The symbols of the faces of the general independent scaleno- 
 hedral form in this system will consequently represent all the 
 permutations of three numbers taken positively and negatively. 
 
 iety and distribution is given in the following table 
 
 TABLE A. 
 
 i. 
 
 iii. 
 
 v. 
 
 vii. 
 
 ii. 
 
 iv. 
 
 vi. 
 
 viii. 
 
 hkl, 
 
 Ill, 
 
 hkl, 
 
 hkl, 
 
 hkl, 
 
 hkl, 
 
 hkl, 
 
 h~kl, 
 
 Ihk, 
 
 Ihk, 
 
 Ihk, 
 
 fhk, 
 
 Ihk, 
 
 Ihk, 
 
 Ihk, 
 
 Thk, 
 
 klh, 
 
 ~klh, 
 
 "klh, 
 
 klh', 
 
 klh, 
 
 klh, 
 
 klh, 
 
 klh', 
 
 hlk, 
 
 hlk, 
 
 hlk, 
 
 hlk, 
 
 hlk, 
 
 hlk, 
 
 hlk, 
 
 hlk, 
 
 khl, 
 
 khl, 
 
 khl, 
 
 khl, 
 
 khl, 
 
 khl, 
 
 khl, 
 
 Thl, 
 
 Ikh, 
 
 Ikh, 
 
 Ikh, 
 
 lkh\ 
 
 Ikh, 
 
 Ikh, 
 
 Ikh, 
 
 Tkh\ 
 
 I \i. Ihk, Ihk, Ihk, Ihk, Ihk, Ihk, Ihk, Ihk, a II. 
 
 Ilia, khl, khl, khl, khl, khl, khl, khl, khl, ft IV. 
 Ikh, Ikh, Ikh, l~kh\ Ikh, Ikh, Ikh, Tkh; 
 
 where the columns represent each an octant ; the first four giving 
 the symbols of the poles lying in the octants XYZ and those 
 attingent to it and adjacent to XYZ; the remaining four 
 columns giving the poles in the octant XYZ and those attingent 
 to it. 
 
 The blocks I p. and IV ^. represent the systematic triangles 
 metastrophic to each other and to the triangle containing the pole 
 hkl, the blocks II a. and III a. represent the triangles antistrophic 
 to the former, but mutually metastrophic. 
 
 The symbols of a form the poles of which lie on a great circle 6" 
 will have a zero in the place of the index corresponding to that 
 axis which is normal to the great circle ; and since all the eight 
 
Types of the systematic triangle. 1 49 
 
 arcs on each of the great circles S are similar and will each 
 carry a pole of the form, the form will have twenty-four poles, in 
 the symbols for which two of the indices are different and one is 
 zero. These then will correspond with the twenty-four various 
 interchanges of position in the indices and of character in their 
 signs of which the general symbol {hko\ is susceptible. 
 
 The poles of a form lying on a great circle 2 will be the poles 
 of planes intersecting with identical intercepts on two of the axes. 
 Two of the indices must therefore be identical ; hence the general 
 symbol for a form the poles of which lie on the great circles 2 will 
 be either {hkk} or {hhl} ; and according as its position lies on 
 one or the other side of the pole of the octahedral form {in}, 
 it will present the one or the other of these types of symbol. 
 In fact, when the pole lies on an arc H its indices for two axes 
 are greater than that for the third and the form is {hhl}, when it 
 lies on an arc D its symbol is {hkk}. The form in either case 
 has twenty-four faces symmetrical to the great circles 2; their 
 symbols interchanging the positions of the indices in each octant 
 and the character and position of their signs from octant to octant. 
 
 General discussion of the systematic triangle. 
 
 128. We have so far considered certain special cases in which 
 a crystalloid polyhedral system may be supposed to be sym- 
 metrical to one or simultaneously to several planes. These cases 
 have included all the possible conditions under which such planes 
 of symmetry may lie in the same zone, and certain conditions 
 under which other planes heterozonal to these may also be planes 
 of symmetry for the system. 
 
 It remains however to determine whether these are the only 
 possible cases of crystalloid symmetry, or whether some polyhedral 
 systems may not exist the law of whose symmetry may have to be 
 represented by a different systematic triangle from any yet con- 
 sidered. And for this we may discuss the general characters of 
 such a triangle and the limits these impose to its variation ; and 
 the most general form of stating this problem will be that of 
 determining under what conditions three great circles may inter- 
 sect at crystallometric angles. 
 
150 Six possible systems. 
 
 129. The six systems of crystallography deduced from the crys- 
 tallometric law. Let S, C, 2 be the sides and s, c, a- be the angles 
 respectively opposite to them of such a spherical triangle. Then 
 as s, c, and <r can neither of them be greater than 90, while their 
 sum must be greater than 180, only the values for these angles in 
 the three first columns of the following table are possible, the 
 corresponding values for the arcs forming the sides of the triangle 
 being those in the three last columns : 
 
 (90, 60, 60; 70 31-7', 54 44-14', 54 44-14 
 54 44-14', 45, 35 15-86 
 
 77 77 77^ 
 
 II. ' 
 
 77 77 77 
 
 9, 3; 2 > -, g ; 
 
 TTT O O O T ^ 77 
 
 III. 90 , 90 , 45 ; -5 
 
 IV. 9 o, 90, 90; - 2 > - 2 > I- 
 
 If to these we add (V) the case of a system symmetrical to a 
 single plane, and (VI) that of a system merely symmetrical to a 
 centre, we shall have represented every possible case in which a 
 crystalloid polyhedral system can be said to be symmetrical at all, 
 It is evident that these are precisely the cases that have been 
 investigated in this chapter. These varieties of symmetry will 
 hereafter be distinguished by the following designations : 
 
 I. 3 planes S, 6 planes 2, Cubic System. 
 
 II. 3 S, 3 2, i plane C, Hexagonal System. 
 
 III. 2 S, 2 ,, 2, T C, Tetragonal System. 
 
 IV. i S, i ,, 2, i C, Orthosymmetric 
 
 System. 
 
 V. i S, Monosymmetric 
 
 System. 
 
 VI. a centre of symmetry, Anorthic. 
 
Cubic involves trigonal symmetry. 1 5 1 
 
 And the planes of symmetry which characterise these several 
 systems, and by their intersections with the sphere of projection 
 in arcs of great circles determine the systematic triangle in each 
 system, will henceforward be designated only as systematic planes. 
 
 130. Relations of the cubic system to the trigonal and hexagonal 
 system. It results from this that the trigonal axes of the first of 
 these systems those namely which represent the normals of the 
 regular octahedron can in no case become axes of hexagonal sym- 
 metry, and therefore the planes of the octahedron cannot be planes 
 of symmetry for that system. Of course planes belonging to a 
 form analogous to the form {mi'n}, the u of Fig. 47 of the 
 trigonal system, and lying in great circles that bisect the angles 
 
 313 
 
 SSL 
 
 Fig. 51. 
 
 at which three planes 2 meet each other, may exist on crystals of 
 this type. But they are essentially trigonal and not hexagonal in 
 the character of their symmetry round the axes 0. It will be 
 furthermore apparent in the comparison of the cubic system with 
 the trigonal type of symmetry, Figs. 50 and 51, that the case 
 alluded to in article 119, where tan r c = 2 cotan A c, is that in 
 which a form in the trigonal system is directly comparable to 
 one belonging to the cubic system ; since then 
 
 rc = ho = 54 44-14' = ff 
 and r t r 2 = h z h^ 90 
 
 in the cubic system, and the three axes are perpendicular. 
 
 131. Symmetry of faces. The features that undergo symmetrical 
 
152 Symmetry of faces. 
 
 repetition in the faces of a polyhedron are the edges that form 
 its sides and the plane angles in which any two of these edges 
 meet so as to form one of the sides of a quoin. 
 
 The character of the boundary line of a face, as being an edge, 
 is determined, not by the length of the side, but by the angle at 
 which the face is inclined on the other face meeting it in the edge ; 
 and generally edges of which the angles are the same are geo- 
 metrically similar and homologous, except in the ambiguous case 
 of such angles being right-angles. 
 
 It is however necessary, especially where they are right-angles, 
 to enquire as to the character of the faces that lying in the zone 
 can intersect each pair of edge-forming faces, and to determine 
 whether two different edges can have faces so replacing them as 
 to cut off the edge in each case at the same angles to the 
 corresponding faces. 
 
 And the symmetry of a face will be known when we know the 
 law of repetition of its edges and angles; and this will obviously 
 depend on the number and nature of the planes of symmetry that 
 may intersect with it perpendicularly. 
 
 When it is not parallel to a zone-plane and is therefore also not 
 parallel to a plane of symmetry it can only be perpendicularly 
 intersected by a single plane of symmetry, and where it is not 
 perpendicular to such a plane of actual or potential symmetry it can 
 have no symmetry at all, except where the system is symmetrical 
 to a single plane, when a face parallel to that plane will be 
 centro-symmetrical. And when we consider any other position 
 that a face may occupy on a crystalloid polyhedron, it is clear that 
 its pole can only lie on a side or at an angle of a systematic 
 triangle, or must be an independent pole ; in which last case it 
 is without symmetry. 
 
 When its pole lies on a side of a systematic triangle, the face is 
 traversed perpendicularly by one plane of symmetry to the trace 
 of which it is euthy-symmetrical ; when the pole lies in an angle 
 of a systematic triangle, and therefore at the intersection of two or 
 of four, of three or of six planes, the face is symmetrical to an axis 
 of symmetry and is ortho-symmetrical or ditetragonal, ditrigonal 
 or dihexagonal in character. 
 
Symmetry of quoins. 153 
 
 Striations and recognisable physical features in the case of a 
 crystal often throw light on the character of the symmetry that the 
 face obeys which exhibits them ; and it is from the aid which such 
 characteristics, as exhibited in important faces, afford for the 
 determination of the type of symmetry, not merely of the face but 
 of the entire crystal, that this study of these features derives its 
 significance : and indeed the information thus attained will be 
 found often to go deeper than a symmetry that is merely 
 morphological, and to involve the symmetry that rules in the dis- 
 tribution of physical properties and underlies the geometrical 
 symmetry. 
 
 132. The symmetry of a quoin is determinate in a similar 
 manner to that of a face. For the quoin is composed of edges 
 and plane angles, and where these recur symmetrically the quoin 
 will be symmetrical to a line which is an axis of symmetry of a 
 corresponding order. 
 
 Thus, where a quoin is symmetrical to a zone-axis, its summit 
 is capable of being truncated by a plane parallel to the corre- 
 sponding zone-plane, and according with it in symmetry ; where it 
 is not symmetrical to a zone-axis, it may be symmetrical to a 
 single plane or it may be altogether without symmetry. And it is 
 evident that the symmetry which a quoin will present will corre- 
 spond with that of the diameter of the sphere which meets its 
 vertex. If, as in the case of the axis of symmetry in the clino- 
 rhombic system, the diameter in question is an axis of diagonal 
 symmetry only, the quoin or a face replacing it is diagonally 
 symmetrical to this axis. If, again, the diameter is a normal to a 
 plane the pole of which lies on a side of a systematic triangle, 
 the quoin is euthy-symmetrical to the systematic plane in which 
 the arc lies that carries the pole. Where the axis of the quoin 
 meets the sphere at the angle of a systematic triangle, the quoin 
 is symmetrical to the zone-axis which at that point meets the 
 sphere. 
 
 In every other case the quoin is devoid of symmetry. And the 
 characteristics of the quoins of a crystal, as in the case of the faces 
 forming them, offer one of the most important features by which 
 to recognise the type of the crystal's symmetry. 
 
J 54 Symbols for truncating planes. 
 
 133. Symbols of truncating and bevilling planes. From the 
 principles established regarding crystalloid symmetry it will be 
 seen that the only cases where the edge formed by two faces is 
 truncated by a third face or bevilled by pairs of faces of the system 
 will occur when the faces forming the edge are adjacent faces of a 
 form symmetrical to a systematic plane, or else lie on a great circle 
 traversing an axis of symmetry in regard to which they are sym- 
 metrical ; so that the pole of a face truncating an edge will lie on 
 an arc and may lie at an angle of a systematic triangle. It further 
 results from the symmetrical character of the axial systems adopted 
 for each crystalloid type that the symbols for two adjacent faces of 
 a form either differ only in the signs of one or of two indices, 
 or else differ by the permutation of certain of their indices. In 
 either case, the ratios of the indices in the symbol of the truncating 
 face is obtained by the addition of corresponding indices in the 
 symbols of the planes whose edge is truncated. 
 
 In the Hexagonal system this rule will be found to hold directly 
 only for those faces of, for instance, the general form [hkl efg} 
 which belong to the trigonal semiform {hkl}, or else to those 
 belonging to the correlative semiform inverse to it {efg} ; for in 
 attempting to apply the rule to adjacent faces belonging, the one 
 to a direct the other to an inverse form, it appears to fail. In fact, 
 however, the principle involved in the rule is only obscured. The 
 symbol of a form derived from the symbol of an inverse form 
 by the same method as the latter is derived from that of a direct 
 form should evidently be the identical symbol of the original direct 
 form ; so that the inverse form to {efg} should be {hkl}. If how- 
 ever it be derived by aid of the formulae given in article 122, this 
 symbol will be found to be not (hkl) but (g& g&gt); whence it is 
 evident that if the condition above asserted is to be fulfilled we 
 should have to consider the indices of the form inverse to hkl, i.e. 
 
 2 (* + /)-*, 2 (l+h)-k, 2 ( + )-/, 
 
 as equivalent not to efg but to %e sfsg, and in order to compare 
 /Wwith efg on, so to say, equal terms, we should take efg as 
 ranking with 3 h 3 k 3 /. In order then to determine the symbol of 
 a face truncating an edge of adjacent faces of a form {h k I efg} 
 we have, where the faces belong the one to a direct the other to 
 
Case of hexagonal forms. 155 
 
 the inverse semiform, to multiply the indices in the symbol of the 
 face of the direct semiform by a common factor 3, and to add 
 them to the indices belonging to the face of the inverse semiform. 
 The rule will then be found to be general, and it applies equally, 
 of course, to other forms besides those of the general scaleno- 
 hedron. 
 
 Thus, for example, the face truncating the edge (hlk) (e/g), 
 that is to say, the edge of the faces (3^ 3/3^), and 
 
 k-2l+h, lzk + k), 
 
 is 4/1 2 k -\- I, k j r l2h, k + l2h, or 2!!. 
 
CHAPTER VI. 
 
 CRYSTALS AS CRYSTALLOID POLYHEDRA. 
 
 SECTION I. Mero-symmetry. 
 
 134. THE properties of a system of planes mutually related 
 by the law of the ' Rationality of Indices' have been so far in- 
 vestigated as a crystalloid system from a purely geometrical point 
 of view ; and, by establishing as one of these properties the 
 principle that the varieties of isogonal zones that can be extant 
 in such a system are limited to four, it has been possible to shew 
 that only a limited number of types or systems of symmetry can 
 be illustrated in crystalloid plane-systems. When we turn to the 
 natural polyhedra presented in crystals in order to determine to 
 what extent these actually accord in their geometrical characters 
 with the crystalloid systems hitherto considered, we cannot fail to 
 recognise that whereas the .crystallographer, guided heretofore 
 solely by observation and experience, referred every crystal to one 
 or other of six crystallographic systems, those systems furnish 
 precisely the several types of symmetry which coincide in their 
 distinctive features with the six crystalloid types of symmetry 
 resulting from the above principle. 
 
 But in order to carry on the enquiry by means of exact ob- 
 servation into the geometrical relations connecting the faces of a 
 crystal, we must have recourse to instrumental methods admitting 
 of the requisite precision. This object is attained by the use of 
 the Goniometer, an instrument constructed for the measurement 
 of the angular inclinations of planes, of which the description and 
 the use will be given in a future chapter. 
 
Crystals as crystalloid systems. 157 
 
 The results that have been accumulated by means of this instru- 
 ment form a body of observations on which crystallography as 
 a science rests ; but in dealing with these results the crystallo- 
 grapher is often more or less embarrassed by errors incidental 
 to the use of instruments, and still more by difficulties due to 
 peculiarities and imperfections presented by the faces of the crystals 
 themselves. 
 
 In proportion however as the errors arising from such sources 
 are diminished, it has been found that the values obtained for 
 the angular inclinations of the faces of a crystal more and more 
 closely accord with those which would result from the crystal 
 being a crystalloid polyhedron ; that is to say, from the Law of 
 the Rationality of Indices being the fundamental law presiding 
 in its construction. 
 
 And that this is true for all temperatures at which the integrity 
 of the crystal is maintained may be assumed, since it is true within 
 the limited ranges of temperature at which such measurements can 
 be effected ; while within these ranges of temperature some crystals 
 are near the highest limit at which they can exist, while others are 
 examined at temperatures far lower than those at which they have 
 been formed. 
 
 The analogy of the law that indices are integral coefficients of 
 the parametral ratios to the fundamental law of chemical com- 
 bination by which bodies unite in simple multiples of their weight- 
 equivalents, can hardly escape notice ; and the inductive method 
 by which each has been arrived at has consisted in an accumu- 
 lation of experimental results scarcely less extensive and exact 
 in the case of the crystallographic than in that of the chemical 
 law. 
 
 135. A significant illustration of the occurrence on crystals of 
 only such forms as are possible in a crystalloid polyhedron, is 
 furnished in the fact that of the five regular solids, three, namely, 
 the cube, octahedron, and tetrahedron, are frequently crystal forms, 
 whereas the dodecahedron and the icosahedron have never been 
 met with on any crystal. The three first figures are crystalloid 
 in their symmetry ; the faces of the two last cannot be expressed 
 by symbols with rational indices, and they furthermore present 
 
158 Mero-symmetrica I crystals. 
 
 a pentagonal symmetry which is of an order impossible, as has 
 been seen, in a plane system obeying the law of rational indices. 
 Experience however has, on the one hand, proved that a crystal 
 is not merely thus externally and geometrically, but is also phy- 
 sically and throughout its substance symmetrical and seolotropic ; 
 while, on the other hand, it has led to the recognition of a natural 
 principle which in a great number of cases limits the complete- 
 ness of the symmetry of the crystal. Thus it not unfrequently 
 happens that certain of the faces on a crystal which, in accordance 
 with the geometrical principles of symmetry laid down in the last 
 chapter, should constitute a form, will present in their physical 
 characteristics differences so marked and occasionally so contrasted 
 that it is impossible to view them all as equally repetitions of the 
 same face. 
 
 136. This partition of the faces geometrically similar into phy- 
 sically dissimilar groups is, however, found to be itself obedient to 
 principles of symmetrical distribution which concord with those of 
 the crystallographic system to which the crystal belongs : and such 
 an interruption in the complete accord of physical and geometrical 
 symmetry will be seen to be a particular case only under a more 
 general law which deals not only with a division of the faces of the 
 crystal into correlative groups, but in general also with the entire 
 suppression of all the faces not belonging to one of the groups. 
 
 137. Mero-symmetry. In the last chapter the character of a 
 form under each of the different types of symmetry were con- 
 sidered, and the trigonal type was treated as a partial and incom- 
 pletely developed variety of hexagonal symmetry. 
 
 While observations directed to such crystals shew that this 
 incompletely developed type is represented abundantly in nature, 
 numerous analogous cases of incomplete symmetry are also met 
 with in other crystallographic systems. 
 
 DEF. A holo-symmetrical form in any system will be the 
 term applied to a form in which all the faces required to complete 
 the symmetry of the system are present, and are physically as well 
 as geometrically similar. 
 
 The term mero-symmetrical will be employed in all cases in 
 which the faces requisite to build a geometrically complete form 
 
Hemi-symmetry ; t-wo kinds. 159 
 
 are partially suppressed, or in which these faces fall into physically 
 contrasted groups ; the suppression of faces or of the features 
 characteristic of the form taking effect however in a certain sym- 
 metrical manner. 
 
 Mero-symmetrical forms may be hemi-symmetrical, and 
 will then present one-half of the faces of the complete form ; or 
 tetarto-sym metrical, presenting one-quarter only of the faces 
 of the, holo-symmetrical form; a form of the hemi-symmetrical kind 
 will be termed a semiform or a hemihedron, one of the latter kind 
 a tetartohedron. And the term merohedral will be reserved for 
 certain cases in which a defalcation is met- with in the faces of 
 a crystal out of accord with any fixed law of symmetry ; though 
 sometimes such a merohedral crystal simulates the mode of 
 grouping of a crystal belonging to a different type of symmetry 
 from its own. 
 
 138. Hemi-symmetry. Now in considering in what ways it may 
 be possible, while conserving the essential idea of each type of 
 symmetry, to suppress one-half of the faces of a form, we have to 
 keep in view the principle that in surrendering or modifying the 
 symmetral character of a systematic plane or zone-line, each cor- 
 responding similar systematic plane or zone-line must simulta- 
 neously undergo the same degree of deprivation of its symmetral 
 character. And again, the kinds of mero-symmetrical forms will 
 be essentially different according as the suppression of their faces 
 is due to the form ceasing to be symmetrical to its centre, or 
 to the symmetral character of a plane or group of planes of 
 symmetry falling into abeyance. 
 
 If the crystal be not centro-symmetrical, each origin-plane will 
 be represented by only one of its two poles on the sphere ; or, 
 to adopt the corresponding fiction of centro-normals, each normal 
 will be represented by a single ray, that is to say, will carry but 
 one face and one pole. 
 
 Where the form is centro-symmetrical, on the other hand, the 
 suppression will affect the faces in parallel pairs; so that only 
 one-half the origin-planes corresponding to faces of the crystal 
 and only one-half the number of normals will be extant, the 
 remaining half being absent. 
 
1 60 Classes of mero-symmetrical forms. 
 
 And a mero-symmetrical form may further be conceived such 
 that alternate normals only should be extant and should each carry 
 but a single face. 
 
 139. Holo- and mero-sysiematic forms. Since failure of sym- 
 metry in the number of normals or of origin-planes belonging to 
 a form in any system can only result from abeyance of symmetral 
 character in one or more of the planes or groups of planes of 
 symmetry which have already been designated as ( proto-, deutero-, 
 trito-) systematic planes, we may avoid the ambiguity in which the 
 terms hemihedrism, tetartohedrism, &c. are involved by the more 
 restricted or wider senses in which different authors have em- 
 ployed them, if we adopt a nomenclature consistent with our use 
 of the former term. 
 
 DEF. A holo-systematic form, then, is a form in which all the 
 origin-planes or normals required by the complete symmetry of the 
 system are extant. 
 
 A hemi-systematic form is a form in which only half the origin- 
 planes or normals are extant, the correlative half being absent. 
 
 In a tetar to- systematic form, only one-fourth of the origin-planes 
 or normals can be considered as extant. 
 
 A diplohedral form will, as before defined, be a form in which 
 every origin-plane is parallel to two faces (or has both its poles 
 extant) ; or in which each normal is made up of two rays or carries 
 two faces and their poles. 
 
 In a haplohedral form each origin-plane or each normal is repre- 
 sented by a single face and its pole. 
 
 140. Kinds of mero-symmetry. Whence there is 
 
 I. Holo -symmetry, where a form is at once holo-systematic 
 
 and diplohedral. 
 
 II. Hemi-symmetry, where a form is 
 
 or 
 
 i. holo-systematic and haplohedral, ) Semiforms 
 ii. hemi-systematic and diplohedral. } hemihedra. 
 
 III. Tetarto -symmetry, where the form is 
 
 i. hemi-systematic and haplohedral, ) . 
 
 i ^ i- i i j i J Tetartohedi 
 or 11. tetarto-systematic and diplohedral. j 
 
L aw of mero-symmetry. 161 
 
 IV. Hemimorphism is the term for a particular case of 
 haplohedral mero-symmetry. One-half or, it may be, one-fourth 
 of the faces of the original form are present in the hemimorphic 
 form: but these all lie on one side of a systematic plane, the 
 symmetral character of which is in abeyance. 
 
 141. The law of mero-symmetry. The conceivable modes in 
 which either one-half of the normals or one-half of the faces cor- 
 responding to the full complement of normals (each represented 
 by a single face) might be suppressed are evidently numerous and 
 varied. And in the resulting polyhedra we should in many cases 
 look in vain for any characteristic features of the symmetry of the 
 system to which the holo-symmetrical form belonged. But in 
 crystals, as has been before observed, some special quality dis- 
 tinguishing the original type of symmetry is always preserved in all 
 their mere-symmetrical forms ; and it is in accordance with this 
 experience that we seek for a geometrical principle that shall 
 embody such a condition. 
 
 Now the only geometrical assumption that can be made 
 regarding the mero-symmetrical suppression of the faces of a 
 system so as to satisfy this condition is one which we find in 
 Nature to include all known cases of mero-symmetry, while 
 without extension of its terms it will also be found to embrace 
 the symmetrical conditions presented by holo-symmetrical forms. 
 It may be stated then in .the form of a general law of crystal- 
 lographic symmetry, that on a crystal the extant or absent 
 features of a form must be extant or absent in the same 
 way in respect to equivalent systematic planes. This is 
 the second fundamental law of crystallography. 
 
 142. The nature of the forms necessitated by this law in the 
 different systems will be discussed hereafter. But certain of 
 the general results which are involved in its application may be 
 pointed out here. 
 
 Thus hemimorphism can only exist in relation to a unique 
 systematic plane, since it could not hold in the case of two con- 
 formable systematic planes. It is thus precluded from every form 
 of the Cubic system. It may, on the other hand, occur on a form 
 otherwise holo- or hemi-systematic, and so be either hemi- or 
 
 M 
 
162 
 
 Tetragonal hemi-symmetry . 
 
 tetarto-symmetrical in its character (i. e. presenting only the half 
 or the fourth of the faces of the complete form). 
 
 143. So, again, a tetragonal axis of symmetry is, so to say, the 
 creature of the two pairs of alternating planes of symmetry S 
 and 2 of which it is the zone-line, and in the holo-symmetrical 
 case the poles of an independent general form {hkl} are grouped 
 ditetragonally round this zone-line. If, now, we suppose one of 
 these pairs of systematic planes say the planes 2, Fig. 52 (i) to 
 fail of being symmetral, the result will be that either the planes S 
 continue planes of symmetry and their zone-axis becomes an axis 
 of ortho-symmetry, as in Fig. 52 (2); or the planes S also fail as 
 planes of symmetry, while the zone-axis retains its character as an 
 
 (3) 
 
 axis of tetragonal but not of ditetragonal symmetry, the grouping 
 of the faces round it being in alternate systematic triangles, as in 
 
 Fig- 52 (3)- 
 
 144. In the same way, Fig. 53 (i), an axis of hexagonal (in the 
 general case dihexagonal) symmetry is the zone-axis of two triads of 
 alternating planes of symmetry; and the mero-symmetrical sup- 
 pression of half the poles of a form can be effected by the suppression 
 either of the symmetral character of one of the triads of systematic 
 planes, or of both triads simultaneously. If, for example, the 2 
 planes are the triad of which the symmetral character is in abey- 
 ance, the grouping of the six faces that remain extant out of the 
 twelve of the original scalenohedral form {hkl} lying on one side 
 of the equatorial plane will be such that the dihexagonal axis 
 becomes a ditrigonal axis; see Fig. 53 (2). 
 
 If, on the other hand, both triads of planes are no longer planes 
 
Hexagonal hemi-symmetry. 163 
 
 of symmetry, the twelve faces may become reduced to six in such 
 a way that the poles lie in alternate systematic triangles, see 
 Fig- 53 (3) ; the zone-axis, then, continues to be an axis of 
 hexagonal but not of dihexagonal symmetry. 
 
 In a trigonal system, indeed, in which the axis of form is an 
 axis of ditrigonal symmetry and is the zone-line of three con- 
 formable systematic planes S lt S z , S 3 , it is evident that the 
 abeyance of the symmetral character must take effect on all 
 three or on none of these planes ; that is to say, the ditrigonal 
 axis becomes a trigonal axis, or else the system, if conceived of as 
 
 (3) 
 
 derived from one originally diplohedral, must become hemimor- 
 phous in respect to the zone-plane [^ S z S 3 ~\. 
 
 145. In the case of an axis of ortho-symmetry which is the zone- 
 axis of two perpendicular planes of symmetry 6* and 2, Fig. 54 (i), 
 
 (0 
 
 Fig. 54- 
 
 (2) 
 
 if the symmetral character of one and one only of these planes is 
 in abeyance, for instance that of the plane 2, their zone-axis will 
 not be an axis of symmetry. In this case, however, it will be 
 
 M 1 
 
164 
 
 Orthogonal hemi-symmetry . 
 
 seen that in a holo-systematic system the zone-axis of the planes 
 S and C (C being in a diplohedral system a third systematic plane 
 perpendicular to S and 2) must be an axis of ortho-symmetry, 
 and that the system will be hemimorphous in regard to the plane 2, 
 see Fig. 54 (2) and (3). On the other hand, the symmetral 
 character of the two planes S and C may both be in abeyance ; 
 and then there is no symmetral plane and each zone-axis becomes 
 an axis of diagonal symmetry, as in Fig. 54 (4). 
 
 (3) 
 
 Fig- 54- 
 
 (4) 
 
 146. In the case of the Mono-symmetric system, obeying only 
 one plane of symmetry of which the normal is an axis of diagonal 
 symmetry, the only available varieties of mero-symmetry are 
 hemimorphism resulting from the abeyance of the symmetral 
 character of this single systematic plane, and the case of a semi- 
 form presenting two poles symmetrical to that plane. For, were 
 the form diplohedral, it would retain but a single normal and its 
 symmetry be undistinguishable from that of an anorthic crystal. 
 
 The varieties of mero-symmetrical partition, which faces of a 
 general independent form may undergo in any particular crystallo- 
 graphic system, will have to be considered in detail in chapters 
 specially devoted to the description of the forms of the different 
 systems. But their character is always determined by the necessary 
 conditions that the distribution of the faces must be consistent 
 both with the law of mero-symmetry, and with the rules which 
 have been laid down in the preceding articles regarding the order 
 of the symmetral influence which an axis of symmetry may retain 
 when some or all of the systematic planes of which it is the zone- 
 axis lose their symmetral character. 
 
Simultaneous abeyance. 
 
 165 
 
 147. That the different forms of the same crystal cannot be 
 simultaneously obedient to different types of symmetry, that is 
 to say, cannot belong to different systems, is as obvious a neces- 
 sity as that the different features of the same form cannot be so. 
 There remains however the question whether a crystal hemir 
 symmetrical in regard to a particular form or forms can exhibit 
 holo-symmetry in regard to other forms; for instance, where 
 geometrically diplohedral forms are found concurrent with haplo- 
 hedral forms, it may be asked whether the possibility of such 
 a concurrence is not proved. 
 
 The general answer to this question is however to be found 
 in the principle that planes of symmetry are such for all features 
 of a crystal, and, where they are in abeyance at all, they are in 
 abeyance for all the forms. 
 
 If the effect of any particular mero-symmetrical principle in 
 influencing the different kinds of forms of the same crystal be 
 considered, it will be seen that what in the case of one variety 
 of form will result in the suppression of the half of its faces, may 
 in another produce not a suppression of any of the faces, but a cor- 
 responding loss of symmetry in the outline or in the physical 
 characters and molecular structure of each and all the faces of the 
 form. The latter case, that namely in which the suppression takes 
 effect, not in the obliteration of half the faces of the form but in 
 that of half the features of each face in fact in the partial suppres- 
 sion of the symmetry which these faces would obey in the holo- 
 symmetrical form occurs wherever the poles of the form lie on 
 any systematic great circles whereof the symmetral character is 
 
 (l) (2) 
 
 Fig. 55- 
 in abeyance. A cube with its alternate quoins truncated by the 
 
1 66 Mero-symmetry : notation. 
 
 faces of a tetrahedron would illustrate such a suppression. Its 
 six faces are all ortho-symmetrical, not tetra-symmetrical. Thus 
 the striations on the faces of the cube in Fig. 55 (i) will be 
 seen to run parallel only to the edges these faces form with 
 those of the tetrahedron which truncate its alternate quoins ; 
 while in Fig. 55 (2) the cube faces are striated parallel to their 
 intersections with alternate pairs of faces of the rhomb-dode- 
 cahedron. Fig. 55 (i) represents a crystal of blende symmetrical 
 to the 2-planes only and haplohedral; Fig. 55 (2) is that of a 
 crystal of pyrites symmetrical to the S-planes only, but diplohedral. 
 In both the symmetry of the cube-faces is seen to be orthogonal, 
 not tetragonal. 
 
 148. A uniform notation to represent the kind of mero-symmetry 
 presented by any correlative pair of hemihedra or group of four 
 tetartohedra is of some importance ; and the more as the Greek 
 letters which have been employed as prefixes to the symbols of 
 holo-symmetrical forms in order to represent their semiforms have 
 received different significations from different authors. 
 
 In the Tetragonal system, for instance, the semiforms represented 
 by Prof. Miller as A {hkl}, K {hkl\, and a \hkl} carry in the treatise 
 of Prof. V. von Lang the symbols y{hkl], \{hkl} and K {hkl} 
 respectively. 
 
 The symbol TT is alone in carrying by general consent a con- 
 stant signification, namely, that the semiform it designates is diplo- 
 hedral ; but as employed in the Cubic and Tetragonal systems, for 
 instance, it represents different ideas of symmetry. 
 
 To avoid confusion, therefore, a notation will be adopted in 
 this treatise such that, in the cases where ambiguity has heretofore 
 arisen, the letters employed as prefixes will recall by their sounds 
 the nature of the symmetry that is not in abeyance and which 
 therefore controls the extant form. 
 
 The following prefixes will accordingly be-employed, in the case 
 of hemi-symmetrical forms, to represent that the faces of these 
 forms which are extant (or absent) are so symmetrically, only 
 
 (as haplohedral and holo-systematic forms) 
 i. to one or more zone-lines of symmetry a 
 
Mero-symmetry : notation. 167 
 
 2. to one or more such resultant zone-lines, and also 
 
 to the (proto-systematic or) S- planes s 
 
 to the (deutero-systematic or) 2 -planes o- 
 
 to the (trito-systematic or) C-plane, see below, <p. 
 
 to the C-plane and the S'-planes x 
 
 to the C-plane and the 2 -planes .. .. f 
 
 to the S- planes and the 2 -planes p 
 
 (as diplohedral and hemi-systematic forms} 
 to the ^-planes and to a centre of symmetry .. .. TT 
 to the 2 -planes and to a centre of symmetry .. .. -fy 
 to the C-plane and to a centre of symmetry .. .. (j> 
 The diplohedral character of the last three kinds of form is sug- 
 gested by letters which involve the sound of TT (opposite poles 
 on the sphere separated by a distance IT being extant in such 
 forms), the meaning of the affix TT being restricted to a special 
 case ; while the double letters x and f serve to recall the letters 
 CS or C2 that represent those of the systematic planes which 
 alone retain an actually symmetral character and thus determine 
 the nature of the symmetry ; p has been used by V. von Lang to 
 represent hemimorphism in general, and is here retained for a 
 frequent case of hemimorphism, that on the trito-systematic plane. 
 The usual mode of representing a tetarto-symmetrical form is 
 that of uniting the prefixes corresponding to two out of the three 
 pairs of hemihedra that may be constituted out of the faces of 
 the four correlative tetartohedra. 
 
 Thus, in the Tetragonal system we have for the mero-symmetry 
 of the general form {hkl} (see Plate II) three hemi-symmetrical 
 pairs of forms, 
 
 s {hkl}, s {khl} ; o- {hkl}, a- {hkl} ; {hkl}, $ {khl} : 
 and these three sets of correlative semiforms may be produced by 
 combining in distinct pairs four tetarto-symmetrical forms, which 
 may be therefore indifferently designated as 
 
 sir {hkl}, s<r{khl}, sir {hkl}, s<r{khl}; 
 or s<t>{hkl}, s<t>{khl}, s<t>{hkl}, 
 
 Or <T 
 
1 68 Analysis and synthesis of forms. 
 
 149. It results, as a consequence, from the law of mero- 
 symmetry, that any holo-symmetrical independent form in the 
 Cubic, Tetragonal, or Hexagonal systems may be considered, from 
 a purely geometrical point of view, as a composite form capable 
 of being analysed into various correlative pairs of hemi-symmetrical 
 forms, while these again may undergo further analysis into tetarto- 
 symmetrical forms; each stage in the analysis necessarily sub- 
 tracting from the completeness of the symmetry of the figure, yet 
 always so that the type of symmetry distinguishing the system is 
 recognisable. Conversely, the various semiforms may be built 
 up by a synthetical process of combining correlative tetarto- 
 symmetrical forms in pairs ; while again by the assemblage of 
 the faces of either pair of corresponding semiforms the complete 
 holo-symmetrical form will be constituted. 
 
 Thus, A, M, A', M', four cor- 
 relative tetartohedra, form three 
 pairs of correlative hemihedra 
 AM and M'A', AM' and MA', 
 A A' and M'M, the faces of each 
 correlative pair constituting if 
 united the holo-symmetrical form. 
 See also plates I to VII. 
 
 150. In discussing, by way of 
 illustration, certain kinds of forms 
 resulting from the disparting of 
 
 AM 
 
 >MA' 
 
 M'A' 
 Fig. 56. 
 
 holo-symmetrical polyhedra into constituent correlative hemihedra, 
 the principle of the partition was not discussed. Evidently each 
 semiform may appropriate one face from the pair belonging to every 
 normal, or, on the other hand, may be built up of pairs of parallel 
 faces belonging to alternate normals. But there are consequences 
 depending on this character of the mero-symmetry which must 
 not be lost to view. Thus, in the haplohedral holo-systematic 
 case the original form is so disparted into two semiforms 
 that, for each face of the one semiform, a face, parallel to it 
 in the complete form, belongs to the other semiform. Whether 
 the two semiforms can be brought into congruence will depend 
 on whether their faces are severally capable of being so, meta- 
 
Enantio-morphous forms. 169 
 
 strophically ; and this, again, on the mode in which these faces are 
 grouped. 
 
 From the general scalenohedron of a system may be derived 
 two kinds of correlative semiforms. These may be such that the 
 poles of one semiform lie in alternate systematic triangles and 
 so, therefore, that the faces of the one semiform are metastrophic 
 to each other, but antistrophic to the faces of the other semiform : 
 or, it may be that metastrophic are united with antistrophic faces 
 in each of the semiforms. But it is impossible to partition the 
 faces of a semiform of haplohedral character between two tetarto- 
 hedral forms, otherwise than so as that each of the latter forms 
 shall have all its faces antistrophic to the faces of the other 
 correlative tetartohedron. The configuration of the one tetarto- 
 hedron will then correspond to that of the other as seen in a 
 mirror. In a word, the two tetarto-symmetrical forms are enantio- 
 morphous. 
 
 And since antistrophic asymmetric faces on a crystal cannot 
 be brought into congruence by the revolution of the crystal round 
 any diametral line, it is not difficult to determine whether in any 
 particular case correlative mero-symmetrical forms are enantio- 
 morphous or tautomorphous ; i.e. cannot be brought into 
 congruence, or can be so brought by revolution round one or 
 more zone-lines. 
 
 If the tetartohedra in Figure 56 that are represented by different 
 letters are antistrophic and those with the same letters metastrophic 
 to each other, of the correlative pairs of hemihedra formed by their 
 combination those represented by union of different letters and 
 formed by antistrophic tetartohedra will be tautomorphous, viz. 
 AM with M'A', AM' with MA' , while the pairs represented by 
 union of the same letters, viz. AA' and MM' ', are formed by meta- 
 strophic tetartohedra but are themselves enantiomorphous. 
 
 151. It has been stated in articles 135 and 136 that, whereas 
 the hemi-symmetrical developement of a form implies in general 
 the suppression of one of the correlative groups of its faces, the 
 two groups may nevertheless be concurrent on the crystal; but 
 that, whether they are so concurrent or are found only on separate 
 crystals, they are often distinguishable from each other by differences 
 
170 Antihemihedrism. Rotatory polarisation. 
 
 in external feature or otherwise in physical character. It may be 
 evidenced by variation in smoothness or lustre, or in roundness 
 or plane character of surface, in the directions, depth and form of 
 striations or of hollows, or in the relative magnitudes habitual with 
 the faces of the respective semiforms ; or, also, it may be associated 
 with polarity of character in the physical properties of the correlative 
 hemihedra, especially under pyroelectric excitement due to changing 
 temperature. Such differences, then, may generally be held to 
 indicate a mero-symmetrical habit, and they often impart a very 
 marked contrast to the faces of the correlative semiforms. It may 
 occur, on the other hand, that the distinctive features of concurrent 
 mero-symmetrical forms may not be apparent or recognisable. 
 That they should nevertheless exist follows as a consequence of 
 the principles of crystallographic symmetry. 
 
 In the case of semiforms which are holo-systematic, and there- 
 fore haplohedral and not centro-symmetrical, each normal of a 
 form would carry either a single face or else two parallel faces 
 which differ in physical features; and the fulfilment of the con- 
 ditions imposed by the mero-symmetry of the system is compatible 
 with the supposition that the crystal is endowed with different or 
 also, in a polar sense, opposite properties in opposite directions 
 of any given line. This character has been designated by V. von 
 Lang as Antihemihedrism. 
 
 When the crystal is at the same time hemi-systematic and hap- 
 lohedral the forms are tetartohedral, and for two of the four 
 quarter- forms which are tautomorphous the character of the dis- 
 tribution of the properties and of the features (as, for instance, of 
 the poles) round corresponding lines in the crystal will be meta- 
 strophic, but will be antistrophic to that round corresponding lines 
 in the two quarter-forms enantiomorphous to the former. 
 
 Along the principal axis of symmetry in particular crystals of the 
 Tetragonal and Hexagonal systems, and along every direction in 
 certain crystals of the Cubic system, a ray of plane-polarised light 
 acquires rotatory polarisation. That this property should be con- 
 fined to haplohedral crystals, hemi-symmetrical or tetarto-sym- 
 metrical, in which all the symmetral planes of the original holo- 
 symmetrical form are in abeyance, will be shown hereafter, when 
 
Ultimate significance of symmetry. 171 
 
 the physical characters of crystals are under discussion, to follow 
 from the laws of symmetry. 
 
 152. In this chapter the subject of mero-symmetry has been 
 treated as involving the presence or the absence of certain faces, 
 consequent upon the abeyance of the actual symmetral character 
 of planes which are otherwise potentially planes of symmetry. But 
 in this treatment of the subject a symmetral influence has still 
 been recognised as, so to say, latent in these dormant systematic 
 planes ; inasmuch as the zone-lines, which may be considered as 
 becoming axes of symmetry by virtue of the original symmetral 
 nature of those planes, generally retain this character in a greater 
 or less degree, notwithstanding the abeyance of a direct symmetral 
 influence in the planes themselves upon the forms of the crystal. 
 By an inverse method of treating the subject of symmetry, it would 
 have been possible to have evolved the laws of symmetry, and 
 deduced those of mero-symmetry, from a discussion of the con- 
 ditions regulating the degrees of symmetry possible round a 
 zone-axis, and to have considered the various planes and groups 
 of planes of symmetry, not as originating axes of symmetry so 
 much as being the results of the symmetral character of such 
 axes. 
 
 It is however evident that either method, and indeed that the 
 whole treatment of crystallographic symmetry on the assumption 
 of planes and axes of symmetry, actual or potential, represents a 
 geometrical abstraction ; an abstraction that needs for its develope- 
 ment and due explanation a complete science of position applied 
 to the molecular mass-centres, competent to embrace not merely 
 the relative distribution inter se the intermolecular distribution 
 of the chemical molecules constituting the crystallised substance, 
 but also the intramolecular arrangement of the atoms, or molecules 
 of secondary order, whereof the molecules of the substance are 
 themselves composed. Then the true significance of the ideal 
 planes and axes of symmetry will be understood ; and they will 
 assuredly retain a place in the explanation of crystalline symmetry, 
 since they rise into recognition directly from the fundamental 
 principle of rationality of indices and are controlled by its con- 
 sequences. 
 
172 Composite crystals. 
 
 SECTION II. On Composite and Twin Crystals. 
 
 153. Crystals have thus far been considered as single structures, 
 each complete in itself: but this structural individuality is far from 
 being the only or even the most frequent form of their occurrence. 
 Whether as minerals or as products of the laboratory, crystals 
 continually present themselves as aggregates ; often, undoubtedly, 
 united by no definite law, but often also so combined that, while 
 corresponding faces of different individual crystals are quite or 
 approximately parallel, they appear as belonging to a single 
 crystal with its faces interrupted by recurring and by re-entering 
 edges, or tesselated by surfaces not lying precisely in a plane. And 
 a large proportion of the crystals that appear as single individuals 
 are thus composite, built up of more or less numerous single but 
 similar crystals, nearly but often not quite parallel in orientation, 
 though continuous in their substance and in optical contact. In 
 many crystals these component individuals exhibit faces, generally 
 minute, to which it is only possible to assign symbols of a complex 
 kind, that is to say, with indices that are high and often only 
 approximate in their ratios; and to this cause numerous pecu- 
 liarities have frequently to be traced, such as belong to forms of 
 which the faces are rounded, tesselated, or terraced, or which 
 exhibit ridged or ' drused' surfaces, wartlike protuberances, re- 
 entrant edges, or hollows whereof the sides and base are facetted 
 with crystallographic planes. 
 
 Instances of these peculiarities are familiar to the mineralogist 
 in apophyllite, tourmaline, quartz, idocrase, galena, blende, dia- 
 mond, calcite, etc. 
 
 But there is another and somewhat more regular manner in 
 which a composite structure asserts itself, that, namely, in which 
 the corresponding faces of the united crystals, though not in the 
 same plane, have a definite relative orientation. A law which, by 
 the geometrical fiction of a rotation of one of the crystals, expresses 
 the character of this orientation will be enunciated in the ensuing 
 article. Such crystals as obey it are termed hemitropic, macled, or 
 twinned crystals. Certain cases, however, which this law does not 
 suffice to explain will be discussed when the forms and combina- 
 tions presented under the different systems are described. They 
 
Twin crystals. 173 
 
 belong for the most part, though not exclusively, to symmetrically 
 grouped tetartohedral forms. 
 
 154. DEF. Twin-crystals. Two crystals are said to be hemi- 
 tropic or twinned when, presenting identical forms, they are united 
 together in such a way that, if we conceive one of them as 
 being turned through half a revolution round a particular line 
 which will be termed the twin-axis, the two crystals would have 
 identical orientation ; that is to say, corresponding faces and 
 edges in the two crystals would become parallel. Or, if the initial 
 orientation of the crystals was the same, such a rotation would 
 bring them into the relative position of twin-crystals. The plane 
 to which the twin-axis is perpendicular is termed the twin-face 
 or twin-plane. 
 
 The twin-axis may be (i) a face-normal, or (2) a zone-line, or 
 (3) it may in certain cases be at once a face-normal and a zone- 
 line : finally (4) there are crystals in which the twin-axis has 
 been stated to be a line perpendicular to a zone-line and parallel to 
 a face of the zone, while not itself either an edge or a normal. In 
 fact, it is the result of observation that in the four first, or ortho- 
 symmetrical, systems, in all of which it has been seen that three 
 lines, at once normals and zone-axes, co-exist perpendicular to 
 each other, the twin-axis is almost invariably a face-normal, and 
 is, in some cases, at the same time a zone-axis. In the Mono- 
 symmetric system the axis of twinning occurs as either a 
 zone-axis or a face-normal, but by the conditions of the system 
 cannot be at once a zone-axis and a normal, if the crystal be 
 holohedral, since the single systematic axis is already an axis 
 of diagonal symmetry. In the Anorthic system the twin-axis 
 is found to be either a normal or a zone-axis; unless, further, 
 as is maintained by some eminent crystallographers, there may 
 be cases where it is neither of these, but forms with a normal and 
 a zone-line perpendicular to each other a third line perpendicular 
 to both. This question will recur for discussion in the section on 
 the twins of the Anorthic system. In no case, however, can an axis 
 of actual diagonal symmetry or one of higher symmetry, except in 
 the case of a trigonal axis, be a twin-axis for the two crystals. 
 Otherwise, the half rotation of either crystal round such a diagonally 
 
174 
 
 Twins : their plane of union. 
 
 symmetral axis would only serve to bring the two crystals again 
 into an identical aspect, and they would as at first merely present 
 the condition of a parallel aggregation, not to be confounded with 
 a twinned structure. It will also be evident that, a twin-axis being 
 thus precluded from being an axis of diagonal symmetry, a face of 
 one crystal can only be parallel to the corresponding face of the 
 other, when it is either parallel to the twin-axis or perpendicular to it. 
 
 Where the twin-plane is a crystallographic plane, it must be 
 parallel to an identical origin-plane or zone-plane for both the 
 crystals, and will have the same symbol whether considered as 
 belonging to one or the other crystal : the symbol, indeed, of the 
 twin-plane will be the same for both crystals even in cases where 
 this is not a crystallographic face and its symbol therefore has 
 not rational indices. 
 
 In Fig. 57 the rotation of the regular octahedron a round the 
 
 Fig. 57- 
 
 normal of the face i il would bring that figure into the position 
 b. The figure c exhibits the. two octahedra a and b united so 
 as to form a twin-crystal : it is the so-called spinel-twin, being of 
 frequent occurrence in the mineral spinel. 
 
 Hemitropic crystals, in cases where they are merely juxtaposed 
 (the cases to which the term hemitropic was more exclusively 
 applied), are frequently but not invariably united at the surfaces of 
 their common twin-plane : in fact, if we assume with Haidinger that 
 the plane of junction must have the same relation to the two crystals, 
 it will be clear from what has been said above in this article that the 
 surface at which the two crystals are in contact must be either 
 the twin-plane or a plane in the zone perpendicular to it. The 
 face or plane in which two crystals are thus in contact is termed 
 the face or plane of union, or also the combination-plane. There 
 
Degrees of inter penetration. 
 
 175 
 
 are cases in which this plane would seem to have irrational 
 indices and thus not to be a face of the crystal. 
 
 155. In certain cases all or some of the faces of one crystal 
 can have rational symbols when referred to the same axes and 
 parameters as the other crystal. Such rationality is however con- 
 fined to cases (i) where the crystal belongs to the Cubic system, or 
 (2) where the crystal belongs to the Tetragonal or the Hexagonal 
 system and the twin-plane is parallel or perpendicular to the 
 morphological axis, i. e. is a face belonging to a zone containing 
 planes of abortive symmetry in both crystals, or is perpendicular to 
 that zone. 
 
 156. The two individual crystals which compose a twin-crystal 
 never have their individuality merged in the resulting structure, 
 the material of the one being never so intimately blended with that 
 of the other that the two cease to coexist independently side 
 by side, however minute the lamina? or intercalated parts may 
 be which represent the several individuals; and there are cases 
 in which the twin-structure recurs in successive parallel repetitions 
 so numerous as to be represented by a series of laminae of the 
 utmost tenuity. 
 
 But the degree of intimacy in combination which twin-crystals 
 may present is very varied. 
 
 Fig. 58 a. Fig. 58 5. 
 
 The two individuals may present a mere contact at a common 
 
 surface or plane of union, as in the case of twins by hemitropic 
 
 juxtaposition ; such is fas juxtaposed twin represented in Fig. 57 c, 
 
 Article 154 : or there may be an interlocking of the crystals, each 
 
1 76 Poly synthetic twins. 
 
 being bedded more or less deeply in the other, as in the (so- 
 called Carlsbad) embedded twin of orthoclase, Fig. 58 a : or again, 
 there may be a complete mutual inter penetration of the individuals, 
 as in the Fig. 58 b of an inter penetr ant twin of galena, generally 
 with plane surfaces of junction, occasionally, however, the material 
 of the one being so intercalated in that of the other that the 
 individuals are united at surfaces with jagged, or waved, or other- 
 wise irregular outlines. And in the case of polysynthetic twins 
 several or almost innumerable hemitropic individual crystals may 
 be combined each individual being formed of a crystalline plate, 
 and all being twinned on the same plane so as to build up an 
 essentially laminated structure, the alternate lamina being in 
 crystallographic orientation parallel to each other but hemitropic 
 to the crystal-laminae that intervene between them. Character- 
 istic groovings or striations are often produced by the outcrop 
 of these alternating laminae where they present edges that 
 are actually or nearly parallel in the two series, but are formed 
 by faces which do not for either series coincide with a 
 plane perpendicular to that of the laminae. Such striations are 
 thus due to ridges formed by the alternate crystal-layers, and 
 they result in a corrugated surface which, when the laminae are 
 very thin, has the appearance of a crystalline face; as in Fig. 59 
 of a polysynthetic twin of albite and Fig. 60 of labradorite. 
 
 Fi g- 59- Fig. 60. 
 
 157. Where the twin-axis is a crystallographic line which in 
 accordance with crystal-symmetry would be repeated in the 
 crystals, it does not follow that the twinning process will be 
 repeated for the other lines, whether normals or zone-axes, homo- 
 logous to that line. Repetition of the twinning on similar twin- 
 
Twins : re-entrant angles. 
 
 177 
 
 faces may indeed occur, and in certain kinds of zones the twinning 
 often does recur, but not as a necessary result of the law of 
 symmetry. Such crystals are triple, quadruple, &c. hemitropes 
 (or triplings, fourlings, &c.). And crystals occasionally exhibit 
 twinning of more than one kind ; that is to say, one or more 
 individuals twinned on a given crystal round one axis may be 
 united with another individual or twin-group twinned on the 
 same nuclear crystal round another axis not homologous with the 
 former axis. 
 
 158. A noteworthy feature in hemitropic crystals is the mode 
 in which the substance of the crystals is distributed in different 
 directions; it being a frequent habit of twins that the thick- 
 ness of the combined crystals in the direction of a twin-axis is 
 no greater and is often less, relatively to the thickness along other 
 directions, than would be the case for a single individual not 
 twinned but otherwise corresponding in dimensions with the twin. 
 
 Fig. 61. 
 
 159. It will be seen that, in the case of two hemitropic diplo- 
 hedral crystals, the twin-plane becomes in a crystallographic sense 
 a plane of symmetry to the twin-structure ; but at the same time 
 each crystal in the hemitropic group retains its individuality, not- 
 withstanding the mutual interpenetration of the crystals. From 
 this it results that two faces of the different individuals may meet 
 in an edge of which the internal angle is greater than TT ; or, in 
 other words, the edge may be re-entrant, which is not the case 
 in a simple crystal. In Fig. 61, if PP be the traces of two 
 parallel planes of one individual, P'P' those of the corresponding 
 planes of the other individual, the trace of the twin-plane being T 
 and the twin-axis /, it will be seen that in the hemitrope position, 
 
 N 
 
178 Twins of semiforms. 
 
 as in the figure, one pair of the faces forms a re-entrant angle 
 P'P 2 and the other pair a salient angle P'P =26. 
 
 Re-entrant edges of this kind then are highly but not exclusively 
 characteristic of twinned crystals ; at the same time they are not 
 always to be met with upon them (see Fig. 62). If in Fig. 61 be a 
 right angle, there can be no geometrical evidence, afforded by 
 the faces PP' or PP', of the twin structure. It may happen 
 that the twin-faces of the two individuals in cases where the 
 plane of junction is perpendicular to these faces may form a 
 single plane, or again, as in Fig. 62 of a crystal of hcematite, 
 that faces belonging to a zone, the zone-line of which is the twin- 
 axis, thus fall into coincidence, and in the crystal in question, Fig. 
 62, both these conditions occur: but it is generally possible, at 
 
 Fig. 62. 
 
 least in the latter case, to trace on the composite face the line 
 of junction of its component faces ; and frequently the needed 
 evidence of twin-structure is afforded by the fact that the direc- 
 tions of the striations on these faces are symmetrically inclined on 
 the line in question. 
 
 160. It follows from the foregoing statements that, in the case 
 of diplohedral forms, whether holo- or hemi-symmetrical, a plane 
 of actual symmetry to the individual crystals is precluded from 
 being the twin-plane of their hemitropic union. It will be seen, 
 however, that in the case of haplohedral crystals a face parallel 
 to a plane of actual symmetry may be a twin-plane, since the 
 normal of such a face is not an axis of diagonal symmetry. 
 
 In the case where a hemi-symmetrical crystal is thus twinned on 
 a systematic plane, the poles of the two individuals will evidently 
 fall into the same positions on the sphere as those of a simple holo- 
 symmetrical crystal. Hence it might be anticipated that cases would 
 occur in which it might be a question whether we had to deal with 
 
Twins : ambiguous cases. 1 79 
 
 a hemi- symmetrical crystal disguised by twin structure, or with 
 a crystal, really holo-symmetrical, but exhibiting in its growth 
 a composite structure simulating the characters of a twinned 
 crystal. And further it might be suggested that in the case of 
 a hemi-symmetrical crystal, on the forms of which the faces of 
 both the correlative semiforms were concurrent, the resulting 
 appearance of a holohedral developement may be due to the 
 twinning of the correlative forms with each other. But in each of 
 the supposed cases the question would in general be answered by the 
 inspection of the crystal itself : since the two crystal-elements com- 
 posing a twin can in fact in almost every case be distinguished 
 from each other, owing to their retaining their individuality side 
 by side, however minute the individuals may be and however com- 
 pletely they may exhibit mutual interpenetration. 
 
 In the case, for instance, of the haplohedral Cubic mineral 
 blende, all the faces of the octahedron, that is to say, the faces o 
 and the faces w of the two correlative tetrahedra, may be present 
 together on a crystal without twin structure, so that the geome- 
 trical character of the crystal would be holohedral ; that the crystal 
 is however of hemi-symmetrical character will be readily recog- 
 nised by the faces of the octahedron in attingent octants being 
 found to differ in physical character from those in the octants 
 adjacent to them. 
 
 The diamond, on the other hand, offers an instance of a Cubic 
 mineral presenting bright plane faces of the regular octahedron, or 
 more or less rounded faces of other forms, developed in exactly the 
 same manner geometrically and physically in all the octants, as if 
 they belonged to a holohedral crystal. The edges, however, which 
 lie in the proto-systematic planes, e. g. those of the octahedron, 
 are frequently furrowed more or less deeply with re-entrant grooves, 
 which are facetted with faces rounded when those adjacent to 
 them are so, or in the case of the simple octahedron, are formed 
 by plane faces parallel to the adjacent ones of the octahedron *. 
 
 A question of great difficulty has arisen whether diamond is 
 to be considered as a crystal of tetrahedral (haplohedral) de- 
 velopement, twinned on the faces of the rhomb-dodecahedron, 
 * SeeSadebeck, Zeitschrift d. dentsch. geoL Gesell. 1878. 
 N 2, 
 
1 8 o Twinning of diamond. 
 
 or whether it is in fact a holohedral crystal, in which the furrow- 
 ing of the octahedral edges is to be explained, not by a principle 
 of twinning, but by a strong tendency to a lamellar developement 
 during its growth ; the crystal having, for so hard a body, a very 
 facile cleavage parallel to the faces of the regular octahedron. 
 The difficulty of answering this question is not diminished by the 
 dissimilarity in habit of the diamond to other crystals; and in 
 support of the latter of the two views the following consider- 
 ations have been put forward : first, the occurrence of a dode- 
 cahedron face as the twin-plane for haplohedral crystals of the 
 Cubic system is otherwise rare, and when such twins occur 
 they have a very marked character as interpenetrating twins of 
 an entirely different habit from the crystals of diamond ; which 
 mineral, on the other hand, frequently occurs in twins of the 
 spinel- twin variety : secondly, that, although simple crystals of 
 diamond exhibiting indubitable tetrahedral habit (e. g. with forms 
 of the hexakis-tetrahedron) do occur, yet they are among the rarest 
 specimens*; and holosymmetrical crystals, such as magnetite, spinel, 
 and gold, are also described as occurring with distinct tetrahedral 
 habit : thirdly, that there is no evidence of any internal structure 
 suggesting the idea of an aggregation of separate individuals : 
 and finally, that there is clear evidence in the crystals of the 
 diamond of a lamelliform structure similar to that which imparts 
 to some unquestionably holo-symmetrical Cubic minerals, such as 
 magnetite, steinmannite and cuprite, re-entrant octahedral edges 
 and grooved surfaces on the planes of the rhomb-dodecahedron, 
 somewhat similar to those which are met with in octahedra and 
 rhomb-dodecahedra on the diamond. 
 
 In cases in which a hemi-symmetrical crystal is twinned on a 
 systematic plane, the normal of which would be at least a dia- 
 symmetral axis were its symmetry not in abeyance, it results 
 that systematic planes of each group as well as equivalent lines 
 will present parallelism in the two crystals. It will also be seen 
 that any one of the systematic planes of the group to which the 
 
 * In the British Museum there are two diamonds in the form of hexakis- 
 tetrahedra, and two others that are hexakis-tetrahedra with faces of the tetra- 
 hedron. 
 
Supplementary twins. 181 
 
 twin-plane belongs may be itself regarded as the twin-plane. 
 Further, as stated above, the two crystals are mutually interpene- 
 trant ; sometimes, indeed, in a manner simulating the features of 
 holo-symmetrical crystals. 
 
 DEF. These combinations of correlative semiforms thus have 
 the character of parallel growths and were termed by Haidinger 
 supplementary twins. Certain correlative tetartohedral forms may 
 be twinned in a manner analogous to that of supplementary twins ; 
 but they will obviously only build up a constructive hemihedral form, 
 and the symmetry will be of the nature characterising that form. 
 
 161. There is a remarkable tendency in crystals to assume 
 by the aid of twin-composition a higher degree of symmetry than 
 that characterising the system to which they belong. And where 
 the twin-plane, as in the case of haplohedral crystals, does not 
 become a plane of symmetry, it will occasionally happen that 
 the instinct, so to say, for symmetry will be satisfied by a face 
 other than the twin-face, but perpendicular to it, being the plane 
 of junction ; this face becoming a plane of symmetry to the 
 composite structure. Formerly Professor Groth, in his definition 
 of twin-structure, virtually restricted it to ' symmetrical-twins^ in 
 which the individuals are symmetrically disposed with respect to 
 a plane which is not a plane of symmetry for either of them; 
 a definition the bearing of which will be considered in a future 
 chapter. 
 
 162. The twin-law, though true in the form in which it has 
 always been enunciated so far as concerns a representation of 
 the nature of the union of hemitropic crystals, appears to permit 
 of considerable divergence from precision in the relative orienta- 
 tion of the crystals subject to it ; the angles obtained by measure- 
 ment from faces on the different individuals giving in many cases 
 results which accord less exactly with those obtained by calculation 
 than is the case with the angles on the individual crystals. On 
 hemitrope crystals of albite, the twin-axis of which was in each 
 case normal to (oio), Des Cloizeaux * found variations in the 
 angle between a pair of faces theoretically parallel which amounted 
 to from 40' to i 40'. 
 
 * Man. de Mineralogie, t. i. p. 320. 
 
1 82 Determination of the twin-plane. 
 
 163. It results from the law of hemitropy that each pair of 
 corresponding faces on the two crystals lies in one zone with the 
 twin-plane and that the faces make equal angles with it. So that, 
 Fig. 63, if P, a pole on one of the individuals, correspond to P', 
 a pole on the other individual, and a second pole Q on the first 
 correspond to a pole (/ on the second, T being the pole of the 
 twin-face, TP = \ PP', TQ = \Q(?; 
 
 and if a pole L on either crystal can be found and determined 
 in the same zone with P and P', and a pole R on the same 
 crystal in a zone with Q and Q', the indices of T may be found, 
 since [QK] and [PL] are tautohedral in T. 
 
 If great circles be drawn through PQ and PQ', we have 
 cosP(/= cos TPcos T(? + sin TP sin T&cosPTQ', 
 cos PQ = cos TP cos TQ + sin TP sin TQ cos PTQ, 
 and thus cosPQ / = 2 cos TPcos TQ-cosPQ ; 
 an equation by means of which the angle between any two 
 faces belonging to the two individuals may be found. 
 
 In order to find the indices of the twin-plane where they are not 
 given directly by inspection or by the zone-rule from recognisable 
 faces, it is necessary in the first place to determine, by measure- 
 ments with the goniometer and by calculation, the position of a 
 face of the second individual as referred to the axes of the first. 
 
 Thus if (7, Fig. 64, be the pole of a face of the second individual 
 and Q, C, A be poles of known faces on the first, Q and Q' being 
 
Determination of the twin-plane. 183 
 
 recognisable as corresponding faces on the two individuals, the 
 angles Q'C and QfCA must be determined by measurement and 
 calculation, so that we have the values of the arcs QC and QC 
 and the angle QCQ', for QCQ'= QCA-QCA : from the triangle 
 QCQf we then obtain Qff, and therefore \QQf or QT, and 
 also CQ@. Hence, further, from the triangle TCQ, in which QC, 
 QT, and CQff are now known, the values of TC and TCA may be 
 found, since TCA = ACQ + TCQ. 
 
 From these two values the indices of T can be obtained by 
 methods which will hereafter be given for finding the indices of 
 a face from the requisite data under each system. 
 
 Fig. 64. 
 
 The values of Q'C and Q'C A can only be deduced from two 
 measured angles, unless it should happen that a zone can be found 
 on the first individual into which the pole (/ of the second 
 individual falls, in which case the measurement of a single angle 
 is sufficient; and where two zones can be found containing (/, 
 no measured angle is required. 
 
 164. The practical method to be pursued in the investigation 
 of a twin-crystal will thus be seen to consist, first, in determining 
 the elements of one of the crystals, which of course involves our 
 knowing those of both. Then, by having recourse to principles 
 of symmetry and to the features characteristic of the faces of 
 particular forms, we proceed to identify those faces which belong 
 to some one or more forms on both the crystals, and to deter- 
 mine the symbols of the faces which lie in zones common to the 
 two. Where a face can be recognised as belonging equally to 
 the two individuals, or when a face on one crystal is parallel to a 
 
184 Investigation of a twin-crystal. 
 
 corresponding face upon the other, this may be the twin-face ; or 
 otherwise it is a face perpendicular to the twin-face, and therefore 
 belongs to a zone the axis of which is the twin-axis. And often a 
 careful inspection enables us to recognise such a plane either 
 as the twin-plane or the plane of junction. Where these methods 
 of attacking the problem wholly or partially fail, recourse must 
 be had to others which may be suggested in each case by 
 the character of the crystal itself, and which will depend on the 
 principles laid down in the last article ; but then especially must 
 measurements be made of all zones containing faces of both 
 crystals. A line traversing a face may be the trace of a plane 
 of junction of two crystals, and often some physical character 
 such as striation evinces differences on either side of the line, and 
 perhaps also a symmetrical disposition in respect to it, thus serving 
 as a guide to the discovery of the twin-plane. And the position and 
 nature of re-entrant edges at one part of the structure, and of faces 
 on another part combined in accordance with a higher symmetry 
 than that which governs an individual crystal of the substance, 
 may help in the solution of the problem. In the case of a 
 transparent crystal, polarised light will often lend invaluable aid 
 in establishing the existence of a composite structure, by shewing 
 in a section or cleavage fragment, or even in an entire crystal, 
 dark and light or differently tinted bands, representing regions in 
 which the directions of the optical principal sections are not the 
 same. 
 
 165. Many crystals belonging to the Ortho-rhombic and Mono- 
 symmetric systems fall crystallographically into one of two 
 groups in which the angles between the faces of the prevalent 
 prism or of a dome have values which approximate either to a right 
 angle, or to an angle of 60. And such crystals, and particularly 
 those of the latter group, often simulate, in the twin-structures 
 they build up, the character of crystals endowed with much higher 
 symmetry than their own. 
 
 Thus copper-glance, aragonite, witherite, and other ortho- 
 symmetrical carbonates isomorphous with the latter minerals, occur 
 as groups of crystals which are constructed by a repeated twinning 
 on the prism plane : and they present a symmetry closely resembling 
 
Simulation of higher symmetry. 
 
 185 
 
 that of a hexagonal crystal. Fig. 65 represents a triplet (or 
 triple twin) of chrysoberyl (alexandrite), the character and relative 
 orientation of the individual crystals that compose it being shewn 
 in the three smaller crystals grouped around it. The lines / x / 2 are 
 the twin axes, 7J T 2 the traces of the twin-planes on which the 
 
 Fig. 65. 
 
 crystal A is twinned so as to assume positions parallel to those 
 exhibited by the figures I and II respectively. 
 
 Faces of the dome { 101 }* are taken in the fig. as the twin-planes : 
 the faces of this dome present the normal-angles i o i on To i = 60 1 4', 
 101 on 100 = 59 53'; tne faces oio of the different individuals 
 
 * The above was Kokscharow's interpretation of these triplets. Though, 
 offering a more simple illustration of the simulation of hexagonal forms, it 
 is possibly less correct than the view of Hessenberg, as lately confirmed by 
 Cathrein, viz. that a face of the form {301} is both twin- and junction-plane 
 (Groth's Zeitschr. 1882, p. 257). 
 
1 86 Simulated symmetry of leucite. 
 
 are perpendicular to those of {101}, and unite to form a plane 
 common to the crystals of the group. The striations of these 
 faces parallel to the edges [oio, 100] of the three individuals 
 intersect in a feathered pattern on this plane. 
 
 166. Leucite offers another striking illustration of the simulation 
 by twin-crystals of a symmetry higher than that of the simple 
 individual. Long treated as a typical crystal of the Cubic system 
 in which it represented the form {211}, (a variety of icosite- 
 trahedron formerly termed the leucitohedron), it was removed 
 by the researches of Prof, vom Rath to the Tetragonal system, 
 the crystals being viewed by him as twins in which two individuals 
 presenting combinations of an isosceles (tetragonal) octahedron 
 {111} and a scalene dioctahedron {421} are twinned on a 
 plane parallel to a face of the form {201}, which is also the 
 plane of union. The incomplete and anomalous character of the 
 cleavage, a decided action on polarised light not easily explained 
 in a cubic crystal, and pecularities in the striations on the faces, 
 some of which occasionally exhibit a re-entrant angle, had pre- 
 viously induced doubt as to leucite being correctly ascribed to 
 the Cubic system. M. Mallard has, subsequently to vom Rath's 
 researches, further investigated crystals of leucite and as a con- 
 sequence removes the symmetry of this mineral still further from 
 that of the Cubic system, the result of his investigation going to 
 shew that the crystals are mono-symmetrical, but capable of being 
 referred to an axial system not widely removed from that of a 
 cubic crystal. And in the same memoir M. Mallard has con- 
 siderably extended the number of minerals in which he recognises 
 the simulation of high symmetry by twinned crystals belonging to 
 systems of comparatively simple symmetry. Among them is 
 boracite, a mineral the forms and angular measurements of which 
 are entirely in accord with the symmetry of a crystal of the Cubic 
 system of haplohedral type: the cube being usually the most 
 prominent form ; and of this form the quoins are truncated by 
 octahedral faces, of which the faces o of the form cr { 1 1 1 } are 
 large and bright; while the faces co of the form o-jln} are 
 either absent or small and of no lustre, and acquire an electrical 
 potential opposite to that of the faces o under changes of tern- 
 
Case of boracite. 187 
 
 perature ; for, with a falling temperature down to 225C., the 
 smooth faces are positive, and then become negative till the tem- 
 perature falls to about i2oC., when they again become positive 
 until the ordinary temperature is reached: the rough tetrahedral 
 faces exhibiting inverse phenomena. M. Mallard, guided in the 
 case of boracite entirely by the phenomena presented by sections 
 of a certain tenuity when examined both in parallel and in diver- 
 gent polarised light, arrived at the conclusion that, notwithstanding 
 their cubic haplohedral symmetry and the apparent accord of their 
 measured angles with those of a cubic crystal, they are complicated 
 structures in which twelve orthorhombic crystals are united into a 
 single pseudo-cubic combination ; each individual being a pyramid 
 with its apex at the centre of the crystal and its base conterminous 
 with a face of a pseudo-rhomb-dodecahedron. 
 
 The advance of crystallography has, indeed, for some time been 
 marked by the gradual removal to lower types of symmetry of 
 crystals previously assigned to systems of higher symmetry. And 
 the tendency of exact investigation seems in the direction of con- 
 tinuing this process of removal ; as well by establishing the existence 
 of slight variations from what had previously been deemed to be 
 rectangularity of the axes or equality of parameters, as also by as- 
 serting a more or less elaborately composite structure in crystals in 
 which heretofore only simple forms and an almost typical repre- 
 sentation of high types of symmetry had been recognised. 
 
 Recent investigations by Mallard and Klein respectively have 
 led to the discovery that the optical characters of boracite and 
 leucite undergo a sudden change at a temperature which, in the 
 former case, is about 265C., and, in the latter, is probably lower 
 than the melting-point of zinc : above these critical temperatures 
 the optical behaviour is in each case that of a cubic crystal. In 
 the passage of boracite through the critical temperature there is an 
 absorption or emission of heat amounting to 4-77 thermal units. 
 Klein considers that the peculiar optical behaviour of these minerals 
 at ordinary temperatures is of a merely secondary character, and 
 a consequence of the contraction of the crystals on cooling. 
 
CHAPTER VII. 
 
 THE SYSTEMS. 
 SECTION I. The Cubic (or Tesseral) System. 
 
 A. Holo- symmetrical Forms. 
 
 167. THE symmetrical characters of a form belonging to the 
 Cubic system have been already considered and the systematic 
 triangle determined (Articles 125, 126, and 127). 
 
 The distribution of the poles of the general scalenohedron, the 
 tetrakis-hexahedron, and the cube is shewn on Fig. 66. Those 
 of the octahedron, the rhomb-dodecahedron, and of the two 
 remaining forms, the triakis-octahedron and icosi-tetrahedron, 
 are exhibited on Fig. 75, Article 173. Fig. 66 \L represents a 
 systematic triangle of the system metastrophic with that con- 
 taining thevpole hkl\ Fig. 66 a a systematic triangle antistrophic 
 to the former. 
 
 It is only requisite to state, in recapitulation, that a crystal 
 belonging to this system, if it be holo-symmetrical, will exhibit 
 symmetry to two groups of systematic planes, see Fig. 50, p. 145 ; 
 viz. to three proto-systematic mutually perpendicular planes S, and 
 to six deutero-systematic planes 2 ; which latter planes intersect 
 
 1. with each other in triads, in four trigonal axes o ; 
 
 2. with each other in pairs, perpendicularly, and with the planes 
 
 S in pairs at 45, in the three tetragonal axes h ; 
 
 3. and each individually, perpendicularly, with a plane S, there 
 
 being one such intersection in each of the six ortho- 
 symmetral axes d. 
 
 The letters h, o, d indicate at once the axial points for these axes 
 and the angular points of the forty-eight systematic triangles into 
 
Holo symmetry. 
 
 189 
 
 which the sphere of projection is divided by the planes S and 2; 
 and they also coincide with the poles 
 
 /i, of the three axial planes ; i.e. of the proto-systematic planes S 
 parallel to the faces of the regular hexahedron or cube 
 {100}, 
 
 o, of the parametral planes parallel to the eight faces of the 
 regular octahedron {in}; and 
 
 d, of the planes, also parametral, parallel to the twelve faces of 
 three similar and concurrent square prisms, which together 
 constitute the form {no}, the rhomb-dodecahedron. 
 These viewed as origin-planes are the deutero-systematic 
 planes 2. 
 
 The origin-planes indicated by the poles o are not symmetral 
 planes ; wherefore the symmetry on the axes o will only be tri- 
 gonal or ditrigonal, and not hexagonal. 
 
 The symbols and mode of distribution of the forty-eight poles 
 
190 
 
 Cubic system. 
 
 of the general scalenohedron {hkl} have already been discussed. 
 They are contained in the subjoined 
 
 TABLE A. 
 
 7 
 viii 
 
 III 
 
 hkl hkl hkl hkl 
 Ihk Ihk Ihk Ihk 
 klh Ylh klh kill 
 
 (hlk Jiik liik hTk 
 
 \khl Yhl "khl kill 
 \lkhTkh Ikh lYh 
 
 hkl hkl hkl hkl 
 Ihk Ihk Ihk Thk 
 
 ~kih klh kil Ylh 
 
 hlk hlk hl~k ~hTk 
 "khl kill khl Yhl 
 Ikh Ikh Ikh Tkh 
 
 all 
 
 where the poles in each column belong to a single octant and the 
 blocks marked a contain the symbols of faces antistrophic to the 
 faces of which the symbols lie in the blocks indicated by /x 
 
 The six remaining forms are those of which the poles lie on 
 one or other of the three sides of the systematic triangles, and 
 those of which the poles lie at the angles of these triangles. They 
 will be represented by symbols in which one or two of the indices 
 are zero, or in which two of the indices are equal; and in the 
 latter case the two equal indices may be greater or may be less 
 than the third, that is to say, they may be represented as {Ihh} or 
 {hkk}. In the parametral octahedron all the indices are equal. 
 
 The following are the seven holo-symmetrical forms of 
 this system : 
 
 1. {hkl}, the hexakis-octahedron or forty-eight scaleno-hedron. 
 
 2. {Ihh}, the triakis-octahedron or the octahedrid pyramidion. 
 
 3. {hkk}, ihe icositetra-hedron or twenty-four deltohedron. 
 
 4. {hko}, the tetrakis-hexahedron or the cube-pyramidion. 
 
 5. {no}, the rhomb-dodecahedron or twelve rhombohedron. 
 
 6. {in}, the regular or equilateral octahedron. 
 
 7. {100}, the cube or square hexahedron. 
 
 The names of these figures are more concisely expressed by 
 the terms derived from the Greek numerals : the alternative 
 English names here proposed for them have the single advantage 
 of representing the contour of the figures by recalling certain 
 
Holo symmetry. 191 
 
 characteristic features : the term pyramidion being employed in 
 the case of forms in which a pyramidion or small pyramid 
 composed of similar isosceles triangles surmounts every face of 
 a simpler figure, the faces namely of a cube, octahedron, or 
 dodecahedron : such a figure is then an isoscelohedron. An 
 isoscelohedron, scalenohedron, deltohedron, rhombohedron, &c. 
 are figures formed severally of similar and congruent (but not 
 necessarily metastrophic) isosceles or scalene triangles, deltoids, 
 rhombs, &c. : an equilateral octahedron is the regular octahedron 
 bounded by equilateral triangles. For brevity, the term ' faced ' in 
 e.g. the forty-eight-faced scalenohedron, twenty-four-faced delto- 
 hedron, &c., usually employed in the nomenclature of these forms, 
 is omitted since no confusion can arise from this adjectival use of 
 the numbers. 
 
 It is obvious that names thus employed to represent some of 
 the characteristics of a crystallographic figure can really only 
 strictly describe it as an ideal figure constructed in equipoise ; 
 a figure that can have only an exceptional occurrence in nature. 
 
 If we bear in mind however the special characteristics of 
 crystallographic symmetry, it will be seen that this is the simplest 
 mode of considering a crystal-form; and consequently it will be 
 employed in the following descriptions, in which the forms will 
 be discussed in the order of the simplicity of their indices. 
 
 168. The square hexahedron or crystallographic cube {100} con- 
 sists of the six faces parallel to the three axial or proto-systematic 
 planes S, the normals to which are the crystallographic axes and 
 the tetragonal axes of symmetry h. The three planes -S 1 being 
 similar, the pairs of faces parallel to them will be simultaneous in 
 their occurrence and similar in their geometrical and physical 
 characters. 
 
 The resulting form, see Fig. 69, when in equipoise, is the cube 
 of geometry, a 'regular solid/ with equal squares for its faces; 
 though as a crystal-form the faces are frequently rectangles. It 
 has twelve crystallographically similar edges, the angle of each 
 being 90. And it has eight similar three-faced quoins, the edges 
 of which meet at right angles, and which are symmetrical to the 
 trigonal axes of symmetry o. 
 
Cubic system. 
 
 The symbols of the six faces of the form are 
 
 IOO OIO OOI 
 
 Too oYo ooT. 
 
 Among minerals presenting the cube form, galena (lead sulphide) 
 is remarkable for the facility of its cleavage parallel to the 
 faces of this form. The metals gold, silver and copper, com- 
 mon salt (NaCl) and sylvine (K Cl), fluor spar, cuprite, 
 occasionally diamond, are among the minerals which present 
 themselves as simple cubes. Pyrites also occurs in cubes, but 
 
 Fig. 67. 
 
 Fig. 68. 
 
 Fig. 69. 
 
 the striations often carried by the crystals prove them to be 
 mero-symmetrical. See Fig. 55(2), p. 165. 
 
 169. The equilateral or regular octahedron {in}, Fig. 67. 
 The equilateral octahedron is the parametral form of the 
 system. The intercepts of its faces are equal on all the axes ; 
 and the eight faces are equilateral triangles when the form is 
 in equipoise. It then represents the regular solid from which it is 
 named. 
 
 The twelve edges of the form are all similar and have the 
 dihedral normal-angle of the regular octahedron, viz. 
 
 70 31-7' = 2 arc H. 
 
 The distance of two alternate poles of the form, i. e, of poles 
 lying in attingent octants or measured over a quoin, is the arc 
 
 2 D = 109 28- 3'. 
 
 The poles of the faces are the axial points o of the tri-symmetral 
 axes, and its six quoins are symmetrical to the tetragonal axes h 
 (in which lie the poles of the cube). The quoins are consequently 
 formed by the meeting of four faces with plane angles of 60, 
 or of four edges of which the normal-angles are each 70 31-7'. 
 
Holo symmetry. 193 
 
 The symbols of the eight faces of the octahedron are 
 in In ill nT ill Til Hi TIT. 
 
 Each edge of the cube subtends an arc formed of two con- 
 secutive arcs H of a zone-circle 2, and each joins two poles of 
 the octahedron : and, similarly, the edges of an inscribed octa- 
 hedron join the poles of the cube; the quoins of the latter 
 figure may thus be truncated by the faces of the octahedron, 
 as may the quoins of the octahedron by the faces of the cube. 
 The octahedron is represented as an isolated form by copper, 
 silver, and by argentite, cuprite, senarmontite, fluor, spinel, 
 chromite, periclase, magnetite, pyrochlore, franklinite, and, very 
 rarely, the diamond. 
 
 170. The rhomb-dodecahedron {no}, Fig. 68. 
 
 A face with the symbol no will evidently be a parametral 
 face parallel to the axis Z, and since the axis Z coincides with 
 one of the tetra-symmetral axes h, the face will be repeated in 
 four faces, no Iio ilo ITo, forming a square prism. 
 
 Similarly, the faces 101 Toi lol YoT will constitute a dome 
 form or horizontal prism parallel to the axis Y, and the faces 
 on oil oil oil a dome parallel to the axis X. And since 
 the axes X, Y, and Z are identical with the tetragonal axes h. 
 and are similar to each other, the three square prisms concur in 
 building a form with the twelve congruent rhombic faces 
 no Tio ilo TTo 
 101 Yoi loY YoY 
 on oYi oiY oYY. 
 
 And the axial points d of the axes of ortho-symmetry are at 
 once the poles of the deutero-systematic planes 2 and of the 
 faces of this form {no}. 
 
 Two contiguous poles d not lying on the same great circle S 
 will be symmetrically situate with regard to an arc D of one 
 of the great circles 2. See projection. Fig. 50, Article 125, 
 
 p. 145- 
 
 The edge D of the corresponding faces will therefore lie in 
 the plane of this great circle 2, and will subtend an arc of it which 
 is the side D of a systematic triangle. 
 
 O 
 
194 Cubic system. 
 
 These edges D will be twenty-four in number, and identical 
 in feature. Contiguous faces with poles on the same great circle 
 S will meet with two other contiguous faces with poles on a 
 great circle S perpendicular to the former, in a quoin symmetrical 
 to a tetragonal axis h. And the three contiguous faces sym- 
 metrical to an axis o will meet on that axis in a trigonal quoin. 
 
 The twenty-four edges of the dodecahedron will therefore 
 meet in six four-faced quoins h, formed by identical acute plane 
 angles, and in eight three-faced quoins o, of which the plane angles 
 are also identical but obtuse. 
 
 The normal-distance of a pole d from a pole h being 45, 
 that between two adjacent poles d on different great circles S, 
 for instance between oil and Tio (see projection in Fig. 75), 
 is obtained by the equation 
 
 cos (oiT, Tio) = cos 2 45 = cos 60 : 
 
 the edge D has therefore a normal-angle of 60. 
 
 And as in the zone-circle [oiT, Tio] there lie the four other 
 poles To i oTi i To and loT, distant 60 from each other, and 
 as the edges of the faces corresponding to these six poles will 
 be parallel to each other and to the zone-axis [ni], it follows 
 that if either of the trigonal axes o of the system, e. g. the axis 
 {in}, be placed vertically, the rhomb- dodecahedron presents 
 the character of a hexagonal prism parallel to that axis, and ter- 
 minated by the faces of a rhombohedron which in the case assumed 
 has the faces 
 
 at the upper end no 101 on 
 
 at the lower end TTo ToT oTT. 
 
 Lines joining contiguous trigonal quoins o will give the edges of 
 a cube. Hence of such an inscribed cube the edges are the short 
 diagonals of the rhomb-dodecahedron in the figure. And the dia- 
 gonals of the faces of this cube are evidently each equal and parallel 
 to a pair of the longer diagonals in the same rhomb-dodeca- 
 hedron. So that the ratio of a shorter to a longer diagonal of a 
 rhomb-dodecahedron = the ratio of the side to the diagonal of a 
 
 square = sin 45 = - : . 
 
Holo symmetry. 195 
 
 Whence for the plane angles of the rhomb, 
 
 o hi 
 
 since cotan - = tan - = 5 
 
 2 2 J 2 
 
 the angle h = 70 31 -7', and 
 
 the angle o = 109 28 -3', 
 
 which are also the normal- distances of two faces of the octahedron 
 in adjacent and attingent octants respectively. 
 
 It will be seen from the position of the edges of the dodeca- 
 hedron that its quoins are truncated by the faces of the cube 
 and octahedron, and that its faces in turn truncate alike the edges 
 of the octahedron and those of the cube. 
 
 The rhomb-dodecahedron is of frequent occurrence in com- 
 bination with other forms, but is more rare in an isolated condition : 
 it is however the habitual form of some varieties of garnet, 
 sodalite, as well as of (haiiyne or) lapis lazuli ; while the minerals 
 blende (Zn S) and lapis lazuli (or haiiyne) present cleavages in six 
 directions parallel to the faces of this form. Diamond, gold, and 
 electrum are among the substances that occur in rhomb-dode- 
 cahedra. 
 
 171. Of the three simple forms hitherto considered, the faces 
 are severally perpendicular to the three sets of axes of symmetry 
 of the system, two or more zone-circles of symmetry being tauto- 
 hedral in their poles. 
 
 The forms next to be considered are those of which the 
 poles lie upon the several sides of the systematic triangles; the 
 form {hko} having its poles on the sides O of the systematic 
 triangles, while those of the form {hhl} lie on the sides H and 
 those of {hkk} on the sides D of those spherical triangles: the 
 edges of the faces of the several forms will consequently lie in 
 the sectors of each zone-plane corresponding to the arcs O, H, or 
 D on which their poles do not lie. 
 
 172. The ietrakis -hexahedron, or the cube-pyramidion [hko], 
 Figs. 70-2. If the poles of the form lie on the arcs O, its faces 
 will form zones of which the tetragonal axes h are the zone-lines, 
 and they will have a zero for the first, the second, or the third 
 
 O 3 
 
196 
 
 Cubic system. 
 
 index of their symbol, according as the zone-plane in which the 
 pole lies is the plane S 3 , or S 2 , or S lt Fig. 50, p. 145. 
 
 In any zone-circle S, a pole of the form {hko} lies on the 
 
 Fig. 70. 
 
 Fig. 71. 
 
 arc separating every two adjacent poles of the forms {100} and 
 {no}; eight such poles lying in each zone-circle. The form 
 will therefore have twenty-four faces symmetrical in pairs to the 
 
 poles of the forms {100} and 
 { 1 10} ; the edges of either of these 
 latter forms would thus be bevilled 
 by faces of the form {hko}. 
 
 The figure presents the aspect of 
 a cube each face of which is sur- 
 mounted by an obtuse pyramid, 
 and it may, on this account, be 
 termed the cube-pyramidion. Since 
 the faces are euthy- symmetrical to 
 a plane S, the triangles that form 
 them are crystallographically iso- 
 sceles, and the figure is a twenty-four-faced isoscelohedron. The 
 thirty-six edges of the form are of two kinds, viz. twelve similar 
 edges H formed by faces which are symmetrical to and meet 
 in the H sectors of the planes 2, and which are therefore 
 coincident in direction with the edges of the cube, and twenty- 
 four edges D that are similar and are formed by faces symmetrical 
 to the D sectors of the zone-planes 2. These D edges meeting 
 in fours form the six terminal quoins of the pyramidions 
 
 Fig. 72. 
 
Holosymmetry. 197 
 
 of the figure symmetrical to the tetragonal axes h; while, 
 of the H edges parallel to the edges of the cube, three, alter- 
 nating with three of the D edges, meet with them to form a six- 
 faced trigonal quoin o and of these quoins there are eight. 
 The symbols of the faces of a form {hko} are arranged in 
 
 TABLE B. 
 
 hkQ hko lik hko 
 
 kho ~khv ~khQ kho 
 
 ohk ohk Qhk ohk 
 
 okh okh okh okh 
 
 koh koh koh koh 
 
 hok hok hok hok 
 
 where h and k may be any integer numbers. The distribution of 
 their poles on the sphere is seen in the projection, Fig. 66, 
 Article 167. 
 
 In the cases of known crystals, the varieties of the cube- 
 pyramidion include the following forms, 
 
 510, 410, 310, 520, 210, 320, 430, 540. 
 
 These are more or less frequent on several crystallised bodies ; 510, 
 for instance, on cuprite, 410 on silver and on gold, 520 on copper 
 and on fluor, 320 on blende, 430 on perowskite, while 310 is a 
 common variety of pyramidion. But the form {210} is by far the 
 most frequent in occurrence, and presents the remarkable property 
 that its H edges have the same dihedral angle as its D edges ; 
 see Fig. 70. In this case it will be proved in Chapter VIII that 
 
 cos H = cos D - j 
 5 
 
 and U= Z>= 3 652'ii-7". 
 
 173. The icositetrahedron^ or twenty -four deltohedron [hkk}> Figs. 
 73-4. This form has its poles situate on the arcs D of the 
 systematic triangle. The edges consequently lie in the H and O 
 sectors of the 2 and planes, and they are of two kinds, viz. twenty- 
 four edges H and twenty -four edges 0. Of the twenty-six quoins, 
 six are four-faced quoins h symmetrical to the tetragonal axes h, 
 
198 
 
 Cubic system. 
 
 eight three-faced quoins o symmetrieal to the trigonal axes o, 
 twelve four-faced quoins d ortho-symmetrical to the axes d. 
 
 The twenty-four faces are euthy-symmetrical to the traces of 
 the sectors D, and have thus the form of symmetrical trapezia or 
 deltoids. 
 
 Fig- 73- Fig- 74- 
 
 The icositetrahedron occurs rarely as an isolated form. Anal- 
 cime, indeed, presents such crystals with the symbol {211}, and 
 the attribution of analcime to the Cubic system, though it has been 
 disputed, is now recognised *. The form {211}, Fig. 73, which 
 is the commonest variety, is met with in a large number of 
 crystals, replacing by its trigonal quoins those of the cube, and 
 occasionally replacing the quoins of the octahedron by its tetra- 
 gonal quoins h. Fluor, garnet, cuprite, and galena are among the 
 minerals that exhibit the form {211}. 
 
 This particular form {211} was termed by Haidinger the 
 leucitohedron, and the general form {hkk} the leucitoid, from 
 the crystals of leucite having been, though erroneously, attributed 
 to this form. The character of these crystals has already been 
 discussed in Article 166. 
 
 This variety of the icositetrahedron is however an important one, 
 not only from its frequent occurrence in combination with other 
 forms, but also from the circumstance that its poles lie in the edge 
 zones, i. e. in the zone-circles perpendicular to the edges, of the 
 rhomb-dodecahedron, so that its faces truncate the edges of that 
 
 * See Arzruni and Koch (Zeitsch. f. Kryst. 1881, p. 483), and Ben Saude 
 (Jahrb. f. Min. 1882, p. 41). 
 
Holosymmetry. 
 
 199 
 
 figure. Thus the two adjacent poles of the dodecahedron (no) 
 and (101) lie in the zone [In]; whence the poles lying on this 
 zone-circle fulfil the condition h + k + l o, or h = k + l, which 
 in the case of the icositetrahedron becomes h = zk and the symbol 
 is therefore {211}. 
 
 Another important icositetrahedron {311}, Fig. 74, occurs on the 
 cubic native metals, on galena, pyrites, fluor, magnetite, and spinel, 
 generally replacing the quoins of the cube by its trigonal quoins o. 
 
 The symbols of the faces of a form {hkk} are arranged in the 
 following 
 
 TABLE C. 
 
 hkk "hkk hk~k hkk hkk hkk hkk h~kk 
 khk Ilk khk khk ~khk khk khk ~kh~k 
 kkh Ykh ~kkh kkh ~kkh kkh kkh ~k~kh. 
 
 iTo 
 
 tfo 
 
 Fig. 75- 
 
 The distribution of the poles of the form {211} on the sphere 
 is shown in the projection, Fig. 75. 
 
 174. The triakisoctahedron, or odahedrid pyramidion {hhl}> Figs. 
 
2oo Cubic system. 
 
 76-7. The poles of this form lie on the arcs H, and its edges con- 
 sequently in the sectors D and O : twenty-four in the former, 
 and twelve longer than these in the latter. Hence the twenty-four 
 faces are isosceles triangles, euthy-symmetrical to the traces on 
 them of the sectors H. 
 
 The figure will have eight trigonal quoins o. 
 ,, six ditetragonal quoins h. 
 
 Its aspect is that of an octahedron, each of the faces of which 
 forms the base of an obtuse pyramid, and the more acute the 
 pyramidion the more nearly it approximates in aspect to a rhomb- 
 dodecahedron. 
 
 Fig. 76. Fig. 77. 
 
 Hence it is a twenty-four-faced isoscelohedron, and may be 
 termed the octahedrid pyramidion. The form {221} exists on 
 the diamond, and this form, as also the form {331}, occurs asso- 
 ciated with the cube, or bevilling the edges of the octahedron, 
 on galena, and occurs also on argentite, spinel, fluor, magnetite, 
 franklinite, and pharmacosiderite. 
 
 It is easily seen that the deltohedron which would truncate 
 the edges of the form {211} must be that with the symbol {332}. 
 In fact the addition of the symbols (211) and (121), which are 
 those of adjacent faces symmetrical to the 2-plane [ooi, no], 
 gives the symbol (332) of the face truncating the edge of the 
 former faces. The form occurs on garnet. 
 
 The symbols of the faces of the form {hhl} are given in 
 
Holosymmetry. 201 
 
 TABLE D. 
 
 hhl III hhl hll hhl hll hhl 111 
 Ihh Yhh Ihh ill Ihh fhh Ihh 111 
 hlh Jilh III hll llh hlh hll HI. 
 
 The distribution on the sphere of the poles of the triakis- 
 octahedron {221} is shown in Fig. 75. 
 
 175. There only remains to be considered the hexakis- 
 octahedron, or forty-eight scalenohedron {hkl}, Figs. 78-80. 
 
 This form has received numerous designations, for the most 
 part recalling the characters of the different figures which result 
 from the indices receiving more or less widely differing relative 
 values. Such are the terms octakishexahedron, hexakisocta- 
 hedron, tetrakisdodecahedron, which may be represented in an 
 English form as the eight-on-six, six-on-eight, and four-on-twelve 
 scalenohedron; and they in fact indicate the different aspects 
 the form assumes, as it approximates to a cube with an eight- 
 faced, an octahedron with a six-faced, or a dodecahedron with 
 a four-faced pyramid on every face. In general these figures 
 cannot be legitimately designated as true pyramidions, since the 
 bases of the pyramids do not actually coincide with the faces 
 of the cube, octahedron, or dodecahedron, respectively : in cases, 
 however, where the indices present the ratio h = k -f- / the basal 
 edges coincide in position with those of the dodecahedron, and 
 the figure, then, is a true dodecahedrid pyramidion. 
 
 The most general designation is the tetrakontaoctahedron (or, 
 forty-eight-faced form); the most usual term is the hexakis- 
 odahedron; the simplest designation for this, the most complex 
 of crystallographic forms, is the forty-eight scalenohedron. Each 
 of the forty-eight systematic triangles contains one pole of the 
 form ; in the case of some of the more frequent varieties of the 
 form the poles lie either on great circles passing through a pole 
 of the octahedron and bisecting the angles of inclination of 
 two adjacent great circles 2, or on great circles passing through 
 two adjacent poles of the rhomb-dodecahedron, situate sym- 
 metrically in regard to a plane S. 
 
 The edges in which a face of the form meets the three faces, 
 
2O2 Cubic system. 
 
 adjacent to it will lie in the three systematic planes containing 
 the sides of the systematic triangle; and the faces themselves 
 will be crystallographically scalene triangles. Hence there will 
 be three sorts of edges, namely, twenty-four edges H and twenty- 
 four edges D, which alternating in triads with each other form 
 eight ditrigonal quoins o, and twenty-four edges 0, which alter- 
 nating in fours with four of the D edges form six ditetragonal 
 quoins h, and again alternating in pairs with pairs of the H edges 
 form twelve ortho-symmetrical quoins d. 
 
 The symbols of the faces of a form {hkl} have been set out in 
 Table A, and the distribution of its poles on the sphere is seen in 
 the projection, Fig. 66, Article 167. 
 
 In considering the particular variety of this form, of which 
 the poles lie on great circles bisecting the angles formed by 
 each pair of great circles 2, it will be seen that poles lying on 
 such intermediate great circles correspond to those which in a 
 Rhombohedral crystal may occur on the great circles of the 
 deutero-systematic planes of the Hexagonal system. In the 
 Rhombohedral type of that system the symmetral character of 
 these deutero-systematic planes is in abeyance, and the poles 
 lying on their great circles belong to forms of trigonal type as 
 they are constrained to do in the Cubic system by the conditions 
 of its symmetry. 
 
 The symbol of a form of this kind in the Rhombohedral system 
 
 would be Ti{hil}, where i ; and that this relation between 
 
 the indices must hold good in the Cubic system also may be 
 shewn by taking the value for the cosine of the arc between two 
 poles in this system, as will be deduced in Chapter VIII, Section 
 2. Thus, for the adjacent poles hkl and hlk meeting in an 
 edge D, it will hereafter be shewn that we have 
 
 and for the poles hkl and khl meeting in an edge H, we have 
 
 Whence the condition for the edges D and H to have the same 
 
Holo symmetry. 
 
 203 
 
 dihedral angle is h 4- / = 2 k. Forms fulfilling this condition 
 would be {321}, {531}, {543}, &c. 
 
 The other noticeable variety of the forty-eight scalenohedron 
 is that the poles of which lie in the edge-zones of the rhomb- 
 dodecahedron, the faces of which consequently bevil the edges of 
 that form. 
 
 We have already seen, Article 173, that the poles of the form 
 { 2 1 1 } lie in such an edge-zone-circle, its faces truncating the edges 
 
 '12 
 
 Fig. 78- 
 
 Fig. 79. 
 
 of the rhomb-dodecahedron: and it was also shewn in Article 
 173 that the indices of a plane lying in one of these zones must 
 fulfil the condition that one of them equals the sum of the other 
 two. The forms {321}, {532}, {431} are among those which 
 belong to this variety of scalenohedron; the form {321}, Fig. 78, 
 is thus remarkable as being common to both the mentioned 
 varieties. 
 
204 
 
 Cubic system. 
 
 Since the faces of any form of the latter variety, e. g. those of 
 the form {431} in Fig. 79, bevil the edges of the rhomb-dodeca- 
 hedron, the complete form has the character of a pyramidion 
 developement of the rhomb-dodecahedron, each face of the latter 
 figure being surmounted by a rhomb-based pyramid, to which it 
 forms a conterminous base. These therefore are the forms that 
 may be correctly designated as tetrakisdodecahedra or dodecahedrid 
 pyramidions. 
 
 The manner in which the quoins and edges of the cube, 
 octahedron, and dodecahedron respectively are modified by asso- 
 ciation with the forms hitherto considered is exemplified in the 
 following figures. 
 
 Fig. 81. 
 
 Cubic System. B. Mere-symmetrical Forms. 
 
 176. In order that mero-symmetrical forms of the Cubic system 
 may satisfy the law of mero-symmetry in respect to the two groups 
 of systematic planes *S* and 2, it is not necessary that either 
 group retain its symmetral character, provided that the character 
 of the symmetry peculiar to the system be preserved in the 
 resulting forms. Where both groups thus fail in directly con- 
 
Mero symmetry. 205 
 
 trolling the symmetry of the forms under consideration they 
 in effect indirectly control it, inasmuch as the axes of symmetry 
 in such a case must continue to regulate the symmetry of these 
 forms. They can however do this, as far as the geometrical sym- 
 metry is concerned, only in the case of the general form of the 
 system, the forty-eight scalenohedron {hkl} : for there must 
 be six faces arranged round a trigonal axis and eight round a 
 tetragonal axis in order that, if half the faces are suppressed, 
 the remaining half may be grouped trigonally in the one case and 
 tetragonally in the other. 
 
 Where only one group of systematic planes is operative as a 
 group of actual planes of symmetry, the character of the resulting 
 forms will be entirely different according as the group is that 
 of the proto-systematic planes, the zone-axes of which are tetra- 
 gonal in their symmetry, or is the deutero-systematic group, the 
 planes of which intersect in threes, in zone-axes that are axes of 
 trigonal symmetry. 
 
 If a group of six faces of the scalenohedron lying in one octant 
 be considered, it will be evident either that all the six faces must 
 be suppressed simultaneously, or that only those three can be so 
 which, lying in alternate systematic triangles, are metastrophic to 
 each other. In the former case the 2 or deutero-systematic 
 planes may be actual planes of symmetry, in the latter case they 
 can only be potentially such. And if an alternate suppression 
 take effect in one octant, a similar suppression must take effect in 
 every octant in the case of a semiform. 
 
 In the case however where all the six faces in one octant 
 are suppressed, the only way in which the forty-eight scaleno- 
 hedron can be divided into two correlative semiforms is by all 
 the faces in any one octant, and in the three octants attingent 
 to it, being simultaneously present or absent. A form of this kind 
 is evidently haplohedral and holo-systematic, and its faces are 
 symmetrical to the planes of the 2 group only; its symbol will 
 be o- {/$//} or <r {/}. 
 
 Where, on the other hand, the alternate faces in a given octant 
 undergo suppression, the faces belonging to the systematic triangles 
 fx in, for example, the first octant, Table A, Article 167, and 
 
206 
 
 Cubic system. 
 
 Fig. 82, might be associated (as extant or as absent faces) either 
 with those belonging to the triangles //. or else with those be- 
 longing to the triangles a in the adjacent octants (ii, iv, and vi) and 
 in the opposite octant viii ; and in either case with the faces be- 
 longing to the triangles p in the attingent octants (iii, v, and vii). 
 The former case then, where all the absent or extant faces are 
 those belonging to the triangles p, yields 
 a haplohedral form ; the latter gives a 
 diplohedral form. The former will be 
 therefore holo-systematic, the latter hemi- 
 systematic in character. The latter kind 
 of partition divides the forty-eight-faced 
 form into two correlative semiforms, the 
 faces of each being symmetrical to the 
 three proto-systematic planes, and not 
 symmetrical to the six deutero-systematic 
 
 Fig. 82. 
 
 planes: their symbols are ir{hkl} and TT 
 
 The former kind of partition is that first considered, in which 
 the zone-axes h, o, and d operate as tetragonal, trigonal, and 
 diagonal axes of symmetry, while neither the S- planes nor 
 the S-planes are effective as planes of symmetry : the symbols of 
 the two semiforms being a {hkl} and a {Ikh}. 
 
 Hence the cases of mero-symmetry of the forty-eight scaleno- 
 hedron, of the three general twenty-four-faced forms of the system, 
 and of the octahedron, are the following : 
 
 I. Holo-systematic haplohedral forms; or holo-tesseral 
 hemihedra. 
 
 1. Asymmetric forms. The pentagonal icositetrahedron or 
 
 twenty -four-pentagonohedron, 
 
 a {hkl} and a{tkh\. 
 
 2. The tetyahedrid forms: 
 
 (a) a {hkl} and a- {hi:!} \ hexakis-tetrahedron. 
 
 (b) v{hhl} and a{hlil} : hemi-triakisoctahedron ; or 
 
 deltoid-dodecahedron, or twelve-deltohedron. 
 
Mer asymmetry. 207 
 
 (c) &{hkk} and 0- \hkk} : hemi-icositetrahedron ; twelve- 
 
 isoscelohedron, or tetrahedrid pyramidion. 
 
 (d) a- {111} and cr {III} : the hemi-octahedron or tetra- 
 
 hedron. 
 
 II. Hemi-systematic diplohedral forms: hemi-tesseral 
 diplohedra. 
 
 TT {hkl} and TT {Ikh} : the dyakis-dodecahedron, or twenty- 
 
 four-trapezohedron : the diplohedron. 
 
 Tt{hko} and ^{okh}: the pentagonal dodecahedron, or 
 twelve-pentagonohedron. 
 
 III. Hemi-systematic haplohedral forms: hemi-tes- 
 seral hemihedra; or, tesseral tetartohedra. 
 
 <m{hkl}: the tetrahedrid twelve-pentagonohedron, or the 
 tesseral tetartohedron. 
 
 The relations of these forms to one another and to the holo- 
 symmetrical forms from which they are derived are illustrated 
 in the stereographic projections of their poles on Plate I. 
 
 177. B. I. Holo-systematic haplohedral forms, i. The 
 asymmetric semiform. Of the hemi- symmetrical forms in which 
 the twenty-four normals of the general form {hkl} are repre- 
 sented, each by a single face, the first to be considered is 
 
 The pentagonal icosUetrahedron (or twenty-four-pentagonohedron), 
 a{hkl] or a {Ikh}, Figs. 83-4. 
 
 Here none of the systematic planes are planes of symmetry, and 
 the faces of which the poles lie in alternate systematic triangles are 
 alone extant. 
 
 The axes of symmetry however preserve the characteristics of 
 the system, but in the absence of planes of symmetry they do so 
 only by a gyroidal (or alternate) distribution of the poles. 
 
 The edges consequently are gyroidally grouped in triads G 
 round the trigonal axes o; and in tetrads V round the axes h, 
 while the axes d would intersect symmetrically a third sort of edge 
 Wj which again does not lie in any systematic plane. 
 
 It will be seen that the faces thus bounded are irregular 
 pentagons, with no edge lying in a systematic plane. Their 
 
208 Cubic system. 
 
 sides are formed of two contiguous edges G, two other contiguous 
 edges F, and a fifth edge W, altogether sixty in number ; namely, 
 twenty-four edges G, twenty-four edges F, and twelve edges W. 
 
 The quoins are thirty-eight in number; viz. the six tetragonal 
 quoins h in which four edges Fmeet, the eight trigonal quoins o 
 formed by the meeting of three edges G, and twenty-four asym- 
 metric quoins k in each of which an edge W meets an edge 
 V and an edge G. 
 
 The symbols of the faces of a form a {hkl} are contained in 
 the first and fourth blocks of Table A, Article 167, those of the 
 
 Fig- 83. 
 
 correlative form in the second and third blocks. Very few crystals 
 have been observed to present this type of mero-symmetry ( 188). 
 
 178. 1.2. Tetrahedrid mero-symmetry. The second case of holo- 
 systematic hemi-symmetry, in which every normal is represented 
 by a single face, is that in which the deutero-systematic planes 2 
 alone have a symmetral character. There being no centre of 
 symmetry to the system, and the proto -systematic planes not being 
 symmetral, the faces of the form are confined to four alternate 
 (or attingent) octants. 
 
 Since the number of normals belonging to the semiform is the 
 same as that belonging to the complete form with corresponding 
 symbol, it will be twenty-four where the indices h, k, and / are all 
 different, and twelve where two of them are equal, i.e. where the 
 poles lie on the arcs H or the arcs D of the systematic triangle. 
 
Merosymmetry. 
 
 209 
 
 The number of normals will be four where all these indices 
 are equal, and the resulting form will be constituted by four 
 alternate faces of the octahedron { 1 1 1 }. 
 
 In the cases in which the poles of the form lie either on a 
 zone-circle S, i.e. on an arc O, or at the angular points h or d 
 of the systematic triangles, none of the faces of the holohedral 
 form will be suppressed : they will however lose their crystallo- 
 graphic symmetry in respect to the planes S. Each face of the 
 cube will nevertheless continue to be symmetrical to its two 
 diagonals, as being the traces on it of two planes 2 ; and therefore 
 also, as in Fig. 55 (i), Article 147, to its normal h as an axis of 
 diagonal symmetry : and similarly, the faces of the dodecahedron 
 will retain their physical symmetry in respect to their shorter but 
 not to their longer diagonals. It is only in this sense, that is to 
 
 no o> 
 
 Fig. 85. Fig. 86. 
 
 say in the sense of a physical hemi-symmetry, that we can speak 
 of a cube, a rhomb-dodecahedron, or a cube-pyramidion pre- 
 senting tetrahedrid hemi-symmetry. 
 
 The tetrahedron or hemi-octahedron, o-{in} or o-jIII}, Figs. 
 85-6, is the simplest of the hemi-symmetrical forms belonging 
 to the section under consideration, and as an isolated figure can 
 only exist with equal faces. 
 
 In the tetrahedron the four alternate faces of the octahedron, 
 
 those namely in alternate octants, are suppressed. The resulting 
 
 semiform corresponds in all respects with the regular tetrahedron 
 
 of geometry. Bounded by four equilateral triangles, as an isolated 
 
 form it can only exist in equipoise. The symbols of its faces are 
 
 for the form o- {in}, in, Hi, III, ill, 
 
 for the form o- {III}, III, nl, ill, In. 
 
 P 
 
2io Cubic system. 
 
 The six edges of the figure will lie in the deutero-systematic planes 
 2, each edge being the trace on the faces forming it of two 
 consecutive D sectors of a 2-plane. 
 
 The four quoins are trigonal ; each axis o meeting a quoin o 
 or CD on one side, and the pole of a face co or o on the opposite 
 side of the origin. The normal-distance between two faces is 
 measured by two consecutive arcs D, and is therefore 
 2D = 109 28-3'. 
 
 Fahlore (or tetrahedrite), blende, eulytine, lead nitrate, and 
 a variety of garnet from Elba are among the substances that 
 present this form. Sometimes, as in helvine and in eulytine, the 
 two correlative tetrahedra concur, but the faces of the one form 
 are distinguished by being more largely developed than those of 
 the other. 
 
 Fig. 87. Fig. 88. 
 
 179. The hemi-triakisoctahedron, or twelve-deltohedron, v {hhl} 
 or <r \hhl}, called also the deltoid- dodecahedron, Figs. 87-8. 
 
 The tetrahedrid semiform of the triakisoctahedron [hhl] will 
 have three faces in each alternate octant corresponding in position 
 and direction but not in form with the three faces with the same 
 symbols belonging to the complete holohedral figure. Its poles 
 therefore lie on the sides H of the systematic triangles. 
 
 Each face will form two edges D with the other two faces in its 
 own octant, and two edges A with two faces lying in attingent 
 octants ; and each face being of course symmetrical to the plane of 
 
Mero symmetry. 211 
 
 the great circle 2 on which its pole lies, the form of the face will 
 be that of a deltoid. 
 
 If the edges D and A were similar the deltoid would become 
 a rhomb, and the symbol become {no}, as the pole would then 
 fall on the axial point d. The rhomb-dodecahedron is thus seen 
 to be the limiting figure of the deltoid dodecahedron ; as it is also 
 of the holohedral form, the triakis-octahedron (or octahedrid pyra- 
 midion). 
 
 And of the fourteen quoins of the form <r {hhl}, four on 
 alternate extremities o of the trigonal axes are dissimilar from the 
 four on the extremities o> of the same axes, alternating with the 
 former. And there are six four-faced quoins h on the tetragonal 
 axes. 
 
 Of the twenty-four edges, the twelve J9-edges meet in threes 
 in the quoins o, and the twelve A-edges meet in threes in the 
 quoins co. 
 
 The symbols of the faces of the twelve-deltohedron a- {hhl} 
 are 
 
 hhl hhl hhl hhl, 
 
 hlh hlh hlh hl\ 
 Ihh Ihh Ihh Ihh. 
 
 Those of the correlative semiform a {hhl} are the faces of the 
 triakis-octahedron supplementary to these; they have an odd 
 number of negative indices. 
 
 The semiform a- {332} occurs onfahlore, and o- {221} on blende. 
 
 180. The hemi-icositetrahedron, or tdrahedrid pyramidion, 
 o- {hkk} or & {hkk}, called also the trigonal dodecahedron, or 
 twelve-hoscelohedron, Figs. 89, 90. 
 
 The suppression of the faces of an icositetrahedron (or twenty- 
 four-trapezohedron) which lie in alternate (i.e. attingent) octants 
 produces a pyramidion figure, namely, a tetrahedron with a three- 
 faced pyramid on each of its faces. 
 
 Since the poles of the form lie on the D arcs of the systematic 
 triangles, its twelve pyramidal edges H lie in the H sectors of the 
 2 planes, and the remaining six edges D coincide in position and 
 direction with those of the tetrahedron. 
 
212 
 
 Cubic system. 
 
 The /Hedges meet in threes in the four quoins o or co ; the 
 remaining four quoins co or o being each formed by the meeting 
 
 Fig. 89. Fig. 90. 
 
 of three edges H with three edges D. The symbols of the faces 
 of the form a {hkk} are 
 
 hkk hlk hkk hTk 
 
 khk Yhk lh~k khk 
 kkh Tkh ~kk~h kkh. 
 
 Those of the form a {hkk} are the symbols of the remaining 
 faces of the trapezohedron ; and in these the negative signs of 
 the indices are one or three in number in each symbol. 
 
 The form a- {32 2} occurs on tennantite; a {211} on fahlore 
 and tennantite ; o- { 3 1 1 } on blende and fahlore. 
 
 181. The hexakis-tetrahedron, a {hkl} or a- \hkl], Fig. 91. The 
 
 hemi-hexakis-octahedron, like the 
 two preceding forms, presents gene- 
 rally a tetrahedrid aspect, each axis 
 o meeting at opposite extremities 
 quoins which are ditrigonal but 
 dissimilar. The six faces in each 
 octant are scalene triangles and 
 meet in six edges, of which alter- 
 nate triads H and D are similar, 
 while the remaining edges of the 
 figure represent broken edges of 
 the tetrahedron and are similar; 
 each successive pair meeting two edges H and D in an ortho- 
 symmetrical four-faced quoin h. 
 
 Fig. 91. 
 
Mero symmetry. 2 1 3 
 
 The form has therefore thirty-six edges ; viz., twelve edges D, 
 twelve edges H, and twelve edges A: fourteen quoins; four 
 obtuse (or acute) ditrigonal quoins o, four acute (or obtuse) 
 ditrigonal quoins o>, and six four-faced tetragonal quoins h. 
 
 The symbols of the faces of the form a- {hkl} are given in the 
 columns i, iii, v, vii; those of the form a {hid} in the columns ii, 
 iv, vi, viii of Table A, Article 167. 
 
 182. B. II. Hemi-systematic diplohedral forms; hemi- 
 tesseral semiforms. 
 
 The second section of tesseral hemi-symmetrical forms is that 
 under which the symmetry is hemi-systematic ; i.e. half only of the 
 normals of the integral form are represented by faces ; but in the 
 hemihedral case each of these normals carries its two parallel 
 faces, so that the form is centro-symmetrical ; at the same time 
 the proto-systematic planes *$* are alone planes of symmetry; 
 and the poles of extant faces of the scalenohedron {hkl} circum- 
 jacent to the trigonal axes must lie in alternate systematic triangles. 
 Furthermore, no forms having their poles on zone-circles 2 can be 
 hemihedral as regards the number of their faces, though every 
 such form while presenting its full complement of faces will reflect 
 in the distribution of their crystallographic (physical and geome- 
 trical) characters a symmetry analogous to that represented in 
 the semiforms is {hkl} and ^{khl}] since in fact this abeyance 
 of symmetrical conditions must be the consequence of the mole- 
 cular structure of the entire crystal. The geometrically hemihedral 
 forms of this class will therefore be confined either to such as 
 have their poles lying on the zone-circles S and symmetrical to 
 the proto-systematic planes, and they will thus be semiforms of 
 the cube-pyramidion ; or else they will be semiforms of the general 
 scalenohedron {hkl}. 
 
 183. I" 1 he pentagon-dodecahedron, T:{hko} or n {kho}, Figs. 92-5. 
 The pentagon-dodecahedron represents the hemi-systematic form 
 derived from the tetrakishexahedron or cube-pyramidion {hko}. 
 The symbols of the faces of one of the semiforms lie in alternate 
 rows in Table B, Article 172, those of the correlative semiform in 
 the rows alternating with these. 
 
 Each face is divided euthysymmetrically by the trace of the 
 
214 
 
 Cubic system. 
 
 systematic plane S in which its pole lies ; and the edge O, formed 
 by two adjacent faces of which the poles are on the same great 
 circle, will lie in a second plane S perpendicular to the former. 
 
 The edges G of adjacent faces, the poles of which lie on 
 different great circles S t will not lie in any systematic plane : each 
 face of the semiform will be adjacent to four such faces, and 
 will thus have five sides formed by one edge O and four similar 
 edges G, as in Figs. 92 to 95. 
 
 Fig. 94 . Fi S-95- 
 
 The semiform then has six similar edges and twenty-four 
 similar edges G gyroidally symmetrical in threes on the trigonal 
 axes o : these meet in eight trigonal quoins o formed each by a 
 triad of G edges, and twelve quoins k formed each by two edges 
 G and one edge O. 
 
 Each face is thus an irregular pentagon symmetrical to the 
 trace on it of a systematic plane S perpendicular to its edge 0. 
 
Mer asymmetry. 215 
 
 In the semiform Tt{hkl} this trace of the systematic plane becomes 
 an edge, each pentagonal face of the form {hko} being 'broken' 
 into two trapezoidal faces. (Compare Figs. 92 and 96.) 
 
 The pentagon-dodecahedron approximates in character to the 
 regular dodecahedron of geometry in proportion as the dihedral 
 angles of its G and edges approach equality, resulting in those 
 edges also approximating to each other in length. 
 
 Were the pentagonal faces to become equilateral and equi- 
 angular the edges and G would be equal, and thus the angle 
 (ohk, ohk) would equal (phk, hko)-} whence, as will be proved in 
 
 Chapter VIII, Section II, hk = tf-k' i and ~= I+ 5 , which is 
 
 /v 2 
 
 irrational. The regular dodecahedron of geometry, thus impossible 
 as a crystallographic form, is the limiting figure between the two 
 classes of pentagon- dodecahedra, in which an edge O either is 
 larger or is smaller in its dihedral angle than an edge G. In 
 proportion as the length of the edge increases, the figure ap- 
 proaches in form to the cube, the ratio j becoming greater: 
 
 K 
 
 where, on the other hand, this value approaches unity, i. e. as h 
 approaches k in value, the figure approximates in form to the 
 rhomb-dodecahedron {no} and the length of the edges is 
 small relatively to that of the edges G. 
 
 The twelve-pentagonohedron is a very characteristic form of 
 certain mineral species which belong to a small group, of which 
 pyrites is a conspicuous member. 
 
 The forms 77(230}, 77(210}, 77(310} occur on pyrites, and 
 the forms 77(410} and (210} on cobaltine. Sixteen other varieties 
 of this form have also been met with on the former mineral. 
 
 184. The dyakis-dodecahedron or twenty-four-trapezohedron ; the 
 diplohedron 77 [hkl], Figs. 96, 97. The poles of this form lie in the 
 alternate triad of systematic triangles in each octant, while in any 
 two adjacent octants the systematic triangles containing extant 
 poles of the form are those which are mutually symmetrical on a 
 systematic plane -S". 
 
 The edges G of the triad of planes in each octant are therefore 
 similar, and, like those of the form 77(^0}, are gyroidal in their 
 
2l6 
 
 Cubic system. 
 
 distribution round the trigonal axes; the edges 12 formed by 
 pairs of contiguous faces in adjacent octants lie in the systematic 
 planes S: and so also do the remaining edges 0, which correspond 
 to the 'broken' edge O of the form Tt{hko], and are formed by 
 other pairs of faces likewise in adjacent octants, but the poles of 
 which do not lie in contiguous systematic triangles. Figs. 96 and 
 97 represent the correlative semiforms TT {231 } and TT {321 }. 
 
 The twenty-four faces of the semiform are therefore quadrilateral 
 figures corresponding to ' broken' faces of the semiform TT {hko}, 
 and, as in that figure, the similar adjacent edges G are not similar 
 to the remaining two ; and of the latter the one represented by an 
 edge 12, is larger and the other represented by an edge is shorter 
 
 Fig. 96. Fig. 97. 
 
 than an edge G. Each face is thus a trapezoid. The form has 
 then twenty-four edges G, twelve edges 12, and twelve edges O] 
 and since the deutero-systematic planes fail of being symmetral, 
 the axes h become axes not of tetragonal but of ortho-symmetral 
 character, the six quoins h of the form being made up of two edges 
 12 alternating with two edges O : the eight quoins O correspond to 
 those in the pentagonohedron and are made up of three edges G, 
 and the remaining twelve quoins are formed by two edges G 
 meeting an edge 12 and an edge O, both of which lie in the same 
 systematic plane S. 
 
 The terms dyakis-dodecahedron and diplohedron have been 
 employed to convey the idea of the form being a doubled or 
 ' broken-faced ' pentagon-dodecahedron. 
 
 The symbols of the faces of the form Tt{hkl} are contained in 
 
Mer asymmetry. 217 
 
 Blocks I and II, those of the form v{khl} in Blocks III and IV 
 of Table A. 
 
 A particular case of the diplohedron occurs where one of 
 the edges G is parallel to an edge 12. In such a case the 
 form 77 {M/} will have e.g. its face hkl in the same zone with 
 Ihk and Ihk, i. e. in the zone [o^//], whence by the doctrine of 
 
 zones h k 
 
 hi = / 2 or r = -7- 
 K I 
 
 The indices of the form 77(421} fulfil this relation. Such a 
 form will have trapezia instead of trapezoids for its faces. 
 
 This form 77(421} occurs on pyrites. Other diplohedra are 
 77(531}, 77(321}, and 77(543}, which are met with on crystals of 
 pyrites, and occur also on those of hauerite and cobaltine. 
 
 185. B. III. Hemi-systematic haplohedral forms; tetar- 
 tohedrism. 
 
 Another case of mero-symmetry in the Cubic system still 
 remains to be considered ; and it presents considerable interest 
 from the association with this peculiar kind of mero-symmetry 
 of the optical property of rotatory polarisation. 
 
 It is the case in which a hemi-systematic form is only haplo- 
 hedrally developed, resulting in a tetartohedral form. Evidently 
 such a form can only exist as a geometrically independent form 
 in a single case, that of the general form {hkl} ; the twenty-four 
 normals characterising which are reduced to twelve. If therefore 
 the half of the faces of each of the twenty-four-faced semiforms 
 o- {hkl} or 77 {hkl} be mero-symmetrically suppressed, the re- 
 sulting figure will be the twelve-faced tetartohedron in question. 
 
 And as this suppression can take effect only on three faces 
 in an octant simultaneously, it must be that three extant faces in 
 alternate octants will be those to be suppressed. 
 
 The resulting tetartohedral solid is the tetrahedrid twelve-penta- 
 gonohedron n<r{hkl}, or the tesseral tetartohedron, Figs. 98, 99. 
 
 Each of the forms -n{hkl} and ir{lkh} may be considered as 
 built up of two tetartohedra, viz. 
 
 77 {hkl} of 770- {hkl} and 770- 
 and if{lkh} of 770- \lkh} and wo- 
 
 and similarly for the forms c-{hkl} and a {hkl}. 
 
218 
 
 Cubic system. 
 
 The faces of this solid are irregular pentagons presenting no 
 symmetry. Their sides are formed by edges of three kinds, viz. 
 two contiguous edges G, two contiguous edges f, and one edge 
 F; the form presenting twelve edges G arranged in triads gy- 
 roidally round the alternate axial points o of the trigonal axes, 
 twelve edges f similarly grouped round the axial points to opposite 
 to the points o of these axes, and six edges V through the centres 
 of which the axes h pass. None of these edges lie in a systematic 
 plane. 
 
 The quoins are also of three kinds, viz. four trigonal quoins o, 
 four trigonal quoins o>, and twelve quoins k in each of which an 
 edge V meets an edge G and an edge f. 
 
 Fig. 98. 
 
 The symbols of the four correlative tetartohedra are contained 
 each in one of the blocks in Table A, Article 167, and Figs. 98 
 and 99 represent two of the four quarter-forms derived from the 
 scalenohedron {321}, viz. TTO- {231 } and TTCT {321}. 
 
 Since the tetartohedral form may be derived from either of the 
 three species of hemi-symmetric forms of the general scalenohedron 
 {hkl}, it may equally well be represented by symbols av{hkl\ or 
 <rir{Ak/\, &c., or -na\hkl} y &c. 
 
 The relations of the tetartohedron to these hemihedra are re- 
 presented in the projections of their poles in Plate I. 
 
 The tesseral tetartohedron has been met with on a very limited 
 number of crystals. 
 
Combinations. 219 
 
 Cubic System. C. Combinations of Forms. 
 
 186. The combinations of forms already observed on crystals 
 belonging to the Cubic system are indefinite in number and 
 variety, and, owing to the completeness of their symmetry and 
 the high number of the faces that consequently represent a form, 
 such crystals are often very complex in their aspect. 
 
 The more frequent, and usually the prevalent forms, however, 
 are those with the simplest symbols. In the Cubic, as indeed in 
 every system, each mineral or artificially formed crystal has its 
 own habit, certain forms being generally predominant; but this 
 habit itself varies with the conditions under which the crystal 
 has been formed; the nature of the parent solution or liquid 
 from which the crystals have been deposited, possibly also the 
 temperature and pressure under which the solidification has 
 occurred, having an influence in determining the character and 
 relative developements of the particular forms that the crystals 
 present. 
 
 It results from this that a sort of individuality of type, or 
 family likeness, as regards forms and even physical features, 
 frequently characterises the crystals of a mineral coming from any 
 given locality, and often approximately serves to indicate that 
 locality ; but, if such minerals afford abundant illustration of the 
 variation of habit resulting from the varying conditions under 
 which the crystals have originated, little, indeed almost nothing, 
 is as yet known regarding those conditions or the methods by 
 which they operate. 
 
 Generally, in tesseral crystals the faces of the three simple 
 forms which have for their normals the axes of symmetry are 
 the most frequent, and the most largely developed: and where 
 two or all three of these forms concur the aspect of the crystal 
 will vary much according as the cube, octahedron, or dodeca- 
 hedron is the more predominant form. The figures 100 and 101 
 present two aspects of the combination of the cube and octa- 
 hedron, Figs. 102 and 103 the combination of the cube and 
 rhomb-dodecahedron, while in Figs. 104 and 105 the three forms 
 concur. 
 
22O 
 
 Cubic system. 
 
 Fig. 100. 
 
 Fig. 102. 
 
 Fig. 104. 
 
 Fig. 101. 
 
 Fig. 103. 
 
 Fig. 105. 
 
 C. I. Combinations of Holo-symmelrical Forms. 
 
 IS7. The octahedrid-pyramidion (or triakisoctahedron) \hhk\ 
 occurs only in combination, except in the case of the form {221} 
 which is stated to occur as a self-existent form on the diamond ; 
 although if that mineral could be considered as haplohedral this 
 form would have to be explained either as a twin of a twelve-faced 
 deltohedron o- {221} twinned on a dodecahedral axis, or as a 
 combination of two independent semiforms o- { 221 } and o- {22!}. 
 
 Among the triakisoctahedrid forms met with in combination 
 {332} occurs on garnet with {211} and {nof; {553} on 
 magnetite; {221} on cuprite and galena, and {331} with {m} 
 and {no} on fluorspar and galena, and both the forms {221} and 
 { 331} concur with the cube and octahedron on crystals of cuprite. 
 
 The icositetrahedron \hkk] is a more frequent form than the 
 octahedrid pyramidion. And it occasionally occurs self-existent. 
 The forms {211} and {311} are thus met with uncombined 
 with other forms, the former in garnet, the latter in native silver, 
 magnetite, and spinel. Otherwise the various icositetrahedra are 
 usually met with in association with and subordinated in importance 
 
Combinations. 
 
 221 
 
 to the cube, octahedron, or rhomb-dodecahedron. Fig. 106, of 
 a crystal of copper, exhibits these three forms associated with 
 the form {411}. On crystals of magnetite from Traversella the 
 form {311} is associated with the two last forms, the dodeca- 
 hedron being predominant. The forms {322} and {433} occur 
 on galena together with, but subordinate to, the cubo-octahedron. 
 The form {833} is met with on fluorspar, as are the forms {311} 
 and {211}, facetting and giving an obtuse aspect to the quoins 
 of the cube. And a similar concurrence with the cube of the faces 
 of the form {311} is met with on crystals of highly argentiferous 
 gold from Transylvania, and of native silver from the Kongsberg 
 
 Fig. 1 06. 
 
 Fig. 107. 
 
 mine in Norway. On amalgam the form {211} is associated 
 with {221}, respectively facetting the edges in which the faces of 
 the octahedron meet those of the cube and dodecahedron. 
 
 The tetrakisheocahedron or cube-pyramidion {hko} occurs as a 
 self-existent form in the case of the form {210} in native copper 
 and native gold, and of the form {310} in fluorspar. Otherwise 
 tetrakishexahedra are found only in association, and frequently so 
 with the cube and with the rhomb-dodecahedron. Not infrequently 
 a form [hko] will be associated with the corresponding icosi- 
 tetrahedron {tikk}. Thus in Fig. 108, of a garnet from Dognaczka, 
 the forms {210} and {320} are combined with the forms {211} 
 and {321}. These forms {210} and {320} occasionally occur 
 with the cube with their faces predominantly developed, but more 
 usually they are the subordinate form, as seen in fluorspar. 
 
 The forty-eight-scalenohedron or hexakisoclahedron occurs as a 
 self-existent form only on the diamond, and in that mineral 
 with faces so curved and striated as to preclude an accurate 
 
222 
 
 Cubic system. 
 
 determination of the symbols of the form or forms. The geometric 
 characters of this form have been discussed, and it is only ne- 
 cessary to add that in combination it not unfrequently concurs 
 with the particular icositetrahedron, of which the faces truncate 
 its edges. Such is the case on the garnet, where the dodeca- 
 hedrid pyramidion {321}, as also the pyramidion {431 }, are asso- 
 ciated with but subordinate to {211}, and with predominant faces 
 of the dodecahedron {no}, which belong to the same zones. 
 
 Fig. 108. Fig. 109. 
 
 Fig. 107 represents a crystal in the British Museum of copper from 
 Lake Superior exhibiting the forms {100}, {530}, and {531}. 
 
 Often, however, the icositetrahedra and triakisoctahedra concur 
 with scalenohedra, to which they are not truncating forms, and 
 the faces of the latter figures then facet the quoin-edges of the 
 cube or dodecahedron with which they may be associated. 
 
 In fluorspar singularly elaborate and symmetrical modifications 
 of the cube-quoins are effected by the faces of scalenohedra some- 
 times two or three in number, generally associated with icositetra- 
 hedra; often also narrow faces of the rhomb-dodecahedron are 
 present truncating the cube-edges ; and on this mineral both sca- 
 lenohedra and the cube-pyramidia occasionally occur with symbols 
 the indices of which rise to very high numbers. Fig. 109 repre- 
 sents one quoin of a crystal of fluor carrying the faces of the cube 
 of the dodecahedron, and of the forms {311 } and { 421 } associated 
 with two other scalenohedra of which the symbols are indeterminable. 
 
Combinations. 223 
 
 C. II. Combinations of Hemi-symmetrical Forms. 
 i. Holo-systematic haplohedral forms. 
 
 (a) Asymmetric Forms. 
 
 188. Though this type of hemi-symmetry has been long recog- 
 nised as theoretically possible, its actual occurrence as a character 
 of crystals has been only recently observed. Tschermak has 
 described artificial crystals of ammonium chloride with small faces 
 of the form a {785} replacing the edges of the form a {211}, the 
 latter of which, as indicated in Art. 176, has its full complement of 
 faces. In Figs, no and in are represented crystals of cuprite 
 
 OOf 
 
 Fig. no. Fig. in. 
 
 from Wheal Phoenix, Cornwall, in which the form a {986} is 
 associated with the cube, octahedron, and dodecahedron, as de- 
 scribed by Miers from specimens in the British Museum. 
 
 (3) Telrahedrid Forms. 
 
 189. Of the tetrahedrid forms of the Cubic system the correlative 
 semiforms are frequently concurrent on a crystal; but when they 
 are it may be said that universally, even though the detection of the 
 differences may be difficult, the different semiforms must be differ- 
 ently endowed in physical character if not in geometrical develope- 
 ment. For, the two directions of the same normal being in the 
 
224 Cubic system. 
 
 case of no tetrahedrid form identical in property, it is not consonant 
 with the idea of crystallographic similarity that the two faces with 
 a common normal but belonging to different though correlative 
 semiforms should be identical in feature and property. And this 
 crystallographic axiom renders the tetrahedrid character of the 
 diamond difficult of acceptance. The evidence in favour of the 
 hemihedrism of the diamond rests on the character of some of 
 the re-entrant edges of its crystals, and on the very rare but un- 
 doubtedly extant crystals which are tetrahedrid in form. The 
 character of the last kind of evidence has been considered in 
 Article 160, and the further treatment of the question will find its 
 place when the twinning of tetrahedrid forms on a dodecahedral 
 axis is discussed. 
 
 The characteristic features that distinguish the faces of the o 
 tetrahedron or \ n i } from those of the o> tetrahedron o- {Yi i } are 
 very marked in the mineral blende. The faces now usually 
 selected from the two correlative semiforms to represent the 
 <r { 1 1 1 } semiform are deeply striated parallel to the edges of the 
 octahedron, and have a dull aspect as compared with the o> faces, 
 which are brilliant in lustre and smooth. These differences are 
 represented in Fig. 125, of a twin-crystal of blende, in which both 
 the correlative tetrahedra are present in association with the faces 
 of the cube. 
 
 In crystals where only one of the correlative forms occurs with 
 the cube, the alternate quoins of the latter figure are truncated by 
 the faces of the tetrahedron, an example of which is afforded by 
 pharmakosiderite (cube-ore) from Cornwall. It occurs also on 
 blende, see Fig. 55 (i). The faces of the co tetrahedron in fahlore 
 are large, and striated parallel to the edges of the tetrahedron ; 
 inverted therefore as compared with those of the o faces of blende, 
 on which the striations are parallel to the edges of the octahedron, 
 that is to say, the lines in which the striations meet correspond 
 to the H sides of the systematic triangle in fahlore, and to the 
 D sides of the triangle in blende ; the o faces are very small in 
 comparison with the to faces. 
 
 The iriakistdrahedra <r {hkk} and <r [Tikk] (tetrahedrid pyra- 
 midia), which are the tetrahedrid semiforms into which the icosi- 
 
Combinations* 225 
 
 hedron {hkk} may be resolved, are only known to occur as 
 self-existent forms in eulytine (bismuth-blende), and in helvine (or 
 rather in achtaragdite, which appears to be a pseudomorph in the 
 form of helvine) ; in each case as the form o- {211 }. 
 
 Crystals which, like fahlore (or tetrahedrite), carry predominant 
 faces of a pyramidion, always retain a tetrahedrid aspect. Associ- 
 ated with the striated tetrahedron o- {III} in that mineral the 
 forms <r {^Ti} striated parallel to the edges of the tetrahedron, 
 o- {3!!}, Q-J955}, o-{4iT} are met with, while the form <r{2ii} 
 
 Fig. 112. 
 
 striated parallel to the edges of the occurring faces, o- { 4 1 1 } and 
 <r {511} are associated with the oo tetrahedron a- { 1 1 1 } in the 
 alternate octants : and the rhomb-dodecahedron is frequently pre- 
 sent. Fig. 112 is a representation of a crystal of fahlore (after Hes- 
 senberg) on which the forms o- {111}, o-(5ii}, v\4ii } and o-J2ii \ 
 concur with or {In}, <r {21}, cr {595}, o- {no} and cr {100}. On 
 blende the more dull and striated tetrahedron o- { TIT } is found asso- 
 ciated with the forms cr {sTfi} and cr { 5"22 }, the faces of the latter 
 form being somewhat rounded: occasionally also o- {^21} and 
 o- {331} are met with. The correlative tetrahedron, on the other 
 hand, viz. <r { 1 1 1}, has its faces more developed and more brilliant 
 and is concurrent frequently with cr {311}, the faces of which carry 
 striations parallel to their intersections with, and therefore to the 
 edges of, the rhomb-dodecahedron: and the forms o-{4ii} and 
 o-{i2 i 1} with a hexakistetrahedron <r{43i} have in some few 
 instances been observed. The dodecahedron is generally present, 
 
 Q 
 
226 
 
 Cubic system. 
 
 the cleavages of the mineral lying parallel to its faces, and occa- 
 sionally it is accompanied by one or more of the forms cr {320}, 
 a- {210}, or or {410}. 
 
 Fig. 113 exhibits the forms o-{ioo}, o-{iio}, 0-^320}, o-{in} 
 with o- {311}; and a {In} with a {"211}. 
 
 The deltohedron a {hhk}, which is the haplohedral form of the 
 octahedrid pyramidion (triakisoctahedron), is unknown as a self- 
 existent form. 
 
 Both correlative semiforms of the form {332} co-exist in 
 fahlore with <r{2io} and cr{iio}, and the faces of <r {332} 
 which are smaller than those of o- {332}. In blende the negative 
 semiforms <r {332} and <r {"22!} occur combined with the tetra- 
 
 Fig. 113. 
 
 Fig. 114. 
 
 hedron crjiiij, the dodecahedron <T{IIO}, and the tetrahedron 
 o-jiii}. 
 
 Of the hexakis tetrahedron a- {hkl} the minerals blende and 
 fahlore offer the prominent examples. Boracite indeed exhibits 
 the form o- {531 } in association with the tetrahedron cr { 1 1 1 } and 
 predominant faces of the cube truncated by those of the dodeca- 
 hedron, while the forms o-J2Ti} and cr {YTT} facet the alternate 
 quoins of the cube. And crystals of diamond occur which are 
 hexaldstetrahedra, but they are extremely rare. Four of these are 
 in the British Museum, and two of them carry faces of a tetrahedron 
 truncating the rounded and six-faced quoins of the hexakistetra- 
 hedron. Fig. 114 represents one of these. There are also three 
 crystals in the same collection in which the hexakisoctahedron and 
 octahedron are combined, but with the octahedron faces in the 
 
Combinations. 227 
 
 alternate octants smaller than the others. On fahlore, a {32!} is 
 concurrent with a- {^11} and o- {TTTj and the dodecahedron. 
 And on blende the correlative forms o-J43i} and o- {43!} are 
 said to concur with o-{m} and cr {TIT} and the faces of the 
 cube which truncate the edges of the larger tetrahedron o- { 1 1 1 } . 
 
 C. II. Combinations of Hemi- symmetrical Forms. 
 ii. Hemi-systematic diplohedral forms. 
 
 190. The twelve-pentagonohedron or pentagon-dodecahedron TT{/>O}, 
 IT {kho}. Pentagonohedra are illustrated almost exclusively by 
 the group of minerals of which pyrites (iron disulphide) is the 
 most familiar member, and hauerite, cobaltite, and gersdorffite, 
 severally the corresponding sulphides or arseno-sulphides of man- 
 ganese Mn S 2 , cobalt Co (S, As) 2 , and nickel Ni (S, As) 2 , are the 
 other distinct minerals. Much discussion and laborious investiga- 
 tion have been devoted by Friedel, Gustav Rose, Brezina, Schrauf, 
 and E. S. Dana to the endeavour to establish some connection 
 between the thermo-electric or the pyro-electric properties of 
 different crystals or parts of crystals of these minerals and their 
 crystalline forms. The result arrived at from the whole series of 
 these investigations may be summarised in the conclusions that 
 (i) pyro-electricity, i.e. the production of opposite electric con- 
 dition in different parts of a body, while its temperature is changing, 
 has no place in the case of the pyritoid minerals; (2) that as 
 regards thermo-electricity, evidenced by a current set up by a 
 difference of temperature at two contacts of one substance with 
 another, the character of the forms of the crystal has no influence 
 on the thermo-electric sign (positive or negative potential) ; and 
 (3) that, though the variations in potential do exist, they have 
 nothing to do with crystallographic symmetry, but are dependent 
 on the relative density, and probably on differences in chemical 
 composition of the crystals or parts of crystals investigated ; the 
 less dense specimens generally representing the electro-positive, 
 the more dense the negative varieties. 
 
 Of the different pentagonohedra known on the pyritoid minerals 
 the only one that is self-existent is the ' pyritohedron ' (of Hai- 
 
228 Cubic system. 
 
 dinger) 77(210} or 77(120}, which is common in pyrites and 
 cobaltite. 
 
 The correlative semiforms being superposable by a quadrant- 
 revolution round a tetragonal axis no distinction can be drawn 
 between the two semiforms as regards crystallographic develop- 
 ment in alternate octants. It is, however, an exceptional occur- 
 rence when two pentagonohedra of symbols 7r{/$/o}and7r {k'h'o } , 
 the poles of which do not lie in the same systematic triangles, 
 concur on a crystal: for usually, where two or more different 
 forms occur together, their symbols are either all of the type 
 i:\hko} or all of the type Tt{khQ}. And the same observation 
 holds regarding the concurrence of diplohedra represented by a 
 general symbol 77 {hkl\ with those of which the symbol would be 
 
 On the other hand, crystallographers have recorded the occur- 
 rence of the correlative semiforms 77 (210} and 77 (120}, 77 (320} 
 and 77(230}, 77(430} and 77(340}, 77(540} and 77(450}, 
 7r{52o| and 77(250}, 77(650} and 77(560}; the symbol being 
 determined by the association of the semiform with others belonging 
 to octants adjacent to those in which the faces of the particular 
 semiform lie. 
 
 Correlative diplohedra with symbols 77(321} and 77(231}, 
 77(421} and 77(241}, 77(432} and 77 {342} have had their exist- 
 ence attested in a similar way by Rome de 1'Isle, Haiiy, Mohs and 
 Struver. Thus, too, Naumann and Zippe describe the union of 
 77(123} with 77(435}. Struver describes five other pentagono- 
 hedra of type IT {hko} associated with 77(120}. And finally 
 G. Rose records 77 (435} an <3 77 (324} as concurrent with 
 
 77 (l2o}. 
 
 But these are few, and comparatively exceptional cases, met with 
 among the very extensive and varied series of crystals that this 
 group of minerals contributes to mineral collections. 
 
 The pentagonohedron presents in the symbols of the forms that 
 have been observed a larger range than does the cube-pyramidion, 
 of which the pentagonohedron is the semiform. Thus the forms 
 * |53}> ^ (43}> 77 (540}, 77 (650}, and 77 (607} are known of 
 which the corresponding pyramidions have not been observed, 
 
Combinations. 
 
 229 
 
 whereas the hemihedron derived from the cube-pyramidion {520} 
 has not yet been met with. Twenty-one distinct pentagonohedra 
 have been recorded as existing on pyrites as well as seven kinds of 
 tetrahedrid pyramidion, three of deltohedron, and thirteen of 
 diplohedron. 
 
 A remarkable combination of pyritohedron and octahedron is a 
 not uncommon form of pyrites and cobaltite. It resembles in 
 aspect the regular icosihedron of geometry, and is formed by the 
 union of the two forms 77(210} and 7r{iii} in equipoise. The 
 eight faces of the octahedron are equilateral triangles, and the 
 twelve faces of the pyritohedron assume also a triangular form ; 
 two sides of each of these triangles being formed by intersection 
 
 Fig. 115. 
 
 Fig. 1 16. 
 
 with adjacent octahedron faces, the third side being the edge 
 between two adjacent faces of the pyritohedron. Consequently 
 the latter are isosceles triangles, and the figure would be a regular 
 icosihedron but for the shorter length of the third side as compared 
 with the sides of the octahedron faces. This combination is 
 represented in Fig. 115. 
 
 The Diplohedron TT \hkl\ or TT \khl\ is known to occur as a 
 self-existent form on pyrites from Traversella with the symbol 
 ^{321}, and on that from Brosso with the symbol ^{421}. 
 The several other diplohedra occur in combination with one or 
 more of the forms 7r{22i},7r{2ii},7r {311}, the cube, octahedron, 
 and dodecahedron. But the forms are so varied, so intricate in 
 aspect, but, withal, so simple in their crystallography, that detailed 
 description of them is quite unnecessary. 
 
 Fig. 116 represents a pyrites crystal with the forms 7r{i2o}j 
 
210 
 
 Cubic system. 
 
 77(231}, 77(543}, and 77(100}; and Fig. 117(0) a crystal after 
 Striiver with the forms 77(241}, TT {5 n 2}, TT {122} with TT {120}, 
 77 { 100 }, and 77 { 1 1 1 }. Fig. 1 1 7 (f) represents a quoin of the last 
 crystal drawn on a larger scale. 
 
 It is only requisite to observe, in regard to those complete forms 
 of which the faces are grouped in triads but not ditrigonally round 
 the trigonal axes, and which therefore do not admit of disparting 
 into correlative semiforms, that these forms equally undergo hemi- 
 symmetrical modification where they are concurrent with hemi- 
 systematic semiforms. They are the cube, dodecahedron, icosi- 
 tetrahedron, octahedrid pyramidion, and the octahedron itself. It 
 has been mentioned in Article 147 that the cube may be striated in 
 
 Fig. ii 7 (a). 
 
 only one direction, but symmetrically to the -S'-planes in the case 
 of its association with diplohedral forms : and that in the case of 
 haplohedral forms it is striated in one diagonal direction symmetri- 
 cally to the 2-planes only. Similar peculiarities of striation or of 
 outline consequent on the abatement of some element of symmetry, 
 and following the extant symmetry of the crystal, will be observed 
 in all these forms. So that we are quite precluded from excluding 
 from their symbols the characteristic o- or 77 which marks the 
 symmetry or abatement of symmetry in the crystal. 
 
 C. III. Combinations of Tetarto-symmetrical Forms. 
 
 191. The tetartohedron of the Cubic system is unknown as a 
 self-existent form ; when met with it is always associated with other 
 forms which are from the geometrical point of view either haplo- 
 
Combinations. 2 3 r 
 
 hedrally or hemi-systematically developed. The forms (nr {351} 
 and O-TT {421} occur on Barium nitrate. The original discovery 
 by Marbach (Pogg. Annalen^ Vols. 91 and 94) of the tetartohedral 
 character and the property of rotatory polarisation belonging to 
 Sodium chlorate, Sodium bromate, Sodium-uranyl acetate long 
 stood by itself. Now, however, the number of substances tetarto- 
 hedral in symmetry has been increased, and includes the nitrates 
 of Barium, Strontium, and Lead ; though in these the rotatory 
 action on the plane of polarisation of plane polarised light either 
 does not occur or is too feeble to be recognised. 
 
 The fact that there can only be a single kind of tetarto- 
 hedron in the Cubic system, that namely resulting from a haplo- 
 hedral hemi-systematic developement of the forty-eight-faced 
 scalenohedron, does not preclude a crystal 
 from having tetartohedral symmetry, even 
 though its forms do not include one of 
 the type O-TT { hkl}. We find, for instance, 
 the faces of a tetrahedron or of distin- 
 guishable correlative tetrahedra associated 
 with those of a twelve-pentagonohedron 
 in crystals of Sodium chlorate. Here the 
 faces of a cube are associated with those 
 of a tetrahedron o- { 1 1 1 } and of the py- Fig. 1 18. 
 
 ritohedron i:\2io}. Their proper sym- 
 bols are therefore cnr {100}, O-TT {m}, O-TT {210}, since they afford 
 evidence of the crystal being, concurrently, haplohedral for one 
 form, and hemi-systematic for another form, where the faces of 
 neither semiform group ditrigonally round the trigonal axes and 
 cannot therefore be disparted into several tetarto-symmetric 
 groups. 
 
 Fig. 118 represents a crystal of Barium nitrate (described 
 by Prof. Lewis) which carries the faces of the octahedron, really 
 of the two tetrahedra, those of the icositetrahedron {311}, which 
 like the former is to be supposed resolved into two tetrahedrid 
 pyramidions, and finally, the tetartohedra o"7r{24i}, <nt \~2i\, 
 and o-7r{35i}, the last form being represented by the faces of 
 but one of the four correlative tetartohedra. 
 
232 Cubic system. 
 
 Cubic System. D. Twinned Forms. 
 
 192. The difficulty of describing and representing in a compact 
 treatise on crystallography the multitude of combinations of forms, 
 which the crystals of any system may exhibit, belongs also to any 
 similar attempt in the case of the twins formed by these combina- 
 tions. But these can be classified and discussed in general terms 
 without entering on the details necessary in the practical work 
 involved in the crystallographic study of any particular crystal. 
 
 Holo-symmetrical crystals in the Tesseral system cannot 
 have a face of the cube or dodecahedron for a twin-plane, since 
 these forms are parallel to planes of actual symmetry at the same 
 time that their normals are axes of actual diagonal symmetry. 
 The faces of the octahedron can be, and almost exclusively are, 
 the twin-faces of holo-tesseral crystals ; the only other recorded 
 variety of such twins being in the case of certain crystals of 
 galena, in which Sadebeck finds the plane of twinning to be a 
 face of the octahedrid pyramidion {441}. In these crystals, how- 
 ever, the twinned individuals are intercalated as thin plates, the 
 edges of which are seen on the faces of cleavage *. 
 
 In hemi-symmetrical crystals the tetrahedrid (haplohedral) 
 semiforms are symmetrical only to the deutero-systematic planes 
 2, and the normals of these planes though potentially axes of 
 diagonal symmetry have their symmetral character in abeyance. 
 This, however, is not the case as regards the normals of the proto- 
 systematic planes parallel to the faces of the cube. There will 
 then be no restriction on the character of the zone-axes and 
 normals that may be twin-axes for such crystals, except that 
 the actual crystallographic axes cannot be twin-axes. In the 
 diplohedral semiforms again, the potentially tetragonal but actually 
 diagonal axes of symmetry (the normals, namely, of the cube- 
 faces), are the only crystallographic lines precluded from being 
 twin-axes. 
 
 But though the idea of symmetry imposes no other restriction 
 
 than that just mentioned on the kind and number of twin-laws 
 
 that might exist in the Tesseral system, it is practically found that, 
 
 excepting in the case of the galena twin alluded to, every twin in 
 
 * Zeitschr. der deutsch. geol. Ges. Band xxvi. S. 617. 
 
Twinned forms. 
 
 233 
 
 this system may be referred either to a twin-law wherein a normal 
 of an octahedron face is the twin-axis, or to a law by which the 
 twin-axis is a normal of a dodecahedron face. The latter law 
 affords twins which are always of the mutually interpenetrant 
 kind. Twins under the former law may be either interpenetrant 
 twins or juxtaposed twins, and in certain cases are embedded 
 twins. 
 
 D. I. Twin-law : a face of the octahedron the twin-face. 
 
 193. Among juxtaposed twins we may at the outset distinguish 
 those in which the twin-face is also the plane of combination from 
 such as have for the combination-plane a face perpendicular to the 
 twin-face. 
 
 Fig. 119. 
 
 Fig. 1 20. 
 
 And again, in crystals of the former class we have the cases of 
 simple twins composed of two individuals, and of complex 
 twins formed by repeated twinning, while the complex twins are 
 further to be subdivided into the cases of parallel repetition 
 (polysynthetic twins), twinned in fact on the same face, and the 
 cases of inclined repetitions, where the repetition takes place on 
 different faces of the same form as twin-faces. 
 
 Holo-symmetrical crystals: twin-plane a face of the 
 octahedron. Of the spinel-twin in which two crystals, on which 
 the octahedron faces alone occur, are in juxtaposition with an octahe- 
 dron-face for twin- and combination-face, a representation has been 
 given in Fig. 57, Article 154. Fig. 119 is an embedded twin of 
 the cube, and Fig. 120 represents a double twin of diamond; the 
 octahedra being compressed in the direction of the twin-axis. It 
 
234 
 
 Cubic system. 
 
 is a frequent twin in the minerals of the spinel group including 
 magnetite, and is distinctly a twin-law for the diamond ; and for 
 those tetrahedrid crystals which affect in the developement of 
 their correlative tetrahedrid forms the appearance of diplohedral 
 crystals, the spinel law holds also. Of this blende will be seen 
 
 Fig. 121. 
 
 Fig. 122. 
 
 to be an example. Of other aspects of crystals twinned on an 
 octahedron-face the following figures are illustrations. Fig. 121 
 is that of a twinned rhomb-dodecahedron of gold, and Fig. 122 
 represents a twin of the form {311}. Fig. 123 represents a twin 
 
 Fig. 123. 
 
 Fig. 124. 
 
 of copper from Lake Superior with the forms .{in}, {511}, and 
 {520} : fig. 124 is another twin of copper with forms {m} and 
 {740}; both from crystals in the British Museum. 
 
 The cases in which the combination-plane is the twin-face are 
 far more frequent than those in which the one plane is perpen- 
 dicular to the other ; the latter have in fact almost the character of 
 
Twinned forms. 235 
 
 exceptions in Tesseral crystals, and in them the combination-plane 
 is a face of the form {211}. 
 
 Besides the twins so far considered, in which two crystals are 
 juxtaposed at the face of an octahedron, or else of the form {21 1 } 
 perpendicular to that face, there exist interpenetrant twins of this 
 class. Of these, cubes of galena (of which Fig. 1 1 9 is an illustration) 
 and dodecahedral crystals of sodalite are examples. 
 
 194. Haplohedral crystals: twin-plane a face of the 
 tetrahedron. Of the simple twins of this kind we have charac- 
 teristic examples in blende and fahlore. 
 
 Fig. 125 represents a twin of blende in which the twin-face 
 (nl) is also the combination-plane. It shows the faces of an o> 
 and an o tetrahedron meeting in an edge in the twin-plane. 
 
 Fig. 125. Fig. 126. 
 
 Fig. 126 represents a twin also of blende with the forms <r {100}, 
 a {no}, o- {311}, and o- {III} twinned on a face (ill). 
 
 While blende usually presents a very holohedral aspect in con- 
 sequence of the preponderance of the rhomb-dodecahedron faces 
 in association with those of both correlative tetrahedra, fahlore, 
 on the other hand, is a mineral with an essentially tetrahedrid 
 aspect, as has already been explained. A twin of fahlore with a 
 tetrahedron-face for its twin-plane is exhibited in Fig. 127. 
 
 Of twins by parallel repetition an example is given in Fig. 128, 
 which represents a crystal of blende from the Binnenthal, the 
 successive combination-planes being parallel to the same face of 
 the tetrahedron a {In }, which is at the same time the twin-face. 
 
 But as regards the crystals in which the combination- 
 plane is perpendicular to the twin-plane, it is to be 
 
2*6 
 
 Cubic system. 
 
 observed that such a plane can be simultaneously a face and 
 a plane of symmetry to the twinned combination. In fact, 
 Groth offered as a definition of a twin-plane that it may be a 
 plane parallel to a face identical for the two crystals, which is 
 not a plane of symmetry to either crystal, but in respect to which 
 the two crystals are symmetrical. That such a face would be a 
 plane of symmetry in the particular case where the plane in the 
 zone [in] belongs to the form {211} will be seen if we consider 
 a stereographic projection of a tesseral crystal on a plane parallel 
 to a face of the octahedron, say to the face {m}. If, then, the 
 crystal is such that the arrangement of the poles of any form is 
 
 Fig. 127. 
 
 Fig. 128. 
 
 ditrigonal round the normal of any octahedron-face, for instance 
 the face (in), and if we suppose the crystal to be twinned round 
 such a normal as a twin-axis, the result will be a dihexagonal 
 distribution of the poles of the united crystals. It will be seen 
 that the faces (no), (Toi), and (oil) of the rhomb-dodecahedron, 
 and the (^11), (i^i) 3 (n^) of the icositetrahedron {112}, and the 
 faces severally parallel to them, are all in the zone [in] and are 
 perpendicular to the face (i 1 1) ; and origin-planes parallel to these 
 faces will be planes of symmetry for the poles of the twinned 
 crystals. 
 
 If another face of the octahedron were taken as the plane of 
 projection and of twinning, different groups of six faces of the 
 forms {no} and {211} would become the planes of symmetry. 
 
 Taking the case first supposed ; let now the plane (112) be the 
 face of combination of two crystals twinned round an axis [in]. 
 The face (no) for both crystals is perpendicular to their common 
 
Twinned forms. 
 
 237 
 
 face (112) and an origin-plane parallel to it, being a deutero-syste- 
 matic plane, is a plane of symmetry for each of the crystals. Hence 
 any pole or face A (say, hkt] of the first crystal, repeated on the 
 second crystal by twinning in A'^ will be symmetrical in respect 
 to the plane (112), with a pole B', i. e. (khl} on the second crystal, 
 which itself is symmetrical with A' in respect to (ilo), or both A 
 and A' are symmetrical with a pole B on the first crystal in respect 
 to the planes (ilo) and (112). Therefore corresponding faces, 
 i. e. faces with the same indices (here, h //) on the two crystals, 
 are those which are symmetrical to an identical face (here, khl} ; 
 the one symmetrical to it over the combination plane (112), the 
 
 IL 
 
 Fig. 129. 
 
 other symmetrical to it over the plane (ilo) perpendicular to the 
 latter. Figs. 130 and 131 (after Sadebeck) represent a crystal 
 of galena twinned on (m) and with combination-plane perpen- 
 dicular to (in). 
 
 The very rare occurrence of crystals in which this kind of 
 twinning is observed, would seem to preclude the definition of 
 a twin that is drawn from it from being admissible as a general 
 statement for twin-crystals. 
 
 Interpenetrant twins, with an octahedron-face for twin-plane, 
 occur with haplohedral crystals in the case of fahlore. Of this 
 Fig. 127 is a representation in the case of a crystal in which the 
 tetrahedron o-jni} is associated with the pyramidions 0-J322} 
 and cr { 2 1 1 }, and with the faces of the rhomb-dodecahedron a- { 1 10 } 
 in subordinate developement. It exhibits the peculiarity that the 
 
238 
 
 Cubic system. 
 
 second individual is smaller than the first, which it penetrates, and 
 does not carry the pyramidion forms but only the faces of the 
 tetrahedron and dodecahedron. 
 
 Between the interpenetrant twins and the juxtaposed twins there 
 are different degrees of interpenetration represented by more or 
 less entirely embedded twins ; and of these an example is cited by 
 Sadebeck in the case of fahlore, in which the composition-plane 
 (21!) is perpendicular to the face of twinning (nT); but instead 
 of the two crystals being in contact merely at the former face, they 
 are so united that a portion of each is, as it were, lost in the other 
 
 Fig. 130. 
 
 Fig. 131. 
 
 but symmetrically to the plane of their junction. Fig. 131 
 illustrates such a growth in the case of a twin-crystal of galena. 
 
 D. II Twin-law: a face of the rhomb-dodecahedron 
 the twin-face. 
 
 195. Haplohedral crystals. Examples of holohedral forms 
 twinned on an axis of actual diagonal symmetry being impossible, 
 we have only to deal with the cases of (i) haplohedral crystals 
 twinned round a normal of the form {no}, and (2) of a diplo- 
 hedral crystal so twinned. Examples of the former kind are 
 extremely rare : eulytine has been cited as one, and pseudo- 
 morphous crystals after fahlore from Bieber in Hanau are instanced 
 by Kopp* and by Grothf. Such twins are interpenetrant, and 
 
 * Neuesjahrb.fur Min. 1877, p. 62. 
 *t* Strassburg Catalogue, p. 4. 
 
Twinned forms. 239 
 
 Fig. 132 represents a twin of eulytine, presenting the pyramidion 
 cr{2n} of Fig. 89, twinned according to this law. Doubts 
 however as to the tesseral symmetry of eulytine have been brought 
 forward by Bertrand, who would refer this mineral to the Hexa- 
 gonal system. 
 
 It is under the head of twins of tetrahedrid forms twinned on a 
 face of the rhomb-dodecahedron that the discussion falls regarding 
 the hemi-symmetrical character of the diamond. 
 
 The grounds urged in favour of the traditional view that the 
 diamond is a hemi-symmetrical crystal, so far as they rest on the 
 occurrence of simple hemi-symmetrical forms and on certain pecu- 
 liarities of conformation in the crystals, have been detailed in 
 Article 160. Of the latter, the most important feature is the 
 very symmetrical furrows or re-entrant edges that are met with 
 
 Fig. 132. 
 
 on certain crystals which carry the faces of the octahedron, or of 
 that form in association with a hexakisoctahedron, as in Figs. 133, 
 134. 
 
 These re-entrant furrows occur at the edges of the octahedron, 
 and channel the crystal in the directions of these edges where the 
 form is an octahedron, or in the directions of the corresponding 
 edges of the hexakisoctahedron, where the latter form is also 
 present. In the former case they are often, as in Fig. 133, 
 merely step-formed, and seem to indicate a lamellar structure 
 resulting in the successive layers of each octahedral face not being 
 completely developed up to the edges of the face. Oftener they 
 
240 Cubic system. 
 
 are facetted on each side of the furrow with faces usually rounded 
 and repeated symmetrically to a proto-systematic (S) plane, to 
 which the furrow and the crystal are alike symmetrical ; of this the 
 diamond figured in Fig. 134 is an illustration. But these faces 
 though it is very difficult to allot them their symbols can be 
 recognised as belonging to forms that differ from, and in fact 
 are shallower in their contour than, the hexakisoctahedral forms 
 extant in association with the octahedral faces in the octants 
 immediately adjacent to them. 
 
 Now to explain this singular, indeed unique, type of crystals, 
 it is supposed that they are in fact composed of correlative tetra- 
 hedra or of hexakistetrahedra, twinned on a dodecahedral axis, 
 
 Fig. 133- Fig. !34- 
 
 but so that faces in the octants containing one form, or set of 
 forms, invariably cover and obliterate the faces lying in the ad- 
 jacent octants ; only a small fringe, as it were, of the latter re- 
 maining unobliterated and forming the narrow margin of the 
 re-entrant furrow. If it be asked where corroborative evidence is 
 to be found in favour of the existence of these otherwise unseen 
 semiforms, the answer is, that there are crystals few indeed in 
 number but indubitable in their character which are actual 
 hexakistetrahedra, sometimes with, sometimes without, the faces 
 of the octahedron belonging to the same octants. And these 
 semiforms are furthermore shallower than the usual forms on the 
 crystals of hexakisoctahedral aspect, and are never channelled by 
 re-entrant edges. Of such crystals, four have been described, and 
 one figured, Fig. 114, Article 189, from the Collection at the 
 
Twined forms. 241 
 
 British Museum. And to these are to be added, as instances of 
 hemihedral developement, from the same collection, one crystal with 
 forms {in} and {hkl}, showing salient edges and faces of the octa- 
 hedron less developed in alternate octants; another crystal similarly 
 developed but with re-entrant edges, and a third similar to the second 
 but with one of the octahedral faces in an alternate octant com- 
 paratively large, indicating a transition to the furrowed octahedron. 
 
 So far, then, the explanation seems complete which treats the 
 channelled crystals as twins on a {no} face, of a crystal pre- 
 senting forms cr {in}, <r {hkl} (belonging to the o octants), with 
 <r{Tii} and <r{h'Kf\ forms (belonging to the <o octants) but 
 almost obliterated by the former. 
 
 It is, nevertheless, difficult to accept this explanation. The re- 
 entrant edges are peculiar to certain forms. The hexakisoctahedron 
 is a frequent form in the diamond, but its edges would seem only 
 to be furrowed when it is united with the faces of the octahedron. 
 The cube and the rhomb-dodecahedron exist as distinct forms; 
 but the edges of the cube are not furrowed, and the dodeca- 
 hedron is entirely regular in aspect. 
 
 Out of 250 crystals in the British Museum there are eight simple 
 octahedra with sharp edges that are entirely holo-symmetric in 
 aspect and present none of the peculiar furrows that have been 
 alluded to. Four of the crystals are cubes and thirteen crystals 
 carry cube with cube-pyramidion forms, but of these none shows 
 a re-entrant edge. There are ten with combined forms of octa- 
 hedron and rhomb-dodecahedron or only the latter form that are 
 without re-entrant edges, but with evident lamellar structure. Of 
 the octahedrid pyramidion, or this form combined with the octa- 
 hedron, there are nine crystals with evident lamellar structure ; six 
 being twinned on an octahedral face. 
 
 Of crystals in which the octahedron is combined with the 
 hexakisoctahedron, there are twenty-two that present no re-entrant 
 edges, seven that show undoubted lamellar structure in a step- 
 formed layer of material deposited on the octahedral faces; and 
 twenty-three such crystals show the same in a less obvious way. 
 And of crystals with these forms that show re-entrant edges in the 
 way described there are sixteen in the collection. 
 
 R 
 
242 Cubic system. 
 
 On the other hand, that the diamond, viewed as a holosym- 
 metrical crystal, is capable of undergoing the ordinary kind of 
 twinning is seen in the many examples of it that occur in the 
 form of the spinel twin. The British Museum collection contains 
 six crystals with forms {m} \hhl\, and ten of the hexakis- 
 octahedron thus twinned, but without any of the facetted channel- 
 lings, though many of them are distinctly lamellar in structure. 
 One large twin octahedral crystal from the Cape has all the faces 
 of a peculiar dull aspect, but the faces of different octants are 
 absolutely indistinguishable from each other. It is of course to 
 be recognised that a collection of the kind referred to does not 
 represent the average character of the crystals of a mineral, but 
 consists for the most part of crystals selected on account of the 
 perfection of their crystallisation or of peculiarities which they 
 illustrate. 
 
 In all the many crystals thus seemingly holo-symmetrical, if we 
 except the three crystals above alluded to, we look in vain for any 
 difference in the faces belonging to adjacent octants, such as 
 ought not unfrequently to manifest itself if these were faces 
 belonging to correlative tetrahedrid forms. Of lamellar structure, 
 and successive step-formed deposition of the laminae that build up 
 the crystal, there is abundant, almost universal, evidence. The 
 question is, whether the furrowed re-entrant edges that seem 
 peculiar to crystals composed of the octahedron, or octahedron 
 and forty-eight-scalenohedron combined, may not be looked on 
 as results of a lamellar structure in which the facetting of the sides 
 of the furrows has been an incident of the manner in which 
 the later-formed laminae of the crystal have been deposited. 
 
 196. Diplohedral crystals. The examples of diplohedral 
 crystals twinned on a face of the rhomb-dodecahedron have to be 
 looked for in the case of the minerals of the pyritoid group. 
 As regards the twinning round such an axis, the normal of a 
 2-plane, it will be seen that the poles of either of the correlative 
 semi-forms of a diplohedron or a pentagon-dodecahedron will, by 
 the process, fall into the positions of the poles of the other cor- 
 relative semiform. But with the relative disposition of the planes 
 themselves this is not the case. In fact, the result in this, as in every 
 
Twinned forms. 
 
 243 
 
 case of twinning on a [no] axis, is an interpenetrant combination. 
 An example of such a twin is illustrated in Fig. 135, which re- 
 presents a crystal of pyrites with the forms TTJIOO}, TTJIII}, 
 TT {210}. 
 
 Fig. 135- 
 
 197. Tetartosymmetrical forms. Crystals of Sodium chlo- 
 rate, presenting the features of supplementary twins, have been de- 
 scribed by Groth. They show only the faces of the form {332}. 
 Fig. 136 represents a simple crystal with the faces of the form 
 ^{SS 2 }? Fig. 137 the crystal described by Groth as a twin: 
 and the two interpenetrant individuals are found by him to 
 
 - 137- 
 
 present opposite characters as regards rotatory influence on the 
 plane of polarisation, and are on that account treated as tetarto- 
 symmetrical with the symbol 0-7TJ332}. The two quarter-forms 
 will be enantiomorphous if of opposite, tautomorphous if of the 
 same rotatory character. But if light incident on the crystal is 
 
 R 2 
 
244 Cubic system. 
 
 plane polarised, the particles of luminiferous ether within the 
 crystal will at any instant be arranged in either a right-handed or 
 left-handed helix according as the individual possesses right or 
 left rotatory polarisation. A right-handed helix is repeated over 
 a diagonal axis of symmetry as a right-handed helix, whereas it is 
 repeated antistrophically in a left-handed helix if it be reflected on 
 a plane of symmetry. Two quarter-forms of opposite rotatory 
 character cannot therefore be twins by rotation through 180 
 round a twin-axis. And it may be reasonably assumed that this 
 difference as regards congruence in physical deportment corresponds 
 to a congruence or incongruence in the molecules themselves. 
 
 In individual crystals of Sodium chlorate, the forms associated 
 with the right-handed rotatory action are such as would be extant if 
 we look on the tetartohedra o-n \hkl\ and O-TT {hkl} as dextro- 
 rotatory, and O-TT \khl\ and a"n{khl\ as Igevo-rotatory. Conse- 
 quently the association of O-TT {332} with 0-77 {m } and CTTT {210} 
 is dextro-rotatory; so is that of CTTT {332} with O-TT {III} and 
 <T7r{2lo}. The laevo-rotatory combinations would be O-TT {in} 
 with O-TT {332} and OTT { 120}; and O-JT {TIT} with O-TT {33"2~} and 
 
 (TTT {T20}. 
 
 The question here arises, as to such enantiomorphous combina- 
 tions, whether they are to be designated as twins or not ; that is 
 to say, how far twins following the ordinary law but formed by two 
 dextro-rotatory or else by two laevo-rotatory quarter-forms are to be 
 included under the same definition with combinations in which the 
 two individuals have opposite rotatory characters. The latter being 
 enantiomorphous cannot be brought into a position of parallelism ; 
 nor, indeed, can they be strictly described as identical molecular 
 structures. The two kinds of structure cannot, in fact, be included 
 under the same definition ; and two alternatives offer themselves : 
 either we may class the tautomorphous combinations in a separate 
 category as parallel growths and define the enantiomorphous com- 
 binations as twins formed by repetition on a systematic plane or 
 planes in this case the 2-systematic planes with abeyant sym- 
 metry; or, on the other hand, we may adhere to the definition of 
 twins as structures formed round an axis or axes of diagonal 
 symmetry in this case the normals of the rhomb-dodecahedron 
 
Twinned forms. 245 
 
 and refer the enantiomorphous combinations to the class of 
 parallel growths having similar systematic lines or planes in 
 parallelism but with the two antistrophic kinds of molecules 
 forming separate portions of the crystalline aggregate. 
 
 The definition on which the former view is founded, viz. that a 
 twin is a structure symmetrical to a plane parallel to a possible 
 face which is not a plane of symmetry for the separate individuals, 
 fails in completeness, since the growths of many holo-symmetrical, 
 and with few exceptions of all haplohedral, crystals undistinguishable 
 from twins have to be excluded from it, as for instance is the case 
 with the twins on a zone-axis in the Anorthic system that will be 
 hereafter considered. 
 
 On the other hand, it is not possible to include all the structures 
 that have been designated as twins, under a definition founded 
 on the law of a half-rotation round a twin-axis, even if such 
 structurally enantiomorphous combinations as those just considered 
 be excluded. 
 
 As a fact, with a single exception, all the other varieties of twin- 
 structure at present known and described as such can be com- 
 pletely and in every case would seem to be more simply explained 
 as rotation-twins. But the single exception of a particular growth 
 of crystals of the tetragonal mineral copper-pyrites precludes this 
 definition from possessing universality of application. That growth, 
 and certain other cases not so difficult of explanation by this defi- 
 nition, will be considered under the systems to which they belong. 
 
 The case of interpenetrant twin-structure is not different in kind 
 from that of juxtaposition ; there has been an intergrowth of each 
 crystal with the other, involving a mutual overlapping of their 
 material as the crystal has grown from the original nucleus of de- 
 parture for the developement of the separate individuals. 
 
 SECTION II. The Tetragonal System. 
 A. Holo-symmetrical Forms. 
 
 198. This system is characterised by a morphological axis 
 which is the zone-axis of two distinct pairs of systematic planes, 
 and is perpendicular also to a systematic plane. The planes of 
 
246 Tetragonal system. 
 
 each pair are perpendicular to each other and inclined at 45 on 
 those of the other pair ; they constitute, the one pair, the proto- 
 systematic planes S, the other pair the deutero-systematic planes 
 2. The equatorial plane, the zone-plane [-5*2], is the trito-systematic 
 plane C (Fig. 138). 
 
 The points in which the normals of these planes meet the sphere 
 of projection are the angular points of the systematic triangles ; 
 a systematic triangle being composed of two sides S and 2, 
 
 which are quadrants, and a third side C, having the value -; 
 
 4 
 the angles of the triangle being s = or = 90 and c = 45. 
 
 The morphological axis cc and the normals j/, ss are taken 
 for the crystallographic axes, and a plane with its pole on an arc 
 2 of a systematic triangle for the parametral plane. The axial 
 system is therefore represented by the expressions 
 
 and offers only a single variable element, namely, the parametral 
 ratio - Of the sixteen systematic triangles which cover the sphere 
 
 two will lie in each octant. 
 
 The distribution of the poles of the various forms under this 
 type of symmetry has been discussed in Chapter V. p. 129. It 
 remains however to consider the nature of the forms themselves, 
 and also of the semiforms which the system admits of under the 
 law of mero-symmetry. 
 
 The seven holo-symmetrical forms of the system will be 
 designated as 
 
 1. [hkl], the scalene dioctahedron, or ditetragonal scaleno- 
 hedron. Its poles lie on no arc of a systematic triangle. 
 
 2. {hko}, the ditetragonal prism. Its poles lie on the arcs C. 
 
 3. {Ao/}, the isosceles proto-octahedron,- or the S- or axial 
 isosceles octahedron. Its poles lie on the arcs S of the systematic 
 triangles. 
 
 4. {100}, the tetragonal proto-prism, or the S- or axial square 
 prism. Its poles lie at the axial points in which the arcs S and C 
 intersect. 
 
Holosymmetry. 
 
 247 
 
 5. [hhl], the isosceles deutero-octahedron, or the 2- or diagonal 
 isosceles octahedron. Its poles lie on the arcs 2 of the systematic 
 triangles. 
 
 6. {no}, the tetragonal deutero-prism, or the 2- or diagonal 
 square prism. Its poles lie at the intersections of the arcs 2 and C. 
 
 7. {ooi}, the tetragonal (basal or C-) pinakoid. Its poles are 
 the axial points in which the arcs S and 2 intersect. 
 
 \/ "* i 
 
 _ \ /Vr AOI 
 
 010 
 
 Fig. 138, 
 
 The relative positions of the poles belonging to a form of each 
 kind are exhibited in the projection, Fig. 138, of a crystal in 
 
 which the parametral ratio = 0-6724 (that of cassiterite), and 
 
 consequently the angle 101,001 = 33 55': the particular forms 
 are {100}, {320}, {no}, {ooi}, {101}, {301}, {in}, {552}, 
 {313}, and {321}. 
 
 199. i. The scalene dioctahedron {hkl}, Fig. 139. 
 
 The sixteen poles of this form are contained in 
 TABLE E. 
 
 jm hkl hkl hkl hkl 
 a khl Ihl khl kill 
 
 hkl hkl hkl hkl 
 
 Ihl khl khl Yhl 
 
248 
 
 Tetragonal system. 
 
 They lie each within a systematic triangle, and un symmetrically 
 therein, that is to say, on none of the great circles bisecting the 
 angles ; hence the faces of this triangle must be scalene triangles, 
 and the edges that are the sides of these triangles will lie in the 
 systematic planes, and will be of three kinds. 
 
 There will thus be eight similar edges lying in the proto- 
 systematic planes S; eight similar edges 2 lying in the deutero- 
 systematic planes 2 ; and eight similar basal edges C lying in the 
 trito-systematic plane C. 
 
 The pyramidal edges -S" and 2 will meet in two quoins c, 
 ditetragonal in symmetry on the morphological axis, forming the 
 vertices of the figure ; and there will be four quoins s, in each 
 of which an adjacent pair of C- edges will meet two S- edges sym- 
 
 001 i^____.p 
 
 120 ! 
 
 Fig. 139- 
 
 -Fig. 140. 
 
 metrical on the plane C, and there will also be four quoins <r in 
 each of which two 2 -edges meet other two adjacent edges C. The 
 quoins of these two latter groups are ortho-symmetrically grouped 
 on their axes j/ and oV respectively. 
 
 A section of the tetragonal scalenohedron perpendicular to the 
 morphological axis at any point is a symmetrical (never a regular) 
 octagon. 
 
 In Fig. 139 is represented the scalene dioctahedron {321} 
 corresponding to the above parametral ratio, and in Fig. 138 are 
 shown the positions of its poles on the sphere of projection. 
 
 200. 2. The ditetragonal prism {hko}, Fig. 140. 
 
 If in the symbols of a series of dioctahedra the values of h or k 
 do not alter, while that of / varies, all sections perpendicular to 
 
Holo symmetry. 249 
 
 the tetragonal axis will be similar symmetrical, but not regular, 
 octagons ; if the index / becomes zero, the -S* and 2, edges of the 
 figures become parallel to the vertical axis, and the form is that 
 of a ditetragonal prism, the poles of which lie on the zone-circle C. 
 Such a prism not being a closed figure can have no quoins and 
 can only exist in combination ; as, for instance, with the pinakoid 
 (ooi }, with which it forms quoins of two kinds (Fig. 140). Its faces 
 are all similar, but the two edges of each face are dissimilar, since 
 adjacent faces are antistrophic to each other. Four similar edges 
 lie in the proto-systematic planes S, and the four similar edges 
 alternating with them lie in the deutero-systematic planes S. The 
 faces are euthy-symmetrically divided by the trace on them of the 
 trito-systematic plane C. The symbols of the faces are 
 
 hko hko hJto liko 
 kho It ho Mo liho. 
 
 201. The isosceles octahedra. The terms proto- and deutero- 
 pyramid have been applied by various writers somewhat am- 
 biguously to the diplo-pyramidal figures, or, in crystallographic 
 language, pyramids, which have been here termed isosceles octa- 
 hedra. It would seem more appropriate to apply, as above sug- 
 gested, the term proto-odahedron to the form the poles of which 
 lie on the proto-systematic zone-circles, and the term proto-prism 
 to the figure of which the faces are parallel and perpendicular to 
 the proto-systematic planes. But Naumann applied these terms, 
 respectively, to the octahedron \hhl\ which has its poles on the 
 deutero-systematic zone-circles, and to the prism {no} which has 
 its faces parallel to the deutero-systematic planes. 
 
 To avoid ambiguity we may adopt a nomenclature which 
 recognises the relative positions of the two classes of forms on the 
 crystal, while recalling also their distinctive character as compared 
 with the analogous forms of the Cubic and Ortho-rhombic systems. 
 The former or proto-systematic forms may be designated the 
 axial, and the latter forms the diagonal isosceles octahedron and 
 square prism ; or for brevity they may be distinguished as the $'- 
 and 2 -octahedron and the S- and 2 -prism. 
 
 The square prisms, whether axial or diagonal, will evidently be 
 
250 
 
 Tetragonal system. 
 
 the limiting forms where the / index in a symbol ^o/ or in a 
 symbol hhl becomes zero by the merging into a single pole of the 
 poles /^o/and holm the one case, and of the poles hhl and hhl in 
 the other case, the resulting forms having the symbols {100} and 
 {no}. If, on the other hand, the h index becomes zero, the 
 symbols hoi and hhl alike become ooi, and represent a face of 
 the basal pinakoid. 
 
 202. 3. The axial isosceles octahedron, or S-octahedron, Figs. 
 141-2, is the form {/zo/}, of which the poles lie on the systematic 
 great circles (Fig. 138), and the edges in the deutero-systematic 
 planes 2 and in the trito- systematic plane C. Fig. 141 represents 
 the form {301} corresponding to the assumed parametral ratio. 
 
 Fig. 141. 
 
 Fig. 142. 
 
 143- 
 
 The faces are euthysymmetrically divided by the trace on them 
 of the systematic planes S, and therefore are metastrophic similar 
 isosceles triangles bounded by sides which are edges of two sorts, 
 namely, eight edges 2 culminating in two quoins c, tetragonally 
 symmetrical on the ^-axis, and four edges C lying in the equa- 
 torial plane and meeting in four quoins o- orthosymmetrical on the 
 systematic axes [2 C~\. 
 
 The symbols of the faces of the form are 
 
 hoi hoi hoi hoi, 
 ohl ohl ohl ohl. 
 
 The form {101} is parametral (Fig. 142), giving the value of 
 the one variable element of the system the parametral ratio - 
 
Holosymmetry. 
 
 203. 4. The tetragonal axial prism, axial square prism, or the 
 S -prism |ioo} 5 Fig. 143. 
 
 This prism is formed of the four faces parallel to the proto- 
 systematic planes. They build a right square prism the faces of 
 which have the symbols 100, oio, Too, oTo; it can only be 
 present in combination, as in Fig. 143. 
 
 204. 5. The diagonal isosceles octahedron or ^-octahedron {hhl}, 
 Figs. 144-5. 
 
 The poles of the diagonal octahedron lie on the deutero-systematic 
 great circles 2 and the edges of the form in the proto-systematic 
 and trito-systematic planes S and C. It is an analogous form to 
 that of the axial octahedron, but the quoins lying in the equatorial 
 plane are ortho-symmetrical on the axes [CS~\. Of the two kinds 
 of octahedra however it is to be noted that an axial and a diagonal 
 
 Fig. 145. 
 
 Fig. 146. 
 
 octahedron can in no case be congruent : were they so, adjacent 
 faces of the two forms would be symmetrical to a new plane of 
 symmetry bisecting the angle formed by two planes S and 2, a 
 condition incompatible with the principles of crystalloid symmetry. 
 
 Indeed it is readily shewn that the symbol of a diagonal octa- 
 hedron \ti ' ti '/'}, the poles of which are equidistant with those 
 of a form \mon\ from the poles {001} and {ool}, will be 
 {h, h, V^l}, where h and / are zero or integral; this is only 
 possible as a form of the system in the former case, namely, 
 where h is zero and the form is the pinakoid {ooi}, or where / 
 is zero and the form is the diagonal prism {no}. 
 
 The symbols of the diagonal octahedron are 
 
 hhl hlil hhl hhl hhl hhl hhl hll 
 
252 Tetragonal system. 
 
 The form {552} corresponding to the above parametral ratio is 
 shewn in Fig. 144, and the arrangement of its poles in Fig. 138. 
 Where the indices h and / are equal the form becomes the para- 
 metral octahedron {111 j- (Fig. 145). 
 
 205. 6. The diagonal square prism or ^-prism {no}, Fig. 146. 
 The character of the faces of the diagonal square prism is 
 
 identical with that of the axial prism, except that, as its poles lie on 
 the great circles 2, the edges of the form lie in the proto-systematic 
 planes, and the faces of the form are ortho-symmetrical on the 
 zone-lines [2C]. The symbols of the faces of the 2-prism are 
 
 no Yio iTo TTo. 
 
 Fig. 146 shews the prism {no} in combination with the form 
 
 {00!}. 
 
 206. 7. The tetragonal pinakoid {001} comprises only the two 
 faces ooi and ool parallel to the trito-systematic plane C. 
 
 It can only exist in combination with other forms, as in 
 Figs. 140, 143, and 146, and must always, where the crystal is 
 holo-symmetrical, present tetragonal or ditetragonal symmetry in 
 regard to its normal. 
 
 Tetragonal System. B. Mero-symmetrical Forms. 
 
 207. In the Tetragonal system the law of mero-symmetry may 
 take effect (i) in the abeyance of symmetral character in either of 
 the dual groups of systematic planes S or 2 ; or, (2) in that of the 
 equatorial systematic plane C, resulting in hemimorphous forms ; 
 or again (3) the symmetral character of all these systematic planes 
 may be dormant, giving rise to asymmetric haplohedral semiforms 
 of which the faces are grouped round a tetragonal (in lieu of a 
 ditetragonal) axis and round diagonal (in lieu of orthogonal) axes 
 of symmetry. And again (4) where the tritorsystematic plane C 
 retains its symmetral character, mero-symmetry requires that no 
 other systematic planes shall be planes of symmetry. 
 
 Since, in the Tetragonal system diplohedrism results alike from 
 centro-symmetry and from an actual symmetral character in the 
 equatorial plane C, it follows that holo-systematic forms may result 
 
Mer asymmetry. 253 
 
 from the first three of the above mero- symmetrical conditions, 
 while hemi-systematic semiforms only take their rise from the last 
 condition. 
 
 The mero-symmetrically distributed poles of a scalene diocta- 
 hedron are those of 
 
 I. Eight faces corresponding to the eight normals: holo- 
 
 systematic haplohedral forms; holotetragonal 
 hemihedra. 
 
 II. Eight faces corresponding to four of the normals : hemi- 
 
 systematic diplohedral forms; hemitetragonal 
 diplohedra. 
 
 III. Four faces corresponding to four of the normals : the 
 
 tetartohedral case, namely that of hemi-systematic 
 haplohedral forms; tetragonal tetartohedra. 
 
 208. The following are the mero-symmetrical forms of the 
 scalene dioctahedron and isosceles octahedra. 
 
 I. Holo-systematic haplohedral forms; holotetra- 
 gonal hemihedra: 
 
 1. Asymmetric forms ; 
 
 a{hkl} and a{khl], the tetragonal trapezohedron. 
 
 2 . Tetrahedrid or sphenoidal forms ; 
 
 (a) s\hkl} and s {khl}, the tetragonal proto- or -S-disphenoid. 
 (6) s {h o /} and s {o hi], the tetragonal axial or -^-sphenoid. 
 
 (c) s {101} and s {on}, the parametral axial sphenoid. 
 
 (d) a{hkl} and o-j/^Z'7}, the tetragonal deutero- or 2- 
 
 disphenoid. 
 
 (e) v{hhl} and {hhl}, the deutero- or 2-sphenoid. 
 
 (f} cr{i 1 1} and cr{TTT}, the parametral deutero-sphenoid. 
 (g) p\hkl\ and p{hkl} } the hemimorphous dioctahedron. 
 
 II. Hemi-systematic diplohedral forms; hemitetra- 
 gonal diplohedra: 
 
 and ${khl}, the tetragonal hemi-dioctahedron. 
 
254 
 
 Tetragonal system. 
 
 III. Hemi-systematic haplohedral forms; tetragonal 
 tetartohedra: 
 
 1. scr\hkl\, the hemidisphenoid. 
 
 2. ap{hkl\i the hemimorphous scalenohedron. 
 
 209. I. Holosystematic haplohedral forms; holotetra- 
 gonal semiforms. 
 
 i. The asymmetric semiform. The tetragonal trapezohedron a{hkl\ 
 or a\khl\, Fig. 147 (a), (c). 
 
 The defalcation of alternate planes round each of the axes of 
 symmetry of the system involves the abeyance of the symmetral 
 character of all the systematic planes, and is only possible in the 
 case of the general form, the ditetragonal scalenohedion \kkl}. 
 The morphological axis [S2] from being an axis of ditetragonal 
 
 Fig. 147 (a}. 
 
 Fig. 147 
 
 Fig. 147 Or). 
 
 symmetry becomes an axis .of tetragonal symmetry only, and 
 the four ortho- symmetral axes \s\ and [o-] become axes of diagonal 
 symmetry. 
 
 The symbols of the faces of the semiform a{hkl\ are those 
 lying in the blocks ju and f/ of Table E, Art. 199, viz. 
 
 hkl hkl likl hkl 
 
 Ihl khl khl 1hl\ 
 
 those of the semiform a {khl} are the remaining symbols of the 
 Table. 
 
 The eight faces of an asymmetric semiform (Figs. 147 a and c) 
 will be quadrilateral figures formed each by two of the similar 
 edges T that meet on the morphological axis, and two dissimilar 
 
Merosymmetry. 255 
 
 edges V and W which like the former lie in no systematic plane. 
 The edges V are traversed at their points of bisection by the 
 crystallographic axes X and Y, and the edges W are similarly 
 traversed by the secondary axes normal to the deutero-systematic 
 planes 2. There are two tetragonal quoins c on the axis Z 
 formed by the edges T y and eight similar three -faced quoins q 
 formed by the meeting of an edge T with an edge V or an edge W. 
 The faces of either form are, by the principle of the construction, 
 metastrophic trapezoids, and those of the correlative form are 
 antistrophic to them, so that the two figures are enantiomorphous. 
 
 210. 2. The letrahedrid or sphenoidal mero- symmetry. 
 
 The eight normals of the tetragonal scalenohedron may carry 
 the eight faces of a semiform in other ways than asymmetrically as 
 regards the systematic planes. Thus the faces may be extant or 
 absent in alternate pairs, symmetrical either on the proto- or on 
 the deutero-systematic planes : and, in accordance with the notation 
 adopted in Art. 148, p. 167, these will be denoted as s{hkl\ if 
 the proto-systematic planes 6" are the planes of symmetry, and as 
 <j{hkl\ where the planes of symmetry are the deutero-systematic 
 planes 2. In neither case is the form symmetrical to the trito- 
 systematic plane or to a centre. 
 
 The axis [-5*2], which is an axis of ditetragonal symmetry in the 
 holo-symmetrical forms, is here one of ortho-symmetry ; the ortho- 
 symmetrical axes o- or [2C] become diagonal axes, and the axes 
 s or \SC~\ lose their symmetral character. These forms are 
 termed sphenoidal, from their wedge-like contour. 
 
 211. (a) The tetragonal proto- or axial or S-disphenoid s\hkl\ 
 or j{A/j,Figs. 148 (a\ (c). 
 
 The symbols for the faces of the first semiform are 
 hftl hkl ~hkl hkl 
 khl khl Ihl Ihl 
 those of the semiform \khl\ are 
 
 khl khl khl Yhl 
 hkl hkl hkl hkl. 
 
 The eight faces bounding either semiform (Figs. 148 a and c) 
 will be triangles and will have two sorts of edges, * and S' } lying 
 
256 
 
 Tetragonal system. 
 
 in the systematic planes S, and grouped ortho-symmetrically on 
 the axis of form. The remaining four edges U are bisected by the 
 diagonal zone-axes [C2] and lie in zigzag across the equatorial 
 systematic plane. 
 
 The quoins are of two kinds ; two ortho-symmetrical quoins 
 formed by the meeting of the edges and S' on the zT-axis, 
 and four lateral four-faced quoins in which two of the edges U 
 meet an edge S and an edge -S". 
 
 The two correlative semiforms are tautomorphous. 
 
 Fig. 148 (a\ Fig. 148 (b\ Fig. 148 (c\ 
 
 (I)) The tetragonal axial or S-sphenoid, s{hol} or s\ohl\, Figs. 
 149 (a), (c), 150 (a), (c). 
 
 The faces of the disphenoid being symmetrical in pairs on 
 the planes S, it is evident that a semiform of which the poles 
 lie on the great circles S will accord with the same condition 
 of symmetry. Such a semiform is the tetragonal axial sphenoid 
 s{hol} t the faces of which are represented by the symbols 
 
 hoi hoi ohl ohl, 
 the correlative ^-sphenoid being composed of the faces 
 
 ohl oil I hoi hoi. 
 
 The sphenoids resemble in general features the tetrahedron of 
 the Tesseral system, and have a double-wedge-like aspect; see 
 Figs. 149 a and c. The four faces are isosceles triangles (whereas 
 those of the tesseral tetrahedron are equilateral), and the two bases 
 of the triangles are edges perpendicular in their directions, being 
 respectively parallel to the X and Y axes. These crystallographic 
 axes pass through the faces of the form but are not axes of 
 symmetry, while the axes [SC], no longer orthosymmetral, retain 
 
Mer asymmetry. 
 
 257 
 
 diagonal symmetry and bisect the four equal edges. The morpho- 
 logical axis becomes only an axis of ortho-symmetry, traversing 
 the points of bisection of the basal edges. The four quoins are 
 
 Fig. 149 (a). 
 
 Fig. 149 
 
 Fig. 149 
 
 similar and three-faced, each being formed by the meeting of two 
 of the four equal edges with one of the basal edges. 
 
 (c) When the indices h and / are equal, the semiforms are the 
 parametral axial sphenoids shewn in Figs. 150 (a) and (<:). 
 
 Fig. 150 (a). Fig. 150 (). Fig. 150 (<:). 
 
 212. (d) The deutero- or diagonal or ^-disphenoid cr{hkl\ or 
 {M/}, Figs. 151 (a), (c). 
 The 2- or diagonal disphenoid is mutatis mutandis identical 
 
 Fig. 151 (a). 
 
 Fig. 151 (0- 
 
258 
 
 Tetragonal system. 
 
 in its general features with the S-disphenoid : but its pyramidal 
 edges lie in the 2-planes which are the planes of symmetry in 
 place of the -5*-planes, and the zigzag edges are bisected by the 
 axes X and Y. The symbols of the faces of the 2-disphenoid 
 {hkl}, corresponding to the left half of Table E, Art. 199, are 
 
 hkl Ykl hkl hYl 
 khl Yhl ~khl khl. 
 
 Those of the correlative semiform a- {hkl} are contained in the 
 right half of Table E, viz. 
 
 "hkl hkl hkl hYl 
 ~khl khl khl Yhl. 
 
 (e) The tetragonal diagonal sphenoid or ^-sphenoid, <r {hhl} or 
 a- {hYl}, Figs. 152 (a), (c), 153 (a), (c), presents similar features 
 to the ^-sphenoid, but its poles lie on the great circles 2 which 
 
 Fig. 152 (a). Fig. 152 (). Fig. 152 (^. 
 
 retain their symmetral character, while that of the proto-systematic 
 planes S is in abeyance. The two wedge-shaped basal edges are 
 parallel to the zone-lines [C2]. 
 
 The symbols of the faces of the form a- {hhl} are 
 
 hhl Yhl hhl hYl; 
 those of the form o- {hhl} are 
 
 hhl hlil hhl hhl 
 
 The crystallographic axes penetrate and bisect the edges of the 
 form. The morphological axis, as in the ^-sphenoid and di- 
 sphenoid, becomes an axis of ortho-symmetry, the [s\ or \SC\ 
 
Merosymmetry. 
 
 259 
 
 axes become axes of diagonal symmetry, but the symmetral 
 character of the [cr] axes is in abeyance. 
 
 (/) When h equals /, the figures are the parametral deutero- 
 sphenoids shewn in Figs. 153 (a), (c). 
 
 Fig. 153 
 
 Fig. 153 (&)> 
 
 Fig. 153 (0- 
 
 Since pro to- and deutero-sphenoidal forms cannot concur on a 
 crystal, and it is at the option of the crystallographer to take either 
 pair of similar axes as crystallographic axes, the analogy with 
 tetrahedrid forms in the Cubic system points to the selection of an 
 axial system such that, when sphenoidal semiforms occur on a 
 crystal, they are treated as deutero-sphenoidal : and in usage 
 therefore they are termed simply sphenoidal semiforms. 
 
 213. (g) Another variety of holo tetragonal mero-symmetry is 
 presented in the case of the hemimorphous dioctahedron p {hkl} 
 wherein the systematic plane C fails of being symmetral. 
 
 And since this kind of hemi-symmetry permits of the other 
 systematic planes being symmetral, we may have hemimorphous 
 forms in which the indices h and k are equal, or one of them is 
 zero, and of which therefore the poles lie on one of the great 
 circles S or 2, or on both. Such semiforms comprise those with 
 the symbols p {hhl\, p{m}, p {hoi}, p{ioi}, /o{ooi}. 
 
 None of these hemimorphous pyramids or prisms can exist 
 except in association with other forms. 
 
 214. II. Hemisystematic diplohedral forms; hemi- 
 tetragonal diplohedra. The tetragonal isosceles octahedron or 
 hemidiodahedron </> {hkl} or </> {khl}, Figs. 154 (a), (c). 
 
 In a hemi-systematic form of the Tetragonal system four normals 
 are absent, and the extant four are diplohedral : a distribution 
 which can only occur in one manner, so that there is only one 
 type of hemi-systematic diplohedral form. Of the faces, as grouped 
 round the morphological axis, only alternate faces are extant, but 
 
 S 2 
 
260 
 
 Tetragonal system. 
 
 the adjacent faces of the form symmetrical on the trito-systematic 
 plane C are concurrent with them, since these are the faces parallel 
 to the former and thus have normals in common with them. The 
 only plane of symmetry is the trito-systematic plane C, the 
 symmetral character of the S and 2 planes being in abeyance ; 
 and the form is obviously centro-symmetrical. The symbols of 
 the correlative semiforms are, of 
 
 4>{hkl), hkl khl h~kl khl, 
 
 ~hkl khl hk~l ~khJ; 
 and of <f> {khl}, 
 
 khl hkl Yhl hkl, 
 
 Yhl hkl khl ~hkl 
 
 The form has the character of a tetragonal octahedron, Figs. 154 (a), 
 (c), its section perpendicular to the Z-axis being a square, the sides 
 
 Fig. 154 0). 
 
 Fig. 154 
 
 Fig. 154(0. 
 
 of which are however not parallel to those of the axial or diagonal 
 forms ; the pyramidal edges are gyroidally grouped in tetragonal 
 symmetry round the ^-axis and are similar; so again are the 
 basal edges. Hence the faces have the character of metastrophic 
 isosceles triangles. The quoins are of two kinds, two gyroidal 
 tetragonal quoins symmetrical on the morphological axis, and four 
 similar four-faced quoins geometrically but not crystallographically 
 ortho-symmetrical on two lines passing through the origin. 
 
 215. The hemidiprism <f> {hko} or (p {kho}, Figs. 155 (a), (c). 
 
 The alternate faces of the ditetragonal prism (Fig. 155 b) pro- 
 duce a prism with a square section which is the limiting form of 
 
 a series of hemidioctahedra having the same ratio for ; they 
 
Merosymmetry . 
 
 261 
 
 are tetragonally instead of, as in the diprism, ditetragonally dis- 
 posed round the morphological axis. 
 
 The faces of the form </> {hko} (Fig. 155 c) are, 
 
 hko ~kho Jiko kTiOj 
 and of the form </> [kh o} (Fig. 155 a), 
 
 kho hko ItliQ hko. 
 
 Their edges are similar, but symmetrical to no plane except the 
 trito-systematic plane to which the faces are euthysymmetrical. 
 And it will be seen that the sections of the forms <f> {hkl\ and 
 {hko} perpendicular to the morphological axis do not accord 
 in the character of their angles with a crystallographic square. 
 
 The association of the hemidioctahedron and hemidiprism with 
 holo-symmetrical forms imparts also to the latter a quasi-mero- 
 
 20 
 
 "S -L 001 ! 
 
 2^3^. 
 
 ! 
 
 i 
 
 
 i 
 
 
 
 
 | 
 
 1 
 
 20 2301 
 
 230 i 
 
 320J 
 
 i 
 
 i 
 
 
 
 L 
 
 j 
 
 530 
 
 Fig- 155 () 
 
 Fig. 155 (). 
 
 Fig. 155 (c). 
 
 symmetral character, since they thereby lose the special symmetry 
 which characterises their edges ; and, as in the analogous case in 
 the Cubic system, it cannot be doubted that this is only the external 
 exponent of an interruption in the symmetry of the physical 
 characters throughout the crystal. 
 
 In this sense such forms as <j>{hhl} } ^{hol}, ${ooi}, 
 (j> {100} and <j) {no} may be recorded as existing on a crystal. 
 
 216. III. Tetartohedral forms; tetragonal tetartohedra. 
 
 The tetartohedra of the Tetragonal system are limited to two 
 kinds, necessarily derived, as distinct geometrical forms, from the 
 groups of faces forming the dioctahedron. As regards the eight 
 normals of this form, there is only one way of suppressing four 
 of them : ditetragonally disposed round the [SI>] axis, they must 
 be reduced by hemi-systematic suppression to four normals tetra- 
 gonally disposed round that axis. But the single face correspond- 
 
262 
 
 Tetragonal system. 
 
 ing to a normal may be extant or absent alternately on one or the 
 other side of the trito-systematic plane, or all four faces may be 
 simultaneously extant or absent on one side of that plane. Hence 
 the two following forms may be originated : 
 (i) The hemidisphenoid, so- {hkl}. 
 
 The faces common to each pair of forms of a proto- and a 
 deutero-disphenoid combine in producing one of the four quarter- 
 forms 
 
 s<r{hkl} t s<r{hll\, 
 sv{khl}, s<r\khl} 9 
 
 each of which is a sphenoid figure with triangular faces which are 
 geometrically but not crystallographically isosceles ; their symbols 
 being, 
 
 for sir {hkl}, hkl, hkl, khl, khl (Fig. 156 c); 
 
 sir {hkl}, hkl hkl khl A /(Fig. 1567); 
 s<r{khl} t khl III Jikl A 7 (Fig. 1560); 
 s<r{khl}, ~khl khl hkl 111 (Fig. 156 d). 
 
 Fig. 156 (a). 
 
 Fig. 156 (3). 
 
 Fig. 156 (0. 
 
 Fig, 156 
 
 Fig. 156 0). 
 
 Fig. 156 (/) 
 
Merosymmetry. 
 
 26 
 
 The wedge-like edges W of the sphenoid parallel to the XY 
 plane are not parallel to the axes X, JT, but lie obliquely to them, 
 as in the case of the diagonals of the hemidiprism. 
 
 The [S"S] zone-axis is an axis of diagonal symmetry. 
 
 Taken in pairs these tetartohedral forms reconstitute the two 
 corresponding disphenoids ; the faces of the forms whose symbols 
 in the above Table are in alternate rows constituting a diagonal, 
 those whose symbols are in the two upper or two lower contiguous 
 rows, an axial-disphenoid ; the symbols in the first and fourth 
 rows constitute the hemidioctahedron (/> \hkl\, those in the second 
 and third rows the correlative form$ {khl}, (see Plate II). 
 
 (2) The hemimorphous hemidioctahedron. The symbols of the 
 faces belonging to the several quarter-forms are ; 
 
 for 
 
 ap\hkl}, 
 
 hkl 
 
 khl 
 
 hkl 
 
 khl, 
 
 ap{khl}, 
 
 khl 
 
 hkl 
 
 Thl 
 
 hkl, 
 
 ap\Ykl}, 
 
 hkl 
 
 khl 
 
 hkl 
 
 Thl t 
 
 ap{Wl], 
 
 Thl 
 
 hkl 
 
 khl 
 
 hkl. 
 
 Each of them is formed by four alternate faces of the diocta- 
 hedron lying above or lying below the trito-systematic plane C, 
 and can only occur in combination with those of other forms. 
 Fig. 157 represents the hemimorphous hemidiocta- 
 hedron ap {321} in combination with the forms 
 ap {100} and ap{Ti~i}. 
 
 Those faces of which the symbols lie in the first 
 and fourth rows would concur to build up a trapezo- 
 hedron a{hkl], those in the second and third, the 
 form a {khl} those in the two upper, or in the two 
 lower rows, would constitute a hemimorphous diocta- 
 hedron p\hkl} or p{7ikl}. Taken alternately, the 
 first with the third or the second with the fourth 
 row, the faces represented by these symbols would 
 by their union produce a hemidioctahedron ($>{hkl} or 
 respectively (see Plate III). 
 
 There is as yet no proof that either of the above tetartohedral 
 developements has been observed on tetragonal crystals : Werther, 
 indeed, has described crystals of urea which presented a single 
 
 Fig. 157- 
 
264 
 
 Tetragonal system. 
 
 pinakoid plane {001} in combination with a square prism {no} 
 and a sphenoid a- { 1 1 1 }, but the occasional absence of a single 
 plane can scarcely be regarded as sufficient evidence of the actual 
 occurrence of tetartohedral crystals. 
 
 Tetragonal System. C. Combinations of Forms. 
 
 217. The simpler character of the symmetry of the Tetragonal 
 system as compared with that of the Cubic system is conspicuous 
 in even the most complex combinations, since the faces of all the 
 forms are seen grouped round a morphological axis with much 
 regularity, and frequently with an approximation to equipoise 
 developement. 
 
 The following figures illustrate the dispositions of the faces of 
 some prominent forms when combined with each other. 
 
 Fig. 159. 
 
 Fig. 1 60. 
 
 Fig. 158 represents the combination of the axial prism with the 
 axial octahedron. 
 
 Fig. 159 shews the diagonal prism and octahedron in com- 
 bination, and Fig. 160 the axial prism and diagonal octahedron. 
 
 /..... lol 
 
 Fig. 161. 
 
 110 
 
 illO 
 
 oil- 
 Fig. 162. 
 
Combinations. 
 
 265 
 
 In Figs. 161-165 are illustrated combinations of several other 
 simple forms. 
 
 Fig. 163. Fig. 164. 
 
 Figs. 1 6 6, 167, 168 represent respectively crystals of anatase, 
 
 321... 
 
 Fig. 165. 
 
 Fig. 1 66. 
 
 / ooi v_;, 
 
 :;, 
 
 g 
 
 38 
 
 , 2 ,\ 
 
 III! 
 
 MMJlp 
 
 10 
 
 jf 
 
 iac 
 
 '\frr~ 
 
 QIC 
 
 120 
 
 no 
 
 210 '! 
 
 si 
 
 i\iai 
 ~~-^-~l 
 
 g 
 
 121 / 
 
 in 
 
 s^ 
 
 Fig. 167. 
 cromfordite, and cassiterite which have been actually observed, and 
 
266 Tetragonal system. 
 
 they illustrate the concurrence in various combinations of the 
 diprism, the dioctahedron, the axial and diagonal forms, and the 
 pinakoid. 
 
 218. (a) Of the mero-symmetrical forms of the Tetragonal system 
 the asymmetric forms are not known to be represented on crystals, 
 but in the case of hexa-hydrated strychnine sulphate, by a process 
 of etching on the faces, Baumhauer developed traces of faces 
 grouped asymmetrically. 
 
 (<5) Of the other types of hemisymmetrically developed forms, the 
 tetrahedrid (or sphenoidal) is illustrated by the minerals copper- 
 pyrites and edingtonite. Copper pyrites occurs as a deutero- 
 sphenoid <T{III}, which differs from the regular tetrahedron by 
 the normal-angle between two faces of the sphenoid, (in, ill) 
 being io84o', and (in, Til) being 109 53'; each of the corre- 
 sponding angles in the regular tetrahedron being 109 28'. 
 
 Again, the angle (ooi, in) is 
 
 in copper pyrites 5 4 20', 
 in the Cubic system 54 44'. 
 
 Fig. 169. Fig. 170. 
 
 The o and co sphenoids often concur on copper pyrites as they 
 do on blende, the positive or o sphenoid a- { 1 1 1 } having its faces 
 dull from a brown incrustation of copper suboxide, and striated 
 parallel to their intersections with the two faces of the form {201} 
 tautozonal with and adjacent to them : they are also larger than 
 the o> faces belonging to the form o- {TIT}, which on the other 
 hand are lustrous and without striation. The crystals therefore 
 shew considerable analogy to those of blende. They sometimes 
 carry also the faces of the deutero-disphenoids cr {316} and 
 <r\3i2\, besides other forms the poles of which lie on the syste- 
 matic zone-circles. Fig. 169 represents a crystal of this mineral. 
 
Combinations. 
 
 267 
 
 Edingtonite (Fig. 170) and strontium-uranyl acetate crystallise 
 in forms which, owing to the manner in which the axial system 
 has been selected, represent the proto-sphenoidal type s {hkl}. 
 
 (c) The hemitetragonal but diplohedral type is represented by 
 several crystalline bodies. Scheelite (calcium tungstate), repre- 
 
 Fig. 171. 
 
 Fig. 172. 
 
 sented in Fig. 171, stoltzite (lead tungstate), and wulfenite (lead 
 molybdate) occur in crystals of this type, as do also fergusonite, 
 shewn in Fig. 172, and sarcolite (Fig. 173), a mineral belonging 
 to the same group with scapolite, which itself exhibits mero-sym- 
 metry of this kind. 
 
 Fig. 173. 
 
 Fig. 174. 
 
 (d) A hemimorphous developement of a holosymmetrical form 
 has been observed by Groth on crystals of an artificial compound, 
 iodosuccinimide (C 4 H 4 NO 2 I), illustrated in Fig. 174. 
 
268 
 
 Tetragonal system. 
 
 Tetragonal System. D. Twinned Forms. 
 
 219. I. Holosymmetrical crystals. The axis of form [ooi], as 
 being an axis of tetragonal symmetry, and the horizontal axes 
 [ioo] [oio] and [no] [Iio], as being axes of diagonal symmetry 
 to the system, are precluded from being twin-axes for holo- 
 symmetrical forms. In fact, the only twin-axes known to occur 
 are such as lie in one or other of the systematic planes but are 
 not normals of such planes. Thus, a face of the form {101} is 
 a frequent twin-plane. 
 
 Fig. 175 is a twin of zircon twinned on a plane of the form 
 {101}, which is also the combination-plane; and similar twins are 
 common in cassiterite (Fig. 176) and rutile. 
 
 Fig- 175- 
 
 Fig. 176. 
 
 Reiteration of the twinning is common in rutile and cassiterite ; 
 sometimes on faces parallel with the first twin-face, resulting in 
 geniculated prolongations of the crystals, as in Fig. 177, which 
 represents a twin-growth of the latter mineral. 
 
 Sometimes the twinning is repeated on each of the eight faces 
 of the form {101} ; or again the crystals are successively twinned, 
 each succeeding crystal on the previous one in the series. 
 
 Pseudo-hexagonal twin-forms may thus arise, as in the case of 
 cassiterite, whereof six individuals are sometimes grouped round 
 a line which is perpendicular to a plane containing the morpho- 
 logical axis of each of the six crystals (Fig. 178); adjacent axes of 
 five of the individuals being inclined to each other at an angle of 
 ii2ic', while the axis of the sixth individual, which is twinned 
 
Twinned forms. 
 
 269 
 
 on the fifth but not on the third, meets the axis of the latter at an 
 angle of 139 10'. 
 
 Fig. 177. 
 
 Fig. 178. 
 
 II. Hemisymmetrical crystals. As in the Cubic, so in the Tetra- 
 gonal system, a twin upon a face of the form {no} or of the 
 form {100}, though impossible in holo -symmetrical crystals, may 
 occur where the normal on such a face is not an axis of diagonal 
 symmetry. Twinning upon a face of the form {no} is not 
 possible, therefore, in the case of a semiform of asymmetrical type 
 a {hkl}, but it is possible with each of the other hemisymmetrical 
 types of the Tetragonal system. 
 
 Fig. 179. 
 
 Fig. 1 80. 
 
 (a) Twins of Diplohedral crystals. The hemitetragonal diplo- 
 hedral type of crystal (Fig. 171) occurs in the twinned condition in 
 certain of the growths of scheelite (calcium tungstate). The twin- 
 plane is a face of the prism {no}, which is also the combination- 
 plane, and the crystals interpenetrate (Fig. 179). Or again, each 
 
270 Tetragonal system. 
 
 is represented by a half-crystal divided by the combination-plane, 
 the two half-crystals being then in juxtaposition. 
 
 220. (3) Twins of Haplohedral crystals. Under the type of 
 holotetragonal haplohedral symmetry copper pyrites affords the 
 only illustrations of twinned forms, but these exhibit considerable 
 variety. Three twin-laws have been ascribed to this mineral ; the 
 twin-plane being in the several cases, (i) a face of the form 
 {in}, (2) a face of the form {no}, and (3) a face of the form 
 fioi}. 
 
 (1) Twin-law ; the twin- plane a face of the form {in}. 
 
 Crystals of copper pyrites twinned under this law present con- 
 siderable analogy to those of blende similarly twinned ; the bright 
 faces co or o-{7T7} of the one individual are seen in juxta- 
 position with the duller ooro-{in} faces of the other crystal, the 
 combination-plane being parallel to the twin-plane. Thus a 
 revolution of one crystal, from parallel orientation, round an axis 
 normal to the face nl would bring the two individuals into a 
 relative juxtaposition analogous to that seen in blende (Fig. 125): 
 such a twin is illustrated in Fig. 180. 
 
 (2) Twin-law; the twin-plane a face of the form {no}. 
 
 This variety of twin-structure is so rare as almost to rest for its 
 authority on the early description given of it by Haidinger in his 
 classical memoir on the crystallisations of copper pyrites. The 
 twin is interpenetrant and is very similar to Fig. 133. 
 
 (3) Twin-law; the twin-plane a face of the form {101}. 
 
 The true character of the crystal-growths that obey this law has 
 been obscured, in part as a consequence of a certain ambiguity in 
 the language in which it was described sixty years ago by Haidinger 
 in one of his papers ' On the Regular Composition of Crystallised 
 Bodies/ but chiefly from the difficulty of finding crystals complete 
 enough, or with faces affording sufficiently good reflections to 
 enable accurate measurements of the angles to be made. Mr. 
 Fletcher has recently pointed out the real significance of the words 
 in which Haidinger described the twin, and by subjecting the 
 crystals in the British Museum collection to a careful examination 
 has established the true character of the growth. That collection 
 possesses several excellent examples of the twin, including a 
 
Twinned forms. 271 
 
 specimen which had been described and figured by Haidinger 
 himself as an illustration of the law. 
 
 The correct description of this particular combination of crystals 
 of copper pyrites possesses additional importance from the fact 
 that the interpretation of its character, following from these ob- 
 servations, either involves so strained an application of the ordinary 
 twin-law as to be quite unsatisfactory, or leads to the alternative 
 that these apparent twins hold an entirely exceptional position 
 among such crystal-growths. They, in fact, present the sole 
 known illustration of a ' symmetrical twin ' of Groth (Article 161), 
 which cannot be completely explained either by the ordinary twin- 
 law as applicable to mero-symmetrical crystals, or by a principle of 
 parallel orientation of equivalent directions where the union is 
 one of enantiomorphous tetarto-forms. 
 
 Haidinger, in 1822*, described the law of these twins as being 
 twin-plane, a face of the octahedron {101}, combination-plane 
 perpendicular to the twin-plane ; but, in a subsequent recital of his 
 earlier conclusion 2 , he described the law as ' regular composition 
 parallel to a face of the octahedron {101} or perpendicular to the 
 terminal edges of the octahedron { 1 1 1 }.' That Haidinger implied 
 the twin-plane of the regular composition, and not the com- 
 bination-plane, by this alternative statement is evident from his 
 referring immediately afterwards to the above as a single kind of 
 regular growth. And such in fact it is, crystallographically 
 speaking, if the twin-plane be in view, for then these statements 
 are identical in their results, as is seen from the following con- 
 siderations. 
 
 The growth due to twinning about the plane (101) is such that 
 both of the normals to the planes (oio), (101), which are at right 
 angles to each other, are axes of ortho symmetry to the com- 
 bination : hence a third line at right angles to both, and therefore 
 parallel to an edge of the octahedron {in}, will likewise be an 
 axis of orthosymmetry to the combination, though not to the 
 separate individuals; and the same combination will thus be pro- 
 duced by a rotation of one of the individuals through two right 
 
 1 Memoirs of Wernerian Society, vol. iv. p. I, 1822. 
 
 2 Edinburgh Journal of Science, vol. iii. p. 68, 1825. 
 
272 Tetragonal system. 
 
 angles around this edge as by a similar rotation round the normal 
 to the plane (101). 
 
 As to the combination-plane, it is, according to Haidinger's 
 view, perpendicular to the face (101) : if this be true, its indices are 
 irrational and the plane is not a possible face of the crystal. It is 
 now found, however, that an erroneous interpretation placed on 
 the above statement of Haidinger by Naumann, and after him by 
 Sadebeck and since generally accepted on their authority, more 
 nearly represents the nature of the combination. This interpre- 
 tation gives the law as twin-plane a face of the form (101) and 
 combination-plane parallel to the twin-plane if the crystals of 
 copper pyrites were holo-symmetrical, this statement of the law 
 would be an accurate description of the twin, and would accord 
 with the results of the measurements recently made. Or, if the 
 faces of the second individual that are brought into contiguity with 
 the faces of the first individual by a rotation round, say, the 
 normal of (xol) presented a contrast in respect to the characteristics 
 of striation, brilliance and magnitude that distinguish the faces of 
 the semiforms <r {hhl} or <r {hkl\ (the o forms) from the faces of 
 the correlative (to) semiforms cr {hJil} or <y\hkl\, the explanation 
 would be complete. But it is indubitable that this is not the 
 case : that in fact it is an o- and not an co-face that is found con- 
 tiguous to an o-face throughout (Fig. 181). 
 
 The explanation by Naumann thus fails as a simple statement of 
 the twin-law. It would require, in order to make it complete, that 
 the original orientation of the two individuals gave correlative faces 
 the same aspect towards space ; such as would arise if from a 
 position of identical orientation one crystal was revolved through 
 two right angles round an axis normal to a face of the form {no} 
 similar, namely, to the revolution under the previously considered 
 twin-law. 
 
 Corresponding systematic planes would be parallel in such an 
 orientation, but the faces with corresponding aspects would belong 
 to opposite octants, in the two individuals. 
 
 Assuming such an initial revolution round a normal of the form 
 {no}, a second revolution of one of the crystals would have to 
 take place round a face-normal of the form {101} in order to give 
 
Twinned forms. 
 
 273 
 
 such an association of the crystals as that under consideration; 
 and the plane of combination would be parallel to the plane of 
 twinning. Evidently, in such a case, this plane of combination is 
 physically, as well as geometrically, a plane of symmetry for the 
 two crystals, the faces of either crystal being reflected in it with 
 identical features. Fig. 182 shows the relative positions of the 
 
 Fig. 181. 
 
 Fig. 182. 
 
 o and co planes of an actual twin-growth according to this law; 
 for the sake of clearness the growth is represented as having been 
 rotated through a right angle, about the vertical axis, from the 
 position shown in Fig. 181. 
 
 Such a combination represents the 'symmetrical twin' of Groth, 
 and it is as a symmetrical twin that it is most simply described, 
 
 SECTION III. The Hexagonal System. 
 
 A. Holo-symmetrical Forms. 
 
 221. Crystals belonging to the Hexagonal system are sym- 
 metrical to a morphological axis (the axis [S2]), which in the 
 case of complete symmetry is the zone-axis of the two triads of 
 systematic planes, that is to say, of the three proto-systematic planes 
 S (Sj_ S z S 3 }, and of the three deutero-systematic planes 2 (2j 2 2 2 3 ), 
 and is perpendicular to the trito-systematic plane C which is the 
 plane of the zone-circle [S"2], (Fig. 183). 
 
 The crystallographic axes OX, OF, OZ lie in the proto-systematic 
 planes and are equally inclined on the axis of form, but are not 
 axes of symmetry. The parameters being by the conditions of 
 
274 Hexagonal system. 
 
 symmetry necessarily equal, the axial system is represented by the 
 expressions 
 
 = n = C < 90, 
 
 a = b = c. 
 
 The zone-axis \_S^\ is an axis of dihexagonal symmetry, and is 
 termed, for brevity, the hexagonal axis. 
 
 The mutual inclinations of the planes of either triad of sys- 
 tematic planes are 60, the inclinations of a plane of one group 
 on the adjacent planes of the other being 90 and 30. 
 
 The normals of the one group coincide with the lines in which 
 the planes of the other group intersect the trito-systematic plane C, 
 and are axes of ortho-symmetry. 
 
 The systematic triangle of the Hexagonal system is formed by 
 two quadrantal arcs S, 2, on great circles *$" and 2, and an arc C 
 
 with the value - on the great circle C; its sides being 
 
 its angles o- = s = 90, c 30. 
 
 222. The relations of two diplohedral plane-systems symme- 
 trical, the one to a trigonal, the other to a hexagonal axis, have 
 already been considered (Chapter V, Cases 3 and 4), and it has 
 been seen that the conditions which need to be fulfilled by the axial 
 elements are identical for the two systems. Further, in treating of 
 the Tesseral system (Case 5), we had to deal with diplohedral forms 
 presenting symmetry of a trigonal type which, under the conditions 
 of that system, had to be considered as holo-systematic forms. 
 
 In the system now under discussion the trigonally symmetrical 
 forms which are possible are numerous ; but they will have to be 
 treated in the character of hemi-symmetrical forms of the Hex- 
 agonal system, rather than as representing an independent type of 
 symmetry. The semi-independent character of a diplohedral trigonal 
 system is, however, kept continually in view in the Hexagonal system 
 by the composite character of the symbols of certain of its forms, and 
 conveniently also in some degree by a nomenclature that recalls the 
 relations of the two systems : indeed this quasi-trigonal character 
 is found to underlie the symmetry of a large number of the most 
 
Holosymmetry. 
 
 2 75 
 
 important of the substances which crystallise in this system in forms 
 of rhombohedral type : of these calcite is a conspicuous example. 
 
 101 
 
 51* 
 
 Fig. 183. 
 
 223. The surface of the sphere of projection is divided by the 
 seven systematic planes into twenty-four systematic triangles, each 
 of which would contain, in unsymmetrical position within it, a pole 
 of the general scalenohedral form (the discalenohedron) of the 
 system, the symbol for which is composite, namely, {hkl, efg}. It 
 is thus a twenty- four-faced figure. 
 
 The seven holo-symmetrical forms of the Hexagonal system will 
 consist of such as have their poles situate 
 
 1. on no systematic great circle, the general independent form, 
 
 {hkl, efg} 
 
 2. on the trito-systematic great circle C only, {pqr} ; 
 
 3. on the proto-systematic great circles -S* only, \hkk, eff}\ 
 
 4. on the deutero-systematic great circles 2 only, \min~] ; 
 
 5. at the intersection of the great circles $and C, {2!!} ; 
 
 6. at the intersection of the great circles 2 and C, {iol} ; 
 
 7. at the intersection of the great circles S and 2, {in}. 
 
 T 2 
 
276 Hexagonal system. 
 
 The relations of the indices in the several symbols of these 
 forms have already been established in the chapter dealing with 
 crystalloid symmetry. 
 
 In fact the indices of the general symbol {hkl, efg} are con- 
 nected by the relation 
 
 g= 
 
 where therefore e +f-\-g = 3 (h + k + /) : h k I are in the order of 
 decreasing magnitude. 
 
 For the symbol {pqr} of a plane in the zone [in] the indices 
 have to satisfy the condition p + q + r o, where the magnitudes 
 of the indices taken absolutely are in the order p > q > r ; and 
 in the symbol {min} the index i is the arithmetic mean of the 
 
 other two indices (i.e. i = - ), of which m is assumed to be 
 
 algebraically the greater. 
 
 The seven holo- symmetrical forms will be designated as follows : 
 
 1. {hkl, efg}, the dihexagonal scalenohedron or discalenohedron. 
 
 2. {pqr}, the dihexagonal prism or diprism. 
 
 3. {hkk, eff}, the (proto-dihexahedron or) dirhombohedron. 
 
 4. \min\, the (deutero-dihexahedron or) hexagonal pyramid. 
 
 5. {2!?}, the hexagonal proto-prism or -S'-prism. 
 
 6. {ioT}, the hexagonal deutero-prism or 2-prism. 
 
 7. { 1 1 1 }, the hexagonal pinakoid. 
 
 The positions of the poles of such forms on the sphere of 
 projection are illustrated in Fig. 183 for the case where 
 
 = n = e=ii2M' 
 
 (as in the mineral willemite). In Table F the poles of the di- 
 scalenohedron are grouped in triads; those of the semiform {hkl\ 
 (to which the axis [-5*2] is an axis of ditrigonal symmetry) taken 
 in the order hkl are indicated by p, and in the order hlk by p' \ 
 the poles of the correlative semiform {efg} are indicated in a 
 similar way by q and q. 
 
 224. i. The discalenohedron {hkl, efg\, Fig. 184. The symbols 
 
Holosymmctry. 
 
 277 
 
 of the faces of the general independent scalenohedron of the Hexa- 
 gonal system are contained in 
 
 TABLE F. 
 p...hkl Ihk klh hlk khl lkh...p', 
 
 ITk Yh kh...p', 
 
 egf feg gfe...q'-, 
 
 p ...hk hk ih 
 
 q...efg gef fge 
 wherein h > k > I and h + k + / ^ o, and any of the literal 
 indices may represent either positive or negative values inde- 
 pendently of the sign in the general symbol. Fig. 184 represents 
 the form {20!, 425} corresponding to the above parametral angle. 
 
 
 J4S 
 
 415 
 
 Fig. 184. Fig. 185. 
 
 The twenty-four alternately metastrophic triangles which com- 
 prise the form are grouped round the morphological axis, and form 
 two pyramids united by a common base in the horizontal plane C } 
 a figure which, though doubly terminated, is by crystallographic 
 usage termed a pyramid. Any section of this figure perpendicular 
 to the morphological axis will be a symmetrical hexagon, of which 
 only the alternate angles are equal. 
 
 Two dihexagonal quoins form the vertices of the pyramids and 
 are composed by edges and 2 alternating with each other, ad- 
 jacent edges representing dihedral angles of different magnitudes. 
 
 The twelve basal edges C common to the two pyramids are 
 similar ; but the six lateral quoins which these edges form by meet- 
 ing pairs of edges S are dissimilar from the six alternating 
 quoins in which they meet pairs of edges 2. These lateral quoins 
 are ortho-symmetrical, the former six on the axes [*S*C], the latter 
 on the axes [2 C]. 
 
278 Hexagonal system. 
 
 225. 2. The dihexagonal prism or hexagonal diprism { 
 185. Since discalenohedra with any given dodecagonal base in 
 common may be indefinite in number, but will differ from each other 
 in the acuteness of the apex of their pyramids, we may consider 
 the case in which the angle formed by the pyramidal edges with 
 the morphological axis [*S"2] vanishes, and the edges themselves 
 become parallel with this axis. 
 
 In such a case the twenty-four edges of the discalenohedron will 
 become twelve, from the coincidence of each pair of edges sym- 
 metrical to the trito-systematic plane; and the resulting form will have 
 twelve faces parallel to the hexagonal axis [S"2]. It will therefore 
 be a symmetrical (not regular) dodecagonal prism, an open form, 
 the section of which perpendicular to its edges is the same as that of 
 the discalenohedra of which it is the limiting form. But it is to be 
 observed that it is, so far as its geometrical characters are con- 
 cerned, also the limiting form of any twelve of the faces of the 
 same figure if they be so selected that no pair is symmetrical on 
 the equatorial plane C. 
 
 If M/be a pole of one of the group of discalenohedra having a 
 common section with the diprism {pqr}, a zone-circle passing 
 through the poles (m) and (hkl} will contain corresponding poles 
 of every discalenohedron of the group and also a pole of the form 
 {pqr} ; and the symbol of the pole (pqr), in which the zone-circles 
 [in, hkl\ and [i 1 1] are tautohedral, will be 
 
 ( 2 h-k-l 2k-l-h 2l-h-k), 
 
 a symbol which fulfils the conditions p -\-q-\-r = o and p > q > r, 
 since h > k > I. If the symbol be derived from the correlative 
 pole efg or the poles Tiki, efg, the two first of these three poles 
 will give the symbol opposite to the face pqr. Thus the zones 
 
 [* I I ]j that is to say, 
 
 -h 2(l+h)-k 2(h + k}-l, in] and [in] 
 are tautohedral in the face 
 
 The hexagonal diprism will have two sorts of edges alternating 
 with each other, and lying respectively in the proto- and in the 
 deutero-systematic planes. Fig. 185 represents the diprism 
 
Holosymmetry. 2 79 
 
 An important feature of the hexagonal diprism results from its 
 faces belonging to a zone, all the planes of which, except the sys- 
 tematic planes and 2, are planes of abortive symmetry. They 
 in fact present a stereotyped feature of unchangeability whether for 
 crystals with different parameters or for different temperatures. 
 
 The following are the normal-angles between the face (pqr) of 
 some of these prisms and the face 2 IT. The angle (2 IT, pqr), 
 where pqr is 
 
 523= 6 35', 3 l2 = io54', 725 =13 54', 4^3 = 16 6'; 
 514=19 6', 615 = 21 3', 7i6 = 2227', 8T7 = 2 3 2 5 '; 
 918 =24 1 i', 101=30, 112; = 60, oil = 90. 
 
 226. 3. and 4. The isosceles dodecahedra. The culminating edges 
 of the discalenohedron being of two kinds, the one symmetrical to 
 the proto-, the other to the deutero-systematic planes, the faces of 
 a form truncating the edges of one kind will be dissimilar from those 
 which truncate the edges of the other kind. Both the forms will 
 be twelve-faced and their faces isosceles triangles, meeting in two 
 vertical quoins symmetrical to the hexagonal axis. Perpendicular to 
 that axis the section of a figure of either kind is a regular hexagon. 
 Thus they are isosceles dodecahedra or hexagonal pyramids. 
 
 The form of which the poles lie on the proto-systematic zone- 
 circles has for its symbol \hkk, eff}, and may be regarded as 
 composed of two correlative rhombohedra, formed by planes 
 having in their symbols the indices h k and ef respectively. And 
 since the rhombohedral type of hemi- symmetry, to which the sca- 
 lenohedron also belongs, is of preponderating importance among 
 all the types of this system, the relation of the hexagonal to this 
 trigonal type is recalled by the use of a terminology for the hexa- 
 gonal forms corresponding to the composite nature of their 
 symbols. 
 
 The hexagonal proto-pyramids will therefore be termed dirhom- 
 bohedra, while the others, of which the faces truncate the 2-edges 
 of the discalenohedron, will be designated simply as hexagonal 
 pyramids, since they belong equally to the hexagonal and trigonal 
 types. The general symbol for the hexagonal pyramids will be 
 \min\\ a symbol in which, as has already been seen, the index 
 
280 
 
 Hexagonal system. 
 
 i is the arithmetic mean of the other two : as in fact is evident, 
 since the symbol for a systematic plane 2, which is also a zone- 
 plane containing four of the poles of a form \mtn\, is that of a 
 plane of the form {1^1}, and thus m 2z'+n = o. 
 
 227. 3. The dirhombohedron {hkk, eff], Figs. 186-7. Tne 
 symbols of the faces of the isosceles dodecahedron {hkk, eff} are 
 
 hkk khk kkh, 
 eff f'f //; 
 
 hkk Yh~k Ykl, 
 
 where e=k + 4k,f = 2h-\-k, since eff is the pole in which 
 the zones [nI, khk~\ and [oil] are tautohedral ; that is to say, is 
 (4k h t 
 
 Fig. 1 86. 
 
 Fig. 187. 
 
 The dirhombohedron may also be treated as the form truncating 
 the *S*-edges of a discalenohedron. If hkl, hlk be two adjacent 
 faces of such a discalenohedron, since the symbols are symmetrical 
 we obtain the symbol of the truncating face by the addition of 
 their indices, i.e. the face in question is (2 h k + l / + /). 
 
 The poles of a dirhombohedron {hkk, eff} which lie on the 
 zone-circle 
 
 [-SJ or [oil] are hkk eff hYk Iff, 
 [S 2 ] or [lol] are khk fef ~kli~k f~ef, 
 [S s ] or [ilo] are kkh ffe Ykli ffe. 
 
Holo symmetry. 281 
 
 Figs. 1 86 and 187 represent the dirhombohedra {4!!, 877} and 
 (100, 122} corresponding to the given parametral angle. 
 
 228. 4. The hexagonal pyramid {min}, Fig. 188. The faces of 
 the hexagonal pyramid will have the following symbols, and lie 
 on the several zones [2] in the subjoined order : 
 
 on zone [2J or [1^1], min nim min nim; 
 on zone [2 3 ] or [112], nmi mni nmi mni\ 
 on zone [2J or pn], inm imn inm imn. 
 
 In Fig. 1 88 is represented the hexagonal pyramid {513} corre- 
 sponding to the above parameters. 
 
 The symbol of the form truncating the 2 -edges of a di- 
 scalenohedron {h k /, efg] may be obtained either by determining 
 the symbols of the face in which a zone containing two adjacent 
 faces of the discalenohedron symmetrical to a plane 2 is tauto- 
 hedral with the zone of which 2 is the zone-plane ; for instance, 
 the face in which \hkl,gfe\ and [I2l] are tautohedral; or by 
 the briefer method of adding corresponding indices of the two 
 symbols hkl gfe when brought into corresponding form, as indi- 
 cated in article 133, p. 154. 
 
 A dirhombohedron and a hexagonal pyramid are precluded by 
 the symmetry of the system from being mutually congruent : nor 
 can a pole of the form {hkk, eff\ be equidistant with a pole of 
 the form \min\ from the pole { 1 1 1 }. A pole of a form {min} equi- 
 distant with a pole (hkk) from (i 1 1) would in fact have the symbols 
 
 the indices of which can only be rational when h = k, and the 
 symbol (min) becomes (m), or when 2k + h = o, and the 
 symbol (min) becomes (in); that is to say, in the cases where 
 the hexagonal pyramid passes over into one of its limiting forms, 
 the pinakoid or the deutero-prism. 
 
 229. The hexagonal prisms. The general symbol \pqr\ has 
 been assigned to such forms as have their poles on the equatorial 
 zone-circle C. Where such a form has three different literal 
 indices the form is the diprism (Art. 225). But since p = (q + r), 
 if q = r the symbol becomes {2 IT}, a particular case of the 
 
282 
 
 Hexagonal system. 
 
 symbol {hkk}, and thus represents a form having its poles on 
 the proto-systematic zone-circles. 
 
 If, again, p = r, then q = o, and the symbol is { ioY}, a par- 
 ticular case of the symbol {mzn}; it thus represents a form 
 having its poles on the deutero-systematic zone-circles. This 
 symbol is in fact that of a face truncating the faces (hkk} 
 and (e/f), i.e. (3^ 3^ 3^) and (h ^k zh + k 2k + k), 
 namely (zhk kh k h\ or (2TT); and similarly a face 
 truncating the edge \rnin nim] is (m n o n m) or loT. 
 
 230. 5. The hexagonal proto-prism {2!!}, (Fig. 189). This form, 
 which may be considered as the form of which the faces truncate 
 the equatorial edges of the dirhombohedron, or as a limiting form 
 
 Fig. 1 88. 
 
 Fig. 189. 
 
 Fig. 190. 
 
 of the figures which may be produced by combining those faces 
 of the dirhombohedron which have only h and k or e and f in 
 their symbols, has for its poles the axial points of the axes [SC] ; 
 on which axes therefore its faces are ortho-symmetrical. The 
 faces are parallel to the deutero-symmetral planes 2, 
 
 those parallel to 2 t being 2!! and 211, 
 those parallel to 2 2 being 12! and ill, 
 those parallel to 2 3 being IT2 and 112". 
 
 Its edges also lie in the planes 2 and are parallel to the hexagonal 
 axis. It is an open form without quoins, but its faces have an 
 ortho-symmetrical character. 
 
Holosymmetry. 283 
 
 231. 6. The hexagonal deutero-prism jioT}, (Fig. 190). This 
 form, which is identical in features with the proto-prism, has for 
 the normals of its faces the ortho-symmetral axes [2C], the faces 
 themselves being parallel to the proto-systematic planes S. 
 
 The faces parallel to S^ are oil and oTi, 
 to S 2 are lol and Toi, 
 
 to S 3 are iTo and Yio. 
 
 The edges of the form also lie in the planes S. 
 
 The horizontal sections of the proto- and deutero-prisms are 
 regular hexagons, and the edges of the one form are truncated 
 by the faces of the other form, and are bevilled by the faces of the 
 diprism. 
 
 232. 7. The pinakoid { u i }. This form, the two faces of which 
 are parallel to the trito-systematic plane C, has the morphological 
 axis for its normal, and is, as its symbol implies, the pyrametral 
 form, meeting the three positive or the three negative arms of 
 the crystallographic axes with equal intercepts. 
 
 The form is further, in holo-symmetrical crystals, dihexagonally 
 symmetrical on the [-S2] axis. Its two faces (in) and (ITT) 
 truncate the terminal quoins of the discalenohedron and the isosceles 
 dodecahedra, and they close the dihexagonal and hexagonal prisms 
 (Figs. 185, 189, 190) with faces that are symmetrical dodecagons 
 in one set of cases and equiangular hexagons in the other. 
 
 Hexagonal System. B. Merc-symmetrical Forms. 
 
 233. The mero-symmetrical forms that are possible in the Hexa- 
 gonal system are numerous and varied in their types ; the faces of 
 the discalenohedron being capable of partition, in accordance with 
 the principles of mero-symmetry, into correlative forms in which 
 they possess one or other of the following kinds of alternate 
 distribution : 
 
 I. Twelve faces corresponding to twelve normals are extant 
 or are absent : holohexagonal haplohedral forms. 
 
 II. Twelve faces corresponding to six normals are so : hemi- 
 hexagonal diplohedral forms. 
 
284 Hexagonal system. 
 
 III. Six faces corresponding to six normals: hemihtxagonal 
 
 haplohcdral forms. 
 
 IV. Six faces corresponding to three normals : tctarto-hexagonal 
 
 diplohfdral forms. 
 
 V. Three faces corresponding to three normals: tdarlo-htx- 
 agonal haploJudral forms. 
 
 In the following table of the symbols of the faces of a discaleno- 
 hedron \hkL cf\, the symbols of the pair of faces belonging to 
 each normal form a column in the upper or in the lower half of the 
 table ; and the triads of faces being for convenience in notation indi- 
 cated by the letters employed for them in Articles 114, 115, 116 and 
 120. pp. 130-138, those designated as />./', q, q' will be mutually 
 metastrophic, and antistrophic to the remaining faces of the form. 
 
 TABLE G. 
 
 p...hkl Ihk klh hlk khl lkh...p', 
 p...hk~l lh~k ~kTh ~hl'k ~khl l~kh ...p' t 
 q...cfg gff fg fgf ffg gf <-<?', 
 
 234. In a holo-systematic haplohedral crystal the poles of a 
 triad /, q, p' or q, cannot concur with the poles of a parallel 
 triad />, q, p' or q', but they may concur with those of any other 
 of the latter triads. 
 
 Thus, we may only have combinations of the following kinds : 
 I. i. p and q with p' and q', and its correlative p qp' /. 
 iL p and p' with q and q' 7 ,. ., pp f q q f . 
 
 iii. p and q' with p' and y, ., ,. pq* ' p' ' q. 
 
 iv. /and/*' with q and q'. ., ., pp'y^- 
 
 II. And. in a hemi-systematic diplohedral cn-stal, any triad 
 /, q. p' or / must be accompanied by the corresponding parallel 
 triad /, q, p' or q*. So that the only possible combinations are : 
 i. / and p with />' and />', and its correlative qqq f q f - 
 ii. p and p with / and q', 
 
 iii. p and p with q and q, 
 
Mer asymmetry. 285 
 
 III. A haplohedral form of a hemi-systematic crystal must belong 
 to one of the tetartohedral types resulting from the concurrence of 
 the following face-triads : 
 
 i. p with p' : and the three correlatives q q', p'p , tf q. 
 
 ii. p with?': ?/,/?> /A 
 
 iii. p with q\ 3P,p'7, // 
 
 iv. p with / : q /, p p', q ?. 
 
 v. p with / : q p', p?,q p' . 
 
 vi. p with q : //, / q , p'tf. 
 
 IV. And since a tetarto -systematic partition of the twelve normals 
 into four groups of three normals is also possible, the discaleno- 
 hedron may yield four diplohedral correlative forms. They are 
 the face-triads that concur in one of the four following correlative 
 groups : 
 
 p with p, q with q, p' with p', and / with q'. 
 
 V. A tetarto-hexagonal haplohedral form would be represented 
 by the faces p, and the discalenohedron would yield seven other 
 forms correlative to it. These octo-forms would be hemimorphous. 
 
 235. The following table exhibits the forms which result from 
 these mero-symmetrical modes of distributing the faces of the dis- 
 calenohedron ; and with them are associated such mero-sym- 
 metrical developements of the other holo-symmetrical forms as are 
 geometrically distinct from the forms from which they may be 
 derived. 
 
 I. Holo-systematic haplohedral forms ; or holohex- 
 agonal haplohedra: 
 
 i. Symmetrical to no systematic plane 
 The ditrapezohedron a (hkl, efg} or a {hlk, tgf\. 
 
 ii. Symmetrical to the C- and ^-systematic planes 
 The di trigonal (j>roto- or} S-pyramid x \hkl\ or x \efg\- 
 The ditrigonal (proto- or) S-prism . x \pqr\ or x [pqr\. 
 The trigonal (proto- or) S-pyramid . x \hkk\ or x \cff\* 
 The trigonal (proto- or) S-prism . . x {211} or x {^n}. 
 
286 Hexagonal system. 
 
 iii. Symmetrical to the C- and 2-systematic planes 
 The ditrigonal (deutero- or) ^-pyramid 
 
 {hkl,gfe} or {hlk,gef}. 
 
 The ditrigonal (deutero- or) HL-prism {pqr} or \prq\. 
 The trigonal (deutero- or) ^-pyramid {min} or f \mni\. 
 The trigonal (deutero- or) ^-prism f { ioT} or f { iYo}. 
 iv. Symmetrical to the S- and 2-planes 
 Hemimorphous hexagonal forms : 
 Discalenohedral hemimorphs p\hkl, efg\ or p{hkl, efg\\ &c. 
 
 II. Hemi-systematic diplohedral forms; or hemihex- 
 
 agonal diplohedra. 
 
 i. Symmetrical ditrigonally to the -^-systematic planes 
 The ditrigonal scalenohedron i:\hkl} or Tt\efg\. 
 The rhomb ohedr on TT {hkk\ or 'R\eff\* 
 ii. Symmetrical ditrigonally to the 2-planes 
 The deutero-scalenohedron -ty \hkl, gfe} or \// {efg, lkh\. 
 The deutero -rhombohedr on ^\f\min\ or \jf\nzm}. 
 iii. Symmetrical to the C-plane 
 Gyroidal forms : 
 
 The tritopyramid ${hkl, efg} or ${lkh, gfe}. 
 The hemi-diprism {pqr} or (j> {rqp}. 
 
 III. Hemi-systematic haplohedral forms; or hemihex- 
 
 agonal haplohedra: each represented by four cor- 
 relative forms : 
 i. Normals symmetrical to the -S-planes - 
 
 The trigonal trapezohedron an {h k I}. 
 ii. Normals symmetrical to the 2-planes 
 
 The trigonal deutero-trapezohedron a\j/ \hkl}. 
 iii. Normals symmetrical to the C-plane 
 
 The skew trigonohedron x$\hkl\. 
 Hemimorphous hemihexagonal forms : 
 iv. Symmetrical ditrigonally to the S-planes 
 
 The hemi-protopyramid PTT \h k /}. 
 
 The hemi-protorhombohedron pir {hkk\. 
 
Merosymmetry* 287 
 
 v. Symmetrical ditrigonally to the 2-planes 
 
 The hemi-deuteropyramid p-ty \h k I}. 
 
 The hemi-deuterorhombohedron p\j/ \m in}. 
 vi. Symmetrical to no systematic plane 
 
 The hemi-tritopyramid /o< {hkl}. 
 
 IV. Tetarto-systematic diplohedral forms; tetarto- 
 hexagonal diplohedra. 
 
 Three extant normals. 
 
 The hemiscalenohedron or skew rhombohedron 7r< \hkl\. 
 
 V. Tetarto-systematic haplohedral forms. 
 
 The hemimorphous hemiscalenohedron pTt($){hkl} ; &c. 
 
 Of these combinations, the third under the holo-systematic type 
 with the merosymmetrical prefix f, the second under the hemi- 
 systematic diplohedral type distinguished by the prefix \f/, and the 
 second under the haplohedral type of the latter division with prefix 
 a\jf have only a theoretical interest, since if any hexagonal crystal 
 should present forms corresponding to either the ditrigonal pyramid 
 or the scalenohedron, the systematic planes to which such forms 
 were symmetrical would offer themselves for choice as the proto- 
 systematic group, for it is contrary to mero-symmetrical principles 
 that, on any crystal, forms symmetrical to only one group should 
 be associated with such as are symmetrical to a different group of 
 systematic planes. 
 
 The consideration of such forms has however a theoretical 
 interest, and their recognition is so far necessary that without them 
 a complete view of the modes of mero-symmetrical dissection of 
 which the hexagonal discalenohedron is susceptible cannot be 
 taken. 
 
 236. I. Holo-systematic haplohedral forms. 
 
 i. The ditrapezohedron a {hkl, efg] or a {hlk, eg/}, Figs. 
 191 (a), (c), is contained by either set of alternate faces of the dis- 
 calenohedron, the other faces forming the correlative figure ; all the 
 faces of one of the semiforms are antistrophic to those of the 
 other, and the semiforms themselves are enantiomorphous. 
 
288 
 
 Hexagonal system. 
 
 The symbols of the twelve faces comprised under the form- 
 symbol a{hkl, efg} are 
 
 hkl Ihk klh 
 
 hll ~klil Tkh 
 efg S e f 
 
 the correlative form a [hlk, eg/} comprising the remaining faces 
 of Table G, Art. 233. Figs. 191 (a) and (c) represent the forms 
 a{2ib, 452} and a (20!, 425}, respectively, derived from the 
 discalenohedron {20"!, 425} represented in Fig. 191 (3). 
 
 The faces are disposed in hexagonal (not dihexagonal) sym- 
 metry round the axis [*S*2], so that the six culminating edges E, 
 which unite in a quoin to form the apex of the figure at either 
 
 Fig. 191 (). 
 
 Fig. 191 (c\ 
 
 end, are similar. But the figure has no plane or centre of 
 symmetry. Its remaining edges F and G are of two kinds 
 traversing the equatorial plane in zigzag. The faces are trape- 
 zoids bounded each by two edges E, an edge F, and an edge G. 
 The twelve lateral quoins are similar, each being formed by the 
 meeting of three dissimilar edges E, F, and G. The forms 
 a (P<I r }> a-{hkk, eff}, a {min}, a {2!!}, a {iol} and a {in} 
 are geometrically identical with the holo- symmetrical forms from 
 which they are derived. No crystal presenting this type of hemi- 
 symmetry has yet been observed. 
 
 237. ii. and iii. To the next two subdivisions under holo- 
 hexagonal merosymmetry belong forms which, by the condition 
 of the division, are not centro-symmetrical. They are, however 
 
Mer asymmetry. 289 
 
 symmetrical to the trito-systematic plane, and to one or other of 
 the triads of systematic planes and 2. 
 
 It has already been observed that in reality we need only take 
 cognizance of one of these types as of actual crystal-forms ; for 
 the two cannot concur, and the occurrence of one type on a crystal 
 would ensure that triad of systematic planes to which it was sym- 
 metrical being selected for the proto-systematic planes S. 
 
 238. ii. The dilrigonal proto-pyramid or S-pyramid x {hkl\ or 
 x \efg\, Figs. 192 (a), (c). In the ditrigonal proto-pyramid the sym- 
 metral character of the deutero-systematic planes is in abeyance. The 
 dihexagonal axis of the original discalenohedron becomes a ditrigo- 
 nal axis by virtue of its being the zone-axis of three similar planes 
 of symmetry S lt S z , S 3 . The plane C being also a plane of sym- 
 metry, the following are the constituent faces of the form x {hkl}, 
 
 hkl Ihk klh hlk khl Ikh, 
 
 e f 8*7 
 
 Fig. 192 (a). Fig. 192 (6). Fig. 192 (c}. 
 
 They correspond to the groups />, />', g, q' of Table G : the form 
 x {^fg} comprises the remaining faces of Table G, viz. those of 
 the groups p,p', <?, q '. Figs. 192 (a) and (c] represent the forms 
 x {201} and ^{425} respectively derived from the discaleno- 
 hedron {20!, 425} represented in Fig. 192 (b). 
 
 The faces of the discalenohedon are thus seen to be extant 
 or to be suppressed in alternate pairs in either hemisphere, and in 
 adjacent pairs on opposite sides of the equatorial plane. They 
 are triangular, and, as their adjacent faces are antistrophic to 
 each other, they are scalene. The edges of each face are therefore 
 
 u 
 
290 
 
 Hexagonal system. 
 
 of three kinds ; two, S and S', are culminating edges lying in the 
 S-planes of symmetry, and, of these, six three of each kind 
 meet on the [S] axis in a quoin at each apex of the figure : the 
 third edge of each face lies in the C-plane, and of these similar 
 edges C the figure has six. Hence any section of the form 
 perpendicular to the ditrigonal axis is a symmetrical hexagon : 
 the aspect of the form is that of a double or crystallographic 
 pyramid having no centre of symmetry ; and its lateral quoins are 
 of two kinds, three of them formed by two edges *S" and two edges 
 C, and three, alternating with these, formed by two edges S' and 
 two edges C\ and they are ortho-symmetrical on the \CS\ lateral 
 axes. The two correlative figures are identical in their aspect, 
 as seen in opposite directions of an axis \SC~\ : and, from the 
 antistrophic character of adjacent faces in each, the figures are 
 tautomorphous, a revolution of either form through 60, or any 
 odd multiple of 60, round the [S] axis bringing it into con- 
 gruence with the correlative form. 
 
 239. The ditrigonal proto-prism or S-prism x {pqr} or x {p'qr}, 
 Figs. 193 (a), (c), may be regarded as truncating the basal edges 
 
 451 
 
 7JS 
 
 
 M 
 
 
 
 && 
 
 
 541 
 
 ~~~~-~^. 
 
 
 
 
 L^ 
 
 'ig. 193 () Fig. 193 (). Fig. 193 (0. 
 
 of a ditrigonal proto-pyramid, or as a limiting case of the same 
 figure in which the faces have become parallel to the ditrigonal 
 axis and pairs of faces symmetrical to the equatorial plane have 
 thus become coincident. It is an open six-sided prism without a 
 centre of symmetry : the edges are of two kinds *S" and S / and 
 lie in the S-planes of symmetry. 
 
 The faces of the form x {pqr} have the symbols 
 pqr rpq qrp prq qpr rqp, 
 
Merosymmetry. 
 
 291 
 
 the symbols of faces of the correlative form x {p q r] being 
 pq? rpq qrp prq q]>? rqp. 
 
 Figs. 193 (a) and (c) represent the forms ^{514} and ^{ 
 respectively, derived from the diprism {5!?} shown in Fig. 
 
 240. The trigonal proto-pyramid or S-pyramid x (h k k} or 
 x { eff}, Figs. 194 (a), (c), may be regarded either as truncating 
 the S- or S'-edges, respectively, or as being a limiting case of the 
 ditrigonal proto-pyramid, in which pairs of faces symmetrical to 
 the *S*-planes of symmetry have become coincident. The figure 
 s thus a double pyramid with an equilateral triangle for base. 
 There are six similar culminating edges S or S / lying in the 
 S-planes of symmetry, and three similar basal edges C lying in 
 the equatorial plane. 
 
 Fig. 194 (a). 
 
 Fig. 194 
 
 Fig. 194 (0. 
 
 The symbols of the faces of the form x [h k k} are 
 hkk khk kkh 
 
 *77 7*7 771 
 
 those of the form x {eff} being 
 
 hYk Yh~k ~k~kh 
 
 eff fef ffe. 
 
 Figs. 194 (a) and (c) represent the forms ^{877} and ^{41!}, 
 respectively, derived from the dirhombohedron {4!!, 877} repre- 
 sented in Fig. 194 (). 
 
 241. The trigonal proto-pr ism or S-prism x {2 11} or x {211}, 
 Figs. 195 (a), (c), is an open prism with an equilateral triangle for 
 
 U 2 
 
292 
 
 Hexagonal system. 
 
 base. The poles lie at the intersections of the S- and C-planes, and 
 the three similar edges -S or S' lie in the *S"-planes of symmetry. 
 The faces of the form x {2!!} are 2i7, 2!, 7i2, 
 those of the form x {ssii} being "211, ili, nI. 
 
 2ia 
 
 all 
 
 Fig. 195 (a.} 
 
 Fig. 195 
 
 Fig. 195 
 
 Figs. 195 (a) and (<:) represent the forms x {211} and x {211}, 
 respectively, derived from the hexagonal proto-prism {2!!} repre- 
 sented in Fig. 195 (d). 
 
 The forms x{min}, x {ioY}, and x {111} are geometrically 
 identical with the holo-symmetrical forms from which they are 
 derived. 
 
 242. iv. Hemimorphous hexagonal forms. 
 
 By the condition of this subdivision only the S- and 2-systematic 
 planes retain their symmetry. Hence the faces of any form be- 
 
 Fig. 196 (a\ 
 
 Fig. 196 
 
 Fig. 196 
 
 longing to it will all lie at one side of the equatorial plane, and 
 the figure will be represented by the upper or lower half of the 
 corresponding holo-symmetrical form. The only forms which are 
 geometrically distinct from those presenting holo-symmetry are 
 
Mev 'o symmetry. 293 
 
 p{hkl, efg}> p{hkk, e//}, p{min], and p{m}, with their 
 correlatives. 
 
 Figs. 196 (a), (b\ (c) represent respectively the forms 
 
 p { 201, 425}, p (411, 877}, and p {513}. 
 
 closed by the form p {HI}- 
 
 243. II. Hemi-systematic diplohedral forms. 
 
 The characteristic distinction of the mero- symmetrical forms of 
 the Hexagonal as contrasted with those of the Tetragonal system 
 is, that in the Tetragonal system centre-symmetry and symmetry 
 to the equatorial plane are concurrent conditions, whereas in the 
 Hexagonal system these two kinds of mero -symmetry are quite 
 independent of each other ; so much so indeed that a centro-sym- 
 metrical semiform in the latter system can only be symmetrical 
 to the equatorial plane in the case of a form which is symmetrical 
 to no other plane than this. 
 
 Of diplohedral semiforms therefore we have two kinds. In the 
 one kind, the normals are symmetrical to one or other but not to 
 both of the systematic plane-triads, and each carries its two faces ; 
 in the other kind, alternate normals, symmetrically distributed round 
 the axis of form as a hexagonal (not dihexagonal) axis, are diplo- 
 hedral. In the former kind the discalenohedron is dissected into 
 two similar scalenohedra, in the latter kind the partition is into two 
 correlative forms of which each is a double pyramid, with a 
 geometrically equilateral hexagonal base. 
 
 The scalenohedra can in theory be of two kinds ; proto-scaleno- 
 hedra symmetrical to the proto-systematic planes S, and deutero- 
 scalenohedra symmetrical to the deutero-planes 2. The three 
 planes, however, to which the scalenohedral forms are symmetrical 
 being those which naturally offer themselves as the proto-systematic 
 planes, we have only to deal practically with the scalenohedron 
 as a proto-systematic diplohedral form. 
 
 In accordance with the notation adopted for such a form its 
 symbol will be Ti{hkl} or -n\efg}\ that of a deutero-scaleno- 
 hedron would be -^ \hkl, gfe} or \jr {hlk, eg/}. 
 
 244. i. The ditrigonal scalenohedron, it {hkl} or TT {efg}, Figs. 
 197 (a), (c). The symbols of the twelve faces of the scalenohedron 
 
294 
 
 Hexagonal system. 
 
 7T {h k 1} symmetrical to the proto-systematic planes S correspond 
 to the groups p, /', p, p' in Table G (Art. 233) and are 
 
 hkl Ihk klh hlk khl Ikh, 
 hYl Thk Yl~h Yl~k Yhl Tkh\ 
 
 those of the form TT {efg} lie in the two lower rows of Table G, 
 and correspond to the groups q, q', g, g'. They are, 
 
 efg e f fS e e gf f e gf e > 
 
 efg g e f fg e e gf f e g gf e - 
 
 Each face of the form is a scalene triangle, of which two of the 
 edges S and S', culminating in a ditrigonal quoin on the axis [*S"], 
 lie in the planes S, while the third edge lies obliquely to the plane C. 
 
 Fig. 197 (a). 
 
 Fig. 197 
 
 Fig. I 9 7 (<:). 
 
 The figure has therefore two ditrigonal quoins forming its vertices, 
 and constituted by three edges alternating with three edges S'. 
 
 It has moreover six lateral edges G, which are similar but lie in 
 zigzag athwart the plane C, and meet in six quoins similar but 
 alternately inverted in position, in each of which an edge -5* and an 
 edge S f meet two of the zigzag edges G symmetrically as regards 
 a plane S. The two correlative scalenohedra are tautomorphous, 
 since adjacent faces on each form are antistrophic to each other. 
 And a revolution of either figure through an angle that is an odd 
 multiple of 60 brings it into congruence with Its correlative form. 
 
 Figs. 197 (a) and (c) represent respectively the forms 71(201} 
 and TT {425} derived from the discalenohedron {20!, 425}. 
 
 245. The ditrigonal prism, TT {pqr}. The zone-circles passing 
 through the pole in and successive poles of a scalenohedron will 
 
Merosymmetry. 
 
 295 
 
 intersect the zone-circle \S~\ in the poles of a prism the faces of 
 which are parallel to the axis of the zone []. 
 
 From a geometrical point of view the figure, considered as an 
 open form, is identical with the diprism {pqr}. Crystallographi- 
 cally however it is not so, as is evident when a ditrigonal form is 
 combined with the prism. The ditrigonal prism is in fact the 
 limiting form of the two correlative scalenohedra, in the case in 
 which their faces become parallel to the morphological axis and 
 their poles fall on the equatorial zone-circle. 
 
 246. The rhombohedron, v{hkk} or TT {e//}, Figs. 198 (#),(<:), 
 199 (a), (c). The relation of the rhombohedron to the scaleno- 
 hedron is analogous to that of the dirhombohedron to the discale- 
 nohedron; and, as the faces of two correlative scalenohedra have, 
 in the one, symbols into which only the literal indices hkl, and 
 
 Fig. 198 (a). Fig. 198 (b). Fig. 198 ). 
 
 in the other, symbols into which only the literal indices efg enter, 
 so the faces of two correlative rhombohedra have in their symbols 
 the indices hkk and eff respectively. 
 
 If the normal-angle of two adjacent faces of the scalenohedron 
 be supposed to become zero, the faces come to lie in a plane, and 
 each pair of antistrophic faces becomes a single quadrilateral face, 
 euthysymmetrical to the line originally representing their edge, 
 which is the trace on the face of a plane of symmetry : the 
 opposite angles are equal in pairs, and the form of the face is 
 geometrically a rhomb. Crystallographically however it is not so, 
 since its adjacent culminating edges which represent the S"-edges 
 of the scalenohedron are not similar to those which represent the 
 lateral or G-edges of the latter figure, though they have become 
 parallel to them. 
 
296 
 
 Hexagonal system. 
 
 The twelve faces of the scalenohedron being represented by six 
 faces in the rhombohedron the figure is a rhombic hexahedron 
 or six-faced rhombohedron, and may be acute or obtuse according 
 as the three-edged quoins which are the vertices on the morpho- 
 logical axis are formed by edges which meet in three acute or 
 three obtuse angles. 
 
 Figs. 198 (a) and (c) represent the correlative acute rhombohedra 
 77 {877} and TT {4!!} derived from the dirhombohedron {4!!, 877} 
 of Fig. 198 (), while Figs. 199 (a) and (c) represent the corre- 
 lative obtuse rhombohedra TT {^22} and TT {100} derived from the 
 dirhombohedron {100, 722} of Fig. 199 (<5). 
 
 Fig. 199 (a). 
 
 Fig. 199 (<:). 
 
 Fig. 199 
 
 The summit-quoins are symmetrical ditrigonally on the [S] axis. 
 The six other quoins are symmetrical to no axis, but only to the 
 -S-planes of symmetry, and are each formed by the meeting of a 
 culminating edge S with two of the six lateral or zigzag edges G. 
 In each lateral quoin of an acute rhombohedron two lateral edges 
 make an acute angle with each other and an -obtuse angle with 
 a summit-edge. In each lateral quoin of an obtuse rhombo- 
 hedron two lateral edges make an obtuse angle with each other 
 and meet a summit-edge in two acute angles. The alternate 
 quoins are directly congruent ; those intermediate to them are also 
 
Mero symmetry. 297 
 
 congruent with them, but are inverted in their orientation relative 
 to the morphological axis. 
 
 The limiting form dividing the acute from the obtuse varieties 
 of the rhombohedron is the cube, the faces of which are geo- 
 metrically squares. The Cubic system is in fact a developement 
 of the Rhombohedral system resulting from the condition that, 
 when the faces of the rhombohedron become squares, the zigzag 
 edges become crystallographically, as well as geometrically, similar 
 to the culminating edges. 
 
 247. The trigonal dihexahedron, ^{min}. Should the poles of 
 a scalenohedron fall on the great circles corresponding to the 
 deutero-systematic planes, the resulting twelve-sided figure would 
 evidently be a limiting form between the two series of correlative 
 scalenohedra. Such a form is geometrically identical with the 
 hexagonal dihexahedron already considered, but its adjacent faces 
 are crystallographically antistrophic. 
 
 The forms TT {2!!}, ^{lol} and -77(111} will likewise be 
 geometrically identical with the corresponding holo-symmetrical 
 forms. 
 
 248. The zigzag edges of a scalenohedron, and therefore also 
 those of a rhombohedron, are truncated by the faces of the hexa- 
 gonal deutero-prism {ioT}. For if the poles of two adjacent 
 faces meeting in one of these lateral edges be considered, the 
 poles hkl and lk~h for instance, in the one form, or the poles 
 h k k li~k Ji in the other, the sum of their indices gives the symbol 
 (ioT) for the face truncating the edge ; and similarly for the other 
 edges of the form. And the lateral quoins in both figures will be 
 replaced symmetrically by the faces of the hexagonal proto-prism 
 {121}, which are perpendicular to the proto-systematic planes to 
 which these quoins are symmetrical. The summit-edges of the 
 scalenohedron will be truncated by the faces of rhombohedra, 
 direct or inverse according as the poles of the scalenohedron 
 faces forming these edges are disposed symmetrically in respect to 
 such quadrants of the systematic great circles S as contain the 
 poles of the form {100}, or to the remaining quadrants of those 
 great circles. In fact, the edge of the faces hkl and h Ik is trun- 
 cated by the face (zh k-\-l & + 1), the symbol of which is obtained 
 
298 Hexagonal system. 
 
 by adding the indices of the two symbols, and has the form 
 h'&k', while the edge of the faces Ihk and / k h is truncated by 
 the face (2! h + k h + k), the symbol of which has the form //'/'. 
 
 249. The indices of a face will all have positive values where 
 the pole of the face lies within the spherical triangle of which 
 the angular points are at the poles 100, oio, and ooi, and they 
 will be all negative when the pole lies in the triangle with its 
 angular points at the poles Too, ooi, oio. The values of one or 
 of two of the indices will be negative in every other symbol. 
 
 250. Forms consisting of two correlative scalenohedra or rhombo- 
 hedra associated together but differing in their physical characters 
 are of very frequent occurrence ; so frequent in fact that the 
 existence of truly holo-symmetrical hexagonal crystals has been 
 disputed : Professor Miller indeed only recognised a Rhombohedral 
 system, holosymmetrical in itself, and referred no crystals to 
 a Hexagonal system. That there exists, inherent in the laws 
 which regulate the coordination of the crystal-molecules, a cause 
 for the great prevalence of trigonal diplohedral symmetry in crystals 
 belonging to this type may be accepted as a result of experience ; 
 but that it is a cause which precludes the recognition of a Hex- 
 agonal system, such as is necessary to the theoretical completeness 
 of the principles of geometrical mero-symmetry, is hardly to be 
 admitted. It would not be so were the evidence against the 
 existence among known crystals of holo-symmetrical hexagonal 
 forms complete; but in fact this evidence is not complete, and 
 there are crystals, of which those of beryl may be taken as an 
 example, which in their cleavage and in the physical as well as 
 geometrical characters of their forms offer no sufficient grounds 
 for referring them to a trigonal rather than a hexagonal type of 
 symmetry. 
 
 251. ii. The deutero-scalenohedron, \|/- \hkl\ and deidero-rhombo- 
 hedron, \ff { m in } need only to be noticed as forms of which some 
 cognizance must be taken in a complete review of the various 
 mero-symmetrical partitions of which the faces of a discaleno- 
 hedron and a hexagonal pyramid are theoretically susceptible. 
 
 Detailed descriptions of these deutero-symmetrical forms are 
 unnecessary, inasmuch as in everything but in the form of their 
 
Merosymmetry. 
 
 299 
 
 symbols and the particular planes to which they would be sym- 
 metrical they are identical in character with the forms considered 
 in the foregoing articles. 
 
 252. iii. Gyroidal forms. There remains a group of centro- 
 symmetrical forms belonging to the hemi-systematic division of the 
 Hexagonal system, which present no symmetry to the proto-or 
 deutero-triad of systematic planes, but which are symmetrical to the 
 trito-systematic plane C. The morphological axis is an axis of simply 
 hexagonal symmetry for such forms, in which therefore of the faces 
 of the discalenohedron which lie on the same side of the equatorial 
 plane only alternate ones can be extant. It is only the discaleno- 
 hedron and the diprism that will present geometrically distinct forms 
 under this type, and from their symmetry to the plane C they have 
 been termed the trito-pyramid and trito-prism or hemi-diprism. 
 
 253. The trito-pyramid, <$){hkl, efg\ or $\lkh, gfe], Figs. 
 200 (a\ (c], differs from the ditrapezohedron in that, while in the 
 
 Fig. 200 (a). 
 
 Fig. 200 
 
 Fig. 200 (c). 
 
 latter figure the whole of the faces of a semiform were meta- 
 strophic, only those on one side of the plane C are so in the trito- 
 pyramid, those on the other side being symmetrical, and therefore 
 antistrophic, to the former. The correlative forms are thus tauto- 
 morphous, being capable of being brought into congruence by a 
 rotation round the morphological axis, after interchange in the 
 positions of the vertices of one of them. 
 
 The figure is formed by two hexagonal pyramids, the base 
 common to which is a regular hexagon. Its two summit- 
 quoins are formed each by six similar edges, and each of its 
 
;oo 
 
 Hexagonal system. 
 
 six lateral quoins by the meeting of two of these edges with 
 two of the basal edges, which are all similar and lie in the 
 equatorial plane. It is evident however from the gyroidal position 
 of its faces in respect to the vertical axis, that none of its summit- 
 edges can undergo truncation, since no two planes of the system 
 can be equally inclined on them. 
 
 The symbols of the faces of the form ${hkl, efg} are contained 
 in the left-hand half of the Table G (Art. 233). They are, 
 hkl Ihk klh 
 
 hkl Ihk 
 fg 
 
 klh, 
 
 Those of the form $\lkh,gfe\ are contained in the right-hand 
 half of that Table. 
 
 Figs. 200 (a) and (c) represent respectively the forms $ {20!, 425} 
 and $ {Io2, 524} derived from the discalenohedron (20!, 425} 
 represented in Fig. 200 (3). 
 
 423 
 
 lag 
 
 14S 
 
 Fig. 201 (a). 
 
 Fig. 201 
 
 Fig. 201 (c). 
 
 254. The hemi-diprism or trito-prism, $>{pqr] or ${prg\, 
 Figs. 201 (a), (c), has the geometrical character of a regular hexa- 
 gonal prism, its edges having normal-angles of 60. But its poles 
 do not lie at angles of the systematic triangles, and the faces are 
 crystallographically symmetrical only to the equatorial plane. Figs. 
 201 (a) and (c) represent respectively the forms $ {54^} an d 
 <J> {514} derived from the diprism {514} shown in Fig. 201 (&). The 
 
 forms 
 
 e//}, ${mtn] 
 
 and 
 
are geometrically identical with the corresponding holo-symmetrical 
 forms. 
 
 255. III. and IV. Tetarto-symmetry. 
 
 The distinct types of tetartohedra that may occur in the Hexa- 
 gonal system, apart from those resulting from hemimorphism, 
 are four in number. 
 
 Three of these types are haplohedral, and in them each of 
 the six alternate normals of the discalenohedron is represented 
 by a single face; they correspond, in fact, to the six different 
 hemi-systematic forms from which they are the haplohedral 
 derivatives. The fourth is a unique case of tetarto-symmetry 
 in which three normals only are represented, but by both their 
 faces: it is therefore of diplohedral tetarto-systematic type (Art. 
 265). 
 
 Four correlative tetartohedra admit of being combined in three 
 different ways, and so can only reproduce three distinct pairs of 
 correlative hemi-symmetrical forms. It will thus be seen that 
 four different tetrads are needed to build up six distinct hemi- 
 symmetrical derivatives from the discalenohedron ; for one pair of 
 hemi-symmetrical forms must be common to any two of the groups 
 that are built up by separate tetrads of tetarto-symmetrical forms. 
 
 All four of the tetarto-symrnetrical forms alluded to must there- 
 fore take part in the construction of the six hemi-symmetrical 
 forms which are not hemimorphous. The projections in plates 
 IV to VII exhibit the relations of these forms to one another. 
 
 The tetartohedral forms of whatever kind may be treated on a 
 principle analogous to that followed for the hemi-symmetrical 
 forms. In the latter, the symmetrical distribution of the poles of a 
 form in respect to the different systematic planes was taken for 
 this purpose. In the tetarto-symmetrical forms it is, further, the 
 distribution of those normals which are conceived as extant that 
 determines the nature of the resulting quarter-forms. 
 
 Thus the normals which are represented each by a pole may 
 be symmetrical to the proto-, to the deutero-, or to the trito- 
 systematic planes ; or, symmetrical to none of these, they may be 
 gyroidally grouped round the axis of form. A form of the last 
 kind is however necessarily hemimorphous in its character. 
 
302 Hexagonal system. 
 
 256. III. Hemi-systematic haplohedral forms. 
 
 If the six normals are to be symmetrical in respect to the 
 proto-systematic planes, they will be the same normals which 
 in a ditrigonal scalenohedron 77 \hkl\ are diplohedral ; or which 
 in the trapezohedron a{hkl, efg\ and ditrigonal 2-pyramid 
 {hkl, gfe] are haplohedral. Now these normals may carry their 
 six faces all on one side of the plane C in a simply hemimor- 
 phous way (Art. 261) ; or, the faces may be distributed partly 
 above and partly below that plane, and this can only occur in one 
 way, that namely indicated in the projections in Plate V. 
 
 257. i. The trigonal trapezohedron, an [hkl\, Figs. 202 (a], (c), 
 (d), (/) The figure corresponding to any one of these four pro- 
 jections has trapezoids for its faces, which meet in two trigonal 
 quoins on the [S] axis, and have in a crystallographic sense three 
 sorts of edges. The faces of each tetartohedron are mutually 
 metastrophic. The symbols of the faces of the four quarter-forms 
 are as follows : 
 
 /x r ,, 7 , (hkl Ihk klh hlk khl Ikh or 
 
 (1) an\hkl\ 1 
 
 P P 
 
 (2) an {hlk} 
 
 (3) *v\'fg\ 
 
 L P 
 
 JiYl Thl ITh hlk khl Ikh or 
 
 p p>_ 
 
 e fg g e f fge e gf feg gfe or 
 
 (4) **{'g/\ {'SS-S'ffg* eg/ /eg gft 
 
 Of these forms the first and third, and the second and fourth, 
 are tautomorphous, but, taken otherwise in pairs, the forms are 
 enantiomorphous. In Figs. 202 (a) and (c} are represented the 
 forms a 77 (20!} and an (2lo}, which together compose the form 
 77(20!} shown in Fig. 202 () : the remaining correlative 
 tetartohedral forms a 77 {425} and a 77 (45 2} shown in Figs. 202 (d) 
 and (f) together compose the form 77 (425} represented in Fig. 
 202 (e). 
 
 The two enantiomorphous groups an {hkl} and a77 (,/} are 
 of great interest from the occurrence of several forms of one or 
 
Merosymmetry. 
 
 303 
 
 other of these particular groups on crystals of certain minerals, 
 of which quartz is a conspicuous example ; the particular group, 
 when it occurs on the crystal, indicating by its presence the 
 nature of a rotatory influence of the crystal on a ray of polarised 
 light traversing it in the direction of the optic axis, which is the 
 morphological axis [*S*2]. In a crystal of quartz the forms 
 a-n{hkl] are indicative of Isevo -rotatory, the forms aTt{hlk} of 
 dextro-rotatory action. 
 
 Fig. 202 (a). 
 
 Fig. 202 (&}. 
 
 Fig. 202 
 
 Fig. 202 (d). Fig. 202 (*). Fig. 202 (/). 
 
 The form an {hkk} is geometrically identical with the form 
 T:{hkk}) and the forms 077(2!!}, a7r{m} with the corre- 
 sponding holo-symmetrical forms; the remaining forms air {pqr}, 
 a-n{min}, and aTrjio!} are geometrically identical with the 
 ditrigonal forms {pqr}, {mz'n}, and f {10!}, and only differ 
 from the forms represented in Figs. 193, 194, and 195 in that they 
 are symmetrical to the 2-planes instead of the -S-planes. 
 
 258. ii. The trigonal deutero-trapezohedron^ a-fy{hkl}. The six 
 normals to which its extant faces belong will be symmetrical to the 
 
304 Hexagonal system. 
 
 deutero- systematic planes, the faces of any two symmetrically placed 
 normals being in opposite hemispheres. The symbols a^t\hkl\, 
 a^{hlk}, a\j/\efg\, a\l/\egf\ would represent the tetrad of 
 forms. Mutatis mutandis, the figure is quite similar to that of 
 the trigonal trapezohedron described in the last article. 
 
 If the extant poles of the pairs of symmetrical normals lay on 
 the same hemisphere, the form would be the hemimorphous 
 tetartohedron p^{hkl], (Art. 263). 
 
 259. Where the six extant normals are in pairs symmetrical 
 to the trito-systematic plane, the faces will either all lie on one 
 side of that plane and form a gyroidal hemimorphous form 
 p$ {hkl}, (Art. 264), or they will be distributed symmetrically in 
 respect to the plane C and form a pyramidal figure in which 
 two trigonal pyramids (of which the faces, though geometrically 
 isosceles, are crystallographically scalene) stand base to base and 
 are trigonally symmetrical on the morphological axis of the 
 system. 
 
 260. iii. The skew trigonohedron, x<\>\hkl\, Figs. 203 (a), (c), 
 (d), (/}. The faces of the form which carries the face hkl are 
 those which are common to the forms <$>{hkl}, x{hkl\, and 
 f \hkl\, their symbols being 
 
 hkl Ihk klh 
 
 e fg 
 
 The summit-quoins are three-faced, formed by trigonally similar 
 culminating edges metastrophic to each other, which can only be 
 replaced by planes which do not truncate them ; so that two of 
 these edges belonging to a face are not alike in their relations to 
 that face. The three lateral quoins are four-faced, two culmi- 
 nating edges alternating with two lateral edges, but they are 
 symmetrical to no axis. In Figs. 203 (a) and (c) are represented 
 the forms oc {20!} and x(f) {425}, which together compose the 
 form </>{2oT, 425} shown in Fig. 203 (d); the remaining cor- 
 relative tetartohedral forms x<p {452} and x$ {210} shown in 
 Figs. 203 (d) and (/) together compose the form (/> {210, 452} 
 represented in Fig. 203 (e). 
 
 The form x$ {pqr} would be a skew three-sided prism with an 
 
Merosymmetry. 
 
 305 
 
 equilateral section, and would consist of the three faces pqr, rpq, 
 qrp\ the forms x${hkk} and x<$> {2!!} would be geometri- 
 cally identical with x{hkk} and x {2!!}, and x$ {mzn}, 
 x<t>{ioi} with {min], f {ioT}: the form x$ {m} would 
 carry both its faces. 
 
 Fig. 203 (a). 
 
 Fig. 203 (). 
 
 Fig. 203 (V), 
 
 Fig. 203 (d). 
 
 Fig. 203 (,?). 
 
 Fig. 203 (/). 
 
 No crystal is as yet known which carries the faces of a skew 
 trigonohedron. 
 
 261. iv. v. vi. Hemimorphous tetartohedral forms. The hemi- 
 morphous derivatives from hemi- symmetrical forms will be repre- 
 sented by three tetartohedral modes of dissecting the general 
 hemimorphous form p{hkl, efg], or p {h ~kl, efg }, and the forms 
 p{hkk,eff}, p{min\, p\pqr}, &c. 
 
 Each tetrad of correlative tetartohedral forms in this division 
 will therefore reproduce the two correlative hemihedral forms with 
 
306 Hexagonal system. 
 
 the above symbols, and also two corresponding correlatives of the 
 six remaining hemi-symmetrical forms. 
 
 Their symbols, taking that one of the four forms which contains 
 a pole hklioi example, are, 
 
 iv. the hemi-protopyramid pit {hkl}, 
 v. the hemi-deuteropyramid p^{hkl}, 
 vi. the hemi-tritopyramid p<$>{hkl}. 
 
 262. iv. The hemi-protopyramid, or, more completely expressed, 
 the hemimorphous ditrigonal protopyramid, as represented by the form 
 PK {hkl}, is the form the normals of which are symmetrical to the 
 S-planes (Fig. 204) : its faces therefore are those of the upper half 
 
 Fig. 204. Fig. 205. 
 
 of a scalenohedron TT {hkl}, or of a ditrigonal ^-pyramid x {hkl}, 
 as well as of p { h k 1} . The symbols of the tetrad are 
 
 pi: {hkl}, pv\e/g}, pv{hkl}, pir{e/g}. 
 
 The hemimorphous ditrigonal prism pit\pqr} is a ditrigonal 
 prism with six faces, forming a symmetrical hexagon, of which 
 the alternate edges are similar and adjacent edges dissimilar, as 
 are the forms which close its opposite ends. 
 
 The hemimorphous rhombohedron and hemimorphous trigonal 
 S-pyramid represent in the forms p7t{hkk} and px {hkk} the 
 upper halves of those figures ; geometrically the same figures, they 
 do not differ crystallographically. Their symbols, taking pit for 
 the designating letters, are 
 
 p*{hkk}, pv{eff}, pv{hYk}, pit {I//}. 
 
 A hemimorphous dihexahedron p f n{min} would be identical 
 with the form p\min\, excepting in its crystallographic association 
 with ditrigonal forms. 
 
Mero symmetry. 307 
 
 263. v. The hemi-deuteropyramid p^{hkl}, hemi-deuteroprism 
 p \fs \p q r } , and hem i- deuterorhombohedron p-fy {mzn}, would be 
 analogous forms to those just described if the deutero-systematic 
 series of forms were to be recognised as actual forms ; but they 
 have only a theoretical interest, for the same reason as that which 
 removes the ditrigonal deutero-scalenohedron and the deutero- 
 rhombohedron from among the forms belonging to practical 
 crystallography. 
 
 264. vi. The normals represented in the hemi-tritopyramid are 
 symmetrical to no systematic plane, but are hemihexagonally disposed 
 round the axis of form. In the form taken as example, they are 
 the normals which are common to the forms p{hkl, efg}, a \hkl}, 
 and $ \h k /}, (Fig. 205). The tetrad is composed of the forms 
 
 p<t>{hkl, efg}, p<j>\hlk, eg/}, 
 p<t>{hkl, e/g\, p<t> \hl~k, ~eg/}. 
 
 The hemimorphous tritoprism p${pqr} differs crystallographi- 
 cally from the hemi-diprism ${pqr} in the different character of 
 the forms closing its opposite extremities, and, so far only, is also 
 geometrically different. 
 
 265. IV. Tetarto-systematic diplohedral forms. 
 
 The hemiscalenohedron or skew rhombohedron TT(f> {hkl}, Figs. 
 206 (a), (c), (d), (f), belongs to the remaining tetarto-symmetrical 
 type, namely, that in which each of the three extant normals carries 
 both its faces. 
 
 The symbols of the faces of the form TT(/> {hkl} are 
 
 hkl Ihk klh 
 
 hkl Ihk klh. 
 
 The figure is geometrically a rhombohedron : it differs from the 
 crystallographic rhombohedron of Art. 246 in that its edges do 
 not lie in systematic planes, and are thus incapable of truncation 
 or bevilment : further, the two three-faced quoins on the morpho- 
 logical axis are trigonally, not ditrigonally, symmetrical. 
 
 In Figs. 206 (a) and (c) are represented the forms TT(#> {20!} 
 and i:(f) {2i"o}, respectively, which together compose the scaleno- 
 hedron TT {20!} shown in Fig. 206 (3), and in Figs. 206 (d) and 
 
 x 2 
 
308 
 
 Hexagonal system. 
 
 are represented the remaining correlative forms TT</> { 45 2} 
 and K<t> {425}, which together compose the correlative scaleno- 
 hedron -TT {425} shown in Fig. 206 (e). 
 
 The forms Ar</>{2ii},^<|>{ioi}, and x $ { 1 1 1 } are geometrically 
 identical with the holo-symmetrical forms {zYT}, (iol},and {in}, 
 and the forms x<$>{pqr], x${hkk}, x${miri\ with the hemi- 
 symmetrical forms < {pqr}, 77 {hkk}, and \|/ {min} respectively. 
 
 Fig. 206 (a). 
 
 Fig. 206 (). 
 
 Fig. 206 (c). 
 
 Fig. 206 (d). 
 
 Fig. 206 
 
 Fig. 206 (/). 
 
 266. V. Tetarto-systematic haplohedral forms. 
 
 The remaining type of mero-symmetry is one in which only 
 three normals are present, each carrying a single face. The 
 symbols of the faces of the form pTr<p [hkl] .are 
 
 hkl Ihk klh. 
 
 The form may be regarded as a hemimorphous developement of 
 the hemiscalenohedron. This type of mero-symmetry has been 
 observed on crystals of sodium periodate (Art. 276). 
 
Combinations. 309 
 
 Hexagonal System. C. Combinations of Forms* 
 
 267. The holo-symmetrical type of the Hexagonal system is 
 represented by so limited a number of examples, in the known 
 range of crystals, as to have induced Professor Miller to treat all 
 crystals presenting hexagonal and trigonal symmetry as being 
 wholly of trigonal symmetry, and, in fact, as belonging exclusively 
 to a Rhombohedral system (Art. 250). 
 
 It is possible to do this by considering a discalenohedron or a 
 dirhombohedron as resulting from the combination of two distinct 
 and independent transverse scalenohedral or rhombohedral forms 
 respectively; or, where the poles of the latter forms lie on a single 
 triad of systematic planes, by viewing them as hexagonal pyramids 
 with symbols of the form \miri\. 
 
 Notwithstanding this high authority for such a treatment of the 
 Hexagonal system, it is more in accord with theoretical considera- 
 tions to regard the Hexagonal system as, potentially at least, a 
 natural one ; and it would still be so, even if the system were not 
 represented in its holo-symmetrical completeness by known crystals. 
 There are crystals, however, of which those of beryl, nepheline, 
 pyrrhotine, and zinc oxide (both artificial and as the mineral spar- 
 talite) are instances, that occur with holo-symmetry as hexagonal 
 forms, with cleavages parallel to a hexagonal prism and the pina- 
 koid, and offer no evidence of a mero-symmetrical structure. 
 Even had not this been the case, the actual occurrence of crystals 
 exemplifying any of the types of hemi-symmetry, other than the 
 rhombohedral type, derivable from a Hexagonal system would 
 necessitate the recognition of the actuality of the holo-symmetrical 
 system from which such types must, in common with the rhombo- 
 hedral type, be derived. And crystals, though their species are 
 few in number, are not wanting to establish the existence of such 
 hemi-symmetry (Art. 271). 
 
 268. (a) Holo-symmetrical forms. The combinations of hexa- 
 gonal forms are as remarkable for symmetrical and equipoised 
 character as are the combinations of forms in the Tetragonal 
 system. This character is conspicuous in beryl, of which Fig. 207 
 is representative, and in connellite, which is illustrated in Fig. 208, 
 
3io 
 
 Hexagonal system. 
 
 the beryl exhibiting discalenohedral and dirhombohedral forms in 
 subordination to the pinakoid- and prism-faces, while in connellite 
 the converse is the case: the faces rz in Fig. 208 belong to the 
 dirhombohedron (100, 122}, the faces 0o> to the discalenohedron 
 {467, 942}, and the faces ba to the proto- and deutero-prisms 
 respectively. 
 
 Fig. 207. 
 
 Fig. 208. 
 
 269. (b) Hemi-symmetrical forms. Of the varieties of hemi- 
 symmetry of which the Hexagonal system is susceptible, that 
 diplohedral type in which only the proto-systematic planes are 
 symmetral, often designated as the Rhombohedral system, is the 
 most frequent, and, indeed, the habitual type to which crystals in 
 the Hexagonal system belong. 
 
 In rhombohedral crystals the trigonal grouping round the mor- 
 phological axis is quite as marked a feature as the hexagonal 
 grouping in the holo-symmetrical crystals, or the tetragonal 
 grouping in the crystals of the system last considered. 
 
 Where prism-forms are extant, the crystals assume a hexagonal 
 aspect ; for the orthogonal projection of the side- or zigzag-edges 
 of a rhombohedron on a plane perpendicular to the trigonal axis is 
 a regular hexagon, the sides of which are the traces on the plane 
 of projection of the faces of the hexagonal deutero-prism {iol}. 
 
Combinations. 311 
 
 Thus the aspect of crystals in which prism-forms are present, even 
 if they be not predominant, differs considerably from that of 
 crystals in which only rhombohedral or scalenohedral forms are 
 extant. Figures 209 (a), 210, 211 (a), 212, represent in ortho- 
 gonal projection on the equatorial plane the relations of a direct 
 
 Fig. 209 (a). 
 
 2ZZ 
 
 Fig. 209 
 
 Fig. 211 (a). 
 
 Fig. 211 
 
 Fig. 212. 
 
 and an inverse rhombohedron to the proto-prism and deutero- 
 prism; Figs. 209 (), 211 () represent combinations of forms 
 corresponding respectively to those of Figs. 209 (a), 211 (a), but 
 projected on a plane distinct from the equatorial plane. 
 
 270. Rhombohedra and scalenohedra are not unfrequently com- 
 bined in tautozonal series, the faces of one rhombohedron trun- 
 cating the edges of another more acute. In the following table 
 any symbol in either of the inner columns is that of a rhombo- 
 hedron, of which the faces truncate the edges of the rhombo- 
 hedron having its symbol in the other inner column and on the 
 
312 
 
 Hexagonal system. 
 
 line next below : a symbol in either of the outer columns is 
 that of a rhombohedron transverse to (i. e. correlative with) the 
 rhombohedron of which the symbol stands on the same line with it 
 and in the inner column : 
 
 T 
 4* 
 
 Z i 
 
 cj 
 
 S" 
 
 1 
 
 Inverse Forms. 
 
 Direct Forms. 
 
 {141717} 
 
 
 {655} 
 
 
 
 {233} 
 
 
 {1077} 
 
 {255} 
 
 
 {211} 
 
 
 
 {Oil} 
 
 
 {4n} 
 
 {122} 
 
 
 {IOO} 
 
 
 
 {III} 
 
 
 {5"} 
 
 {755} 
 
 
 (3"} 
 
 
 
 {533} 
 
 
 {1777} 
 
 {31 17 17} 
 
 
 " 55 
 
 
 Thus the form {122} is an inverse rhombohedron transverse to 
 the direct rhombohedron {100}, the faces of which truncate the 
 edges of the inverse rhombohedron {Tn}, and the edges of which 
 are themselves truncated by the faces of the inverse rhombohedron 
 {on}. 
 
 Figs. 213-218 represent combinations of forms occurring in 
 calcite. 
 
 271. The other variety of hemi-systematic diplohedral forms 
 exemplified on known crystals is the gyroidal type of which 
 apatite, a phosphate and chloride or fluoride of calcium, offers the 
 most conspicuous, if not the only illustration: a corresponding 
 mero-symmetrical character has been developed by an etching 
 process on faces of pyromorphite, an analogous compound in 
 which lead takes the place of calcium. The poles of a gyroidal 
 
Combinations. 
 
 313 
 
 form ${hkl t efg] are those of the hexagonal discalenohedron in 
 the symbols of which the indices all follow either the direct order 
 h kl, efg, or else the inverse order Ikh, gfe ; in either hemisphere 
 the distribution of the poles is thus in asymmetric hexagonal (not 
 dihexagonal) symmetry, and at the same time the extant faces of 
 the form lie symmetrically with respect to the equatorial plane. 
 Fig. 219 illustrates such a form, namely (j> {524, To2}, in asso- 
 ciation with other forms having a holo-symmetrical hexagonal 
 
 Fig. 213. 
 
 Fig. 214. 
 
 Fig. 215. 
 
 Fig. 216. 
 
 'ISO 
 
 Fig. 217. 
 
 Fig. 218. 
 
 aspect. In fact crystals of apatite usually present a considerable 
 number of forms quite hexagonal in their symmetry, while only 
 a few forms exhibit, in the defalcation of their alternate faces, the 
 gyroidal hemi-symmetry, which, however, must be held really to 
 dominate the structure of the entire crystal. 
 
 272. Hemimorphous forms are not rare in the Hexagonal 
 system, but they occur most often as hemimorphs of hemi- 
 symmetrical types of crystal, and are then tetartohedral in their 
 symmetry. But as representing the hemimorphous type of a 
 
314 Hexagonal system. 
 
 crystal otherwise holosymmetrical, greenockite, the cadmium sul- 
 phide, occurs in crystals carrying generally hexagonal pyramids (or 
 dirhombohedra) arranged in series tautozonal with the pinakoid and 
 the hexagonal deutero- (or proto-) prism ; but the faces of some of 
 these forms exist only on one side of the equatorial plane, and the 
 crystals often exhibit on the other side of that plane only a single 
 
 Fig. 219. Fig. 220. 
 
 pinakoid-face (Fig. 220). Artificial crystals of cadmium sulphide 
 have the parameters of greenockite, but present both rhombo- 
 hedral and scalenohedral forms. Perhaps with greenockite may 
 be associated the hexagonal zinc sulphide, wurtzite. 
 
 273. (c] Tetar to-symmetrical forms. Of the four types of tetarto- 
 symmetry that, exclusive of hemimorphs of hemi-symmetrical 
 forms, are possible in this system, two are illustrated in known 
 crystals : they may both be derived from the rhombohedral type of 
 hemi-symmetry, the one as a haplohedral, the other as a diplo- 
 hedral derivative. The trapezohedron is the general tetartohedron 
 of the former type ; of the latter, or tetarto-hexagonal type, the 
 general form is the hemiscalenohedron or skew rhombohedron. 
 
 Of the trapezohedral tetartohedron, quartz and sodium periodate 
 afford examples in their crystals : of the hemiscalenohedron, in- 
 stances are met with on certain crystals of phenakite, dioptase, and 
 ilmenite. 
 
 274. As already mentioned in Art. 257, the trapezohedra that occur 
 on quartz belong to two correlative groups associated, the one with 
 Isevo-rotatory, the other with dextro-rotatory polarisation : and the 
 forms of the two groups, if conceived as occurring simultaneously, 
 would be symmetrical with respect to the protosystematic planes. 
 
 When examining with a nicol-prism divergent light originally 
 
Combinations . 315 
 
 plane-polarised that has traversed a quartz-plate of a certain thickness 
 with its faces cut perpendicularly to the axis, it will be found that 
 on turning the analysing prism to the right (as the hands of a 
 watch move) the isochromatic rings contract and their centre passes 
 through tints in the order blue, plum-colour (the ' sensitive ' tint), 
 orange, red, when the crystal is 'left hand ;' the rings dilate and 
 the tints follow the order red, orange, plum-colour, blue, when the 
 crystal is 'right hand:* the phenomena being reversed in either 
 case if the analyser be turned towards the left. 
 
 Fig. 221 (A.). Fig. 221 (/>). 
 
 In quartz the predominating faces of the laevo-rotatory forms 
 aTi{hkl} lie in zones [102], [210], [021] on the upper or positive 
 hemisphere, and in zones [120], [012], [201] on the negative 
 hemisphere ; those of the dextro-rotatory are in the latter zones on 
 the positive, in the former zones on the negative hemisphere. These 
 zones are tautohedral in faces of the rhombohedron air {100}, the 
 S-prism air {n^}, and the hexagonal pyramid an {412}, (Figs. 
 2 2 1 A and p). 
 
 In the case of quartz the faces of the form a 77 {41 2:} or 
 a IT \4~2i} are small and rhomboidal in shape: they are usually 
 striated parallel to their intersections with faces of the forms 
 aTTJioo}, a TT { 112 j, (see plate V). Edges of the predominating 
 trapezohedra (the so-called plagihedral forms) are parallel to those 
 edges of the form air {412} which run athwart the striation. 
 
 A crystal of quartz, when so placed that the faces 100 oio ooi 
 are at its upper end, will, if right-handed, present on three alternate 
 
316 
 
 Hexagonal system. 
 
 quoins at each end of the prism plagihedral faces arranged in the 
 form of a right-handed screw. Furthermore the two plagihedral 
 faces, belonging to a single form, which are associated with any of 
 the three prism-faces 2!! 12! 112 will be to the right of the 
 observer as he looks at the prism-face (Fig. 221 r). On a left- 
 handed crystal the plagihedral faces are arranged in the form of 
 a left-handed screw and the two associated with any of the above 
 prism-faces are seen to the left of the observer (Fig. 221 /). The 
 faces of the form air {122} denoted in the figures by the letter z 
 are generally smaller and more dull than the faces of the form 
 a7r{ioo} denoted by the letter r: in Figs. 221 (/) and 221 (r) 
 the s faces are the rhomb-faces of the form air {4^}, and the x 
 faces are those of the plagihedral forms air {4^1} and air 
 respectively. 
 
 Fig. 221 (/). 
 
 Fig. 221 (r). 
 
 275. A group of dithionates (or hyposulphates) offers examples 
 of tetartohedrism of the type under consideration ; the lead salt in 
 particular (Pb S 2 O 6 . 4 H 2 O) exhibits trapezohedral forms in con- 
 junction with rotatory action on plane polarised light. These forms 
 are, however, so distributed that of the two prevalent correlative 
 rhombohedra the more developed would have to be taken as the 
 form CLTT {122} transverse to the axial rhombohedron cm {100} in 
 order that the situation of the plagihedral faces relative to the latter 
 form might correspond with that presented by crystals of quartz, 
 that is to say, that their faces might be gyroidally to the left of the 
 primary rhombohedral faces in laevo-rotatory, and to the right in 
 dextro-rotatory crystals. The more developed rhombohedron, how- 
 ever, is usually taken as air {100}, so that the gyroidal distribution 
 of the plagihedral forms becomes inverse to that obtaining in quartz. 
 
Combinations. 
 
 276. Sodium periodate crystallised with three molecules of water 
 also has a rotatory action on plane-polarised light, and carries 
 forms of trapezohedral type indicative of the direction of the 
 rotatory action ; the crystals exhibit the additional peculiarity that 
 they are hemimorphous in developement. Fig. 224 represents one 
 end of a crystal of sodium periodate projected on the equatorial 
 plane : at the other end only the pinakoid plane is present. 
 
 These crystals will be further considered under the subject of 
 twin-crystals belonging to this type of mero-symmetry. 
 
 277. The hemiscalenohedron, when present, is subordinate to 
 rhombohedral forms, as in Figs. 222 and 223. The first of these 
 represents a crystal of dioptase in which the hemiscalenohedron 
 
 \ioi\\xto 
 
 Fig. 222. 
 
 Fig. 223. 
 
 7T< {310} is combined with the rhombohedron TT$ {100} and the 
 deutero-prism 7r<|>{ioT}: in Fig. 223 is illustrated a crystal of 
 ilmenite presenting the hemiscalenohedron ir(j) {31!} in combina- 
 tion with the rhombohedra Tr<f) {100}, TTCJ) {no}, and TT</> {nT}. 
 
 Fig. 224. 
 
 Fig. 225. 
 
 278. Of the hemimorphous developement of crystals of rhombo- 
 hedral type tourmaline is a conspicuous example, different forms 
 
Hexagonal system. 
 
 being present on the opposite sides of the equatorial plane. A not 
 infrequent accompaniment of hemimorphism, the pyroelectric 
 character, is strongly illustrated in this remarkable mineral. Fig. 
 225 represents a crystal of tourmaline in which the lower extremity 
 of the morphological axis is the so-called analogous pole, and the 
 upper the antilogous pole, being respectively positive and negative 
 in electrification when the temperature is rising. 
 
 Hexagonal System. D. Twinned Forms. 
 
 279. I. Twins of holo-symmetrical crystals. Simple holo-sym- 
 metrical crystals are of such rare occurrence in this system that 
 few examples of twin-structure of holo-symmetrical types are to 
 be expected, and in fact the only one on record is a minute crystal 
 of zinc oxide, obtained accidentally as a furnace product, which 
 has been described by vom Rath *. The twin-plane was very near 
 to (100), although the measurements obtained indicated a face of 
 more complex symbol. It seems quite possible that there is in 
 this case only an accidental simulation of twin-structure. 
 
 280.11. Twins of hemi-symmetrical crystals . Hemi-symmetrical 
 crystals afford abundant examples of twin-structure : their great 
 
 sil 
 '. 
 
 (2U) 
 
 (&) 
 
 Fig. 226. 
 
 Fig. 227. 
 
 variety in aspect is however due rather to the diversities exhibited 
 by individual crystals, even of the same substance, than to any 
 great variety in the twin-laws under which crystals belonging to 
 
 * Pgg- Annal. vol. 144. p. 580. 
 
Twinned forms. 
 
 this system are united. It is in the numerous crystals of rhombo- 
 hedral type that we must look for examples of such unions ; and 
 
 Fig. 228. 
 
 of these examples the multiform mineral calcite presents the most 
 conspicuous and varied illustrations. 
 
 281. The twin -laws governing the union of rhombohedral 
 crystals are the following : 
 
 i. Twin-plane a face of a direct rhombohedron. 
 
 (a) Twin-plane a face of the form {100}. 
 
 Figures 226, 227, 228 represent respectively the rhombohedron 
 TT{IOO}, the hexagonal proto-prism 7r{2TY}, and the scalenohedron 
 TT {20!} of calcite twinned upon a face (olo) of the form TT {100}, 
 the twin-plane being likewise the face of union. 
 
 282. (d) Twin-plane a face of the form TT { 2 1 1 } . 
 
 Fig. 229. 
 
 Fig. 230. 
 
 Fig. 229 illustrates a crystal of pyrargyrite twinned about the 
 plane (i~2 T), which is parallel to the plane truncating the edge cl 
 
320 Hexagonal system. 
 
 formed by the faces (no) (on) : in this case the twin-plane is the 
 face of union of the two individuals. Sometimes the twin-plane is 
 at right angles to the face of union and three individuals are in 
 twin-position relative to a central one, as shown in Fig. 230. If 
 the line Ir be at right angles to the edge cl it is the twin-axis for the 
 individuals A B : hence the edges k lc^ of the two crystals fall in 
 the same straight line and the faces no (on) have the same 
 direction; the faces (no) on on the other side of the edges k lc^ 
 are likewise co-planar. The planes truncating the edges cm en 
 are respectively twin-planes for the pair AC and the pair AD. 
 
 If the above be the true expression of the law of this twin- 
 structure, the face of union, having its pole in a proto-systematic 
 plane and being at right angles to a plane of the form TT {211}, 
 cannot be a plane with rational indices. 
 
 283. 2. Twin-plane a face of an inverse rhombo- 
 hedron. 
 
 (a) Twin-plane a face of the form {on}. 
 
 Fig. 231 represents a simple rhombohedron 77 {100} of calcite 
 twinned on the plane (Ho) which truncates the edge between the 
 faces loo olo : the twin-plane is also the face of union. Four 
 cleavage-planes of the twin-structure, namely oio, 100, (Too) and 
 (olo), form a prism, the angle between the planes 100 oio and 
 also between the planes (Too) (olo) being 105 5', and the angle 
 between the planes 100 (Too) and also between oio (olo) being 
 74 55 /: f tne remaining cleavage-planes ooT and (ooi) form a 
 re-entrant angle of 38 8', while ooi and (ooT) form a salient angle 
 of the same magnitude. 
 
 In scrutinising a transparent block of the calcite from Iceland, 
 known as Iceland spar, there is frequently to be found a very 
 narrow twin-lamina or twin-plate of calcite intercalated in the mass 
 of the crystal in accordance with this law. And in other cases of 
 its occurrence, as at the Rathhaus-berg near Gastein, and at Auer- 
 bach, Hesse-Darmstadt, the intercalated laminae are so thin and 
 so numerous as to give quite a lamellar character to the crystal- 
 aggregate. 
 
 Crystals of the metal bismuth also occur twinned in obedience 
 to this law. 
 
Twinned forms. 
 
 321 
 
 284. (3) Twin-plane a face of the form {m}. 
 
 In Fig. 232 is illustrated a scalenohedron 7r{2o7} of calcite 
 twinned on a face YiY of the inverse rhombohedron {Tn} the 
 edges of which would be truncated by the faces of the primary 
 rhombohedron TT{IOO}. The face ill itself truncates the edge cm 
 
 Fig. 231. Fig. 232. 
 
 between the faces 2 To . ol2. In the twin-crystal the edges cm 
 c^ m^ are parallel to each other and to the twin-plane which is like- 
 wise the face of union. 
 
 285. 3. Twin-plane a face of the pinakoid {in}. 
 
 (a) Face of union coincident with the twin-plane. 
 
 Fig- 234. 
 
 Twinned scalenohedra of calcite from Derbyshire, similar to 
 Fig. 233, are familiar illustrations of this twin-structure. The 
 aspect of the twin is similar to that which would result if the 
 scalenohedron TT {20!} were cut through its centre by a plane 
 
 y 
 
322 Hexagonal system. 
 
 parallel to the faces of the pinakoid -TT { 1 1 1 } and then one-half 
 of it turned round the morphological axis through 60, or any odd 
 multiple of that angle. The twin-plane is the face of union : the 
 edges which lie in this plane and are formed by the meeting of the 
 two individuals are salient and re-entrant in alternate pairs. 
 
 (Z>) Face of union perpendicular to the twin-plane. 
 
 The crystal of haematite illustrated in Fig. 62, page 178, is an 
 example of this kind of twin-growth. 
 
 (c) Interpenetrating crystals. 
 
 Rhombohedra of dolomite, and notably of chabazite, twinned about 
 the morphological axis and mutually interpenetrating afford illus- 
 trations of this variety. Fig. 234 represents a twin-growth of the 
 latter mineral : the former mineral is now regarded as being tetarto- 
 symmetrical in its structure. 
 
 286. III. Twins of tetarto-symmetrical crystals. 
 
 In considering the twin-growths of tetarto-symmetrical crystals 
 we shall be almost entirely limited to those of quartz, the crystals of 
 which besides being the most common of mineral products are 
 seldom entirely simple in their structure. 
 
 A remarkable class of regular growths of quartz, simulating the 
 aspect of simple crystals, was first explained by Rose. As already 
 stated in Art. 274, a right-handed crystal of this mineral presents 
 on three alternate quoins at each end of the prism plagihedral 
 faces arranged in the form of a right-handed screw, and if the 
 crystal is so placed that the faces 100 oio ooi are at the upper 
 end, the two plagihedral faces belonging to a single form, which 
 are associated with any of the three prism-faces 2TT 2? and TT2, 
 are to the right of the observer as he looks at the prism-face 
 (Fig. 235 a). In the illustrative figures the faces of the rhombo- 
 hedron air {100} are indicated by the letter r and those of the 
 rhombohedron air {122} by the letter z\ the plagihedral faces are 
 denoted by the letter x : suffixes to these letters refer to the position 
 of the simple crystal to which the corresponding faces belong. 
 The crystal represented in Fig. 235 (a), after half a revolution 
 round the morphological axis takes the position represented in Fig. 
 235 (*") : the faces r l and z l are parallel respectively to the faces 
 s and r of the first position, and the prism-faces (afn) (*"*2) l 1 ^ 1 ) 
 
Twinned forms. 
 
 323 
 
 of Fig. 235 (<r) have positions which correspond to those of the 
 prism-faces 2!! 112 I2l of Fig. 235 (a); it will be seen that in 
 this position of the crystal the two plagihedral faces associated 
 with a face corresponding to any of the above three prism-faces 
 2ll 12! IT 2 of the crystal in its first position are now to the left 
 of the observer. 
 
 An intergrowth of two crystals with a common morphological 
 axis and the relative positions shown in Figs. 235 (a), 235 (c) may 
 thus have for result an apparently simple crystal without re-entrant 
 angles. But in such a compound crystal the plagihedral faces, 
 instead of presenting themselves systematically on alternate quoins, 
 may be either present or absent on any of the quoins according 
 as the quoin belongs to a crystal with the first or second position. 
 
 Fig. 235 (a). 
 
 Fig- 235 
 
 Fig. 235 (<:). 
 
 And in the same way, any one of the six faces forming the pyra- 
 midal termination of the compound crystal may belong wholly to 
 the form a?r {100} or wholly to the form cnr {^22} according as it is 
 part of a crystal with the first or second position, and on the other 
 hand it may be a compound face, one portion of it belonging to 
 the form cnr {100} of one position and the remainder to the form 
 ctTr {122} of the other: as the r faces are generally brighter than 
 the z faces, this compound nature is usually apparent. Fig. 
 2 35 () represents a possible form of a compound crystal resulting 
 from the intergrowth of simple crystals having the positions shown 
 in Figs. 235 (a) and (c). 
 
 287. Similar growths of two left-handed crystals are met with. 
 Fig. 236 (c) represents the position of a left-handed crystal after 
 half a revolution round the morphological axis from the position 
 
 Y 2 
 
324 
 
 Hexagonal system. 
 
 shown in Fig. 236 (a): Fig. 236 (<5) illustrates a possible form of 
 a compound crystal resulting from the intergrowth of crystals 
 having these positions. 
 
 Fig. 236 (a). 
 
 Fig. 236 (b). 
 
 Fig. 236 (<:). 
 
 288. Another kind of intergrowth, described by Rose as a twin- 
 growth, does not come within the definition of a twin given in 
 Art. 154, but yet is so similar in aspect to those just explained that 
 it may be here most conveniently mentioned. It consists of a left- 
 handed and a right-handed crystal with the relative positions 
 shown in Figs. 237 (a) and (<:) respectively, the similar rhombo- 
 hedral planes having corresponding positions in the two individuals : 
 the crystals may be described as having homologous positions ; as 
 
 Fig- 237 () 
 
 Fig. 237 
 
 Fig. 237 (0. 
 
 already pointed out, crystals presenting incongruent forms cannot be 
 parallel (Art. 197). Adjacent quoins at either end of such a com- 
 pound crystal, if modified at all, present plagihedral faces of opposite 
 kinds : the pyramidal terminations are in their physical characters 
 not to be distinguished from those of simple crystals. Fig. 237 (b) 
 illustrates a compound crystal of this kind resulting from the inter- 
 
Twinned forms. 
 
 325 
 
 growth of crystals having the positions shown in Figs. 237 (a) and 
 (<:). That crystals, apparently simple but yet presenting both right 
 and left plagihedral faces, are really composite, as suggested by 
 Rose, was proved by Groth, who found that a slice from such 
 a compound crystal rotates the plane of polarisation oppositely in 
 different parts and in accordance with the right or left position of 
 the adjacent plagihedral faces. 
 
 289. The surface of junction of two right-handed or left- 
 handed individuals or of a right-handed and a left-handed indi- 
 vidual is generally a very irregular one : the difference in the 
 physical characters on opposite sides of the junction often renders 
 it possible to trace the boundary on the faces of the compound 
 crystal; in Fig. 238 is illustrated a crystal belonging to the British 
 
 Fig. 238. 
 
 Fig. 239. 
 
 Museum. Growths of right and left crystals similar to the one 
 shown in Fig. 237 (3) are sometimes twinned about the morpho- 
 logical axis as if they were simple crystals : Fig. 239 illustrates a 
 growth of this character presenting re-entrant angles. 
 
 290. Since the time of Rose, composite crystals of right and left 
 quartz, with the general aspect of simple crystals but with a higher 
 symmetry, have been described in which the r faces of the one 
 individual are co-planar with the z faces of the other, a relative 
 position of the two crystals which may be regarded, from a purely 
 geometrical point of view, as due to a half-revolution of one indi- 
 vidual round the morphological axis from a position in which it 
 was homologously disposed to the other. Figs. 237 (a) and (c) 
 represent a left-handed and a right-handed crystal in homologous 
 positions : Figs. 240 (a) and (c) represent the relative positions of 
 
326 
 
 Hexagonal system. 
 
 the two individuals after the left-handed crystal shown in 237 (a) 
 has been turned through half a revolution round the morphological 
 axis. 
 
 In a composite crystal resulting from the intergrowth of two 
 crystals with these relative positions three alternate quoins at either 
 end of the prism can- present no plagihedral faces at all, while 
 each of the remaining three quoins may present either right or left 
 plagihedral faces or both, according as the quoin is simple and 
 belongs to the right-handed or to the left-handed crystal, or is 
 
 Fig. 240 (a). 
 
 Fig. 240 
 
 Fig. 240 (<:). 
 
 composite and belongs partly to the one and partly to the other. 
 Fig. 240 (i) illustrates a possible disposition of the plagihedral 
 faces on a growth of this kind. 
 
 291. Growths of a right and a left crystal with the above 
 relative positions but presenting re-entrant angles and an evident 
 composite character have likewise been described. As both the 
 individuals did not present plagihedral faces, the real nature of the 
 growth was misunderstood until an optical investigation revealed 
 the right and left character of the crystals. 
 
 292. Another class of regular growths, one of which was first 
 described by Weiss in 1829, is presented by certain rare specimens 
 of quartz from La Gardette, Traversella, Munzig, and Japan. 
 These growths are of two kinds, which are so far similar that in 
 both of them the morphological axes of the two individuals are in 
 a 2-plane and have a mutual inclination of 84 34': but whereas, 
 relatively to the plane bisecting the above angle between the mor- 
 phological axes and at right angles to their plane, the r faces of 
 one individual are symmetrically disposed to the r faces of the 
 
Twinned forms. 
 
 327 
 
 other in the first kind, they are symmetrically disposed to the z 
 faces of the other in the second kind. 
 
 293. The relative positions of the two individuals may also be 
 imagined in the following way. In Fig. 241 (a) the line 0/ x is in 
 a 2-plane and inclined at an angle of 4743 / to the morpho- 
 logical axis; it is therefore at right angles to a terminal edge ca 
 
 Fig. 241 (a). 
 
 Fig. 241 
 
 of the ordinary six-sided pyramid rz and also to the plane of the 
 form a TT {52!} which would truncate that edge: the line 0/ 2 is 
 in the same 2-plane and perpendicular to the line 0/, : it is there- 
 fore parallel to the edge ca and makes an angle of 42 17' with the 
 morphological axis. 
 
 Fig. 242 (a}. Fig. 242 (). 
 
 A half revolution of the crystal represented in Fig. 241 (a) 
 round the line ot^ will bring it into the position shown in Fig. 
 241 (Z>) : the morphological axes of two crystals with these relative 
 positions will be inclined to each other at an angle of 84 34' and 
 will lie in a 2-plane ; further, the r faces of the two individuals 
 will be symmetrically disposed relative to a plane bisecting the 
 
328 Hexagonal system. 
 
 angle 84 34' between the morphological axes, the prism-faces 
 ii? .(TT2) will be co-planar, and the edge ca of one individual 
 will be parallel to the edge cd of the other. A growth of two 
 individuals with these relative positions will thus be one of the first 
 kind (Fig. 242 a). 
 
 294. Again, a half- revolution round the line 0/ 2 of the crystal 
 represented in Fig. 241 (a) will bring the crystal into the position 
 represented in Fig. 241 (c): as before, the morphological axes will 
 be inclined to each other at an angle of 84 3 4' and will lie in 
 a 2-plane ; but now the r faces and the z faces will be sym- 
 metrically disposed relative to a plane bisecting the angle 84 34': 
 the prism-faces nl .[TT2] will be co-planar, and the edge ca of 
 one individual will be parallel to the edge ca of the other. A 
 growth of two individuals with these relative positions will thus be 
 one of the second kind. 
 
 A crystallographically identical result would be obtained by a 
 half-revolution, round the morphological axis, of one of the two 
 individuals of a growth of the first kind (Fig. 242 a). 
 
 295. The positions of the plagihedral faces of a crystal of 
 quartz being defined by those qf the r and z faces and the right 
 or left character of the crystal (Art. 274), the above growths may 
 be classed as twins, always falling under the definition given in 
 Art. 154, whether the two individuals be both right-handed or both 
 left-handed, and may still be geometrically explained by a reference 
 to a single half-revolution even when one individual is right- 
 handed and the other left-handed, if the pair be first placed in 
 homologous positions, as explained in Art. 288. 
 
 In the majority of the few known examples of this class of 
 regular growths, it is practically difficult to decide, owing to the 
 absence of plagihedral faces, as to the right or left character of 
 both individuals : still specimens have been described in which the 
 individuals are regarded as being both right-handed or both left- 
 handed, or one right-handed and the other left-handed in character. 
 
 296. In no case, however, has the simple character of the indi- 
 viduals themselves been satisfactorily demonstrated : we know, 
 indeed, that the observations of Des Cloizeaux have led him to 
 infer that apparently simple crystals of quartz from La Gardette 
 
Twinned forms. 
 
 329 
 
 are not exclusively composed of either the right or the left variety, 
 and that a homogeneous simple crystal of quartz is one of the 
 greatest of mineralogical rarities : further, vom Rath, in his exami- 
 nation of the regular growth from Japan, though he failed to find 
 any evidence as to the right or left character of the individuals, 
 found abundant proof of their compound structure. It is thus 
 quite possible that we have here essentially a single kind of twin- 
 growth, geometrically explained by a half-revolution, round the 
 normal to a plane of the form 077(52!}, of one individual from 
 a position in which it is parallel or homologously disposed to the 
 other, and that the second kind of growth is merely a result of the 
 
 Fig- 243. 
 
 fact that one or both the individuals is itself a twin-growth of the 
 kind described in Arts. 286-290. 
 
 297. An idea of the actual developement of the individuals and 
 of the difficulty of deciding as to their true character may be 
 obtained from an examination of Fig. 243, which represents a very 
 fine British Museum specimen of a twin-growth of this class from 
 La Gardette. The individual A carries largely-developed plagi- 
 hedral faces characteristic of a simple left-handed crystal. On the 
 edgey^- of the individual B are small notches ss of which the sides 
 are parallel to those faces of the rhombohedron and prism which 
 are connected by the edge, and at the bottom of which is a plane 
 of the form s or a 77 {412} : the plane at the bottom of one of the 
 larger notches carries striations indicative of a right-handed crystal. 
 
330 Hexagonal system. 
 
 On the lower part of the same individual there are small faces ssx 
 likewise characteristic of a right-handed crystal. It will be seen, 
 however, from Art. 286 that the disposition of these faces on the 
 individual B is that of the faces of a composite crystal twinned 
 about the morphological axis. As the limits of the individuals of 
 which B is composed cannot be traced, and it is difficult in this 
 specimen to distinguish the r from the z faces by means of their 
 physical characters, we cannot be certain as to which of the 
 rhombohedral faces of B belong to the forms a-7r{ioo} and 
 a TT {22} respectively. 
 
 Some of the rhombohedral faces of both A and B show dull 
 patches, such as are usual in composite crystals twinned about the 
 morphological axis. 
 
 The boundary of the individuals A and B is of a zigzag shape, 
 as is seen from its trace on the co-planar prism-planes. 
 
 298. Of other tetarto-symmetrical minerals than quartz, phena- 
 kite is remarkable for its twin-growths. The individuals are 
 twinned about the morphological axis, and are mutually interpene- 
 trant : owing to the subordinate character of the planes which are 
 tetarto-symmetrically developed the twin-growths have an aspect 
 somewhat similar to that of chabazite twins (Fig. 234). but further 
 exhibit the planes of the hexagonal prism truncating the zigzag 
 edges of the rhombohedron. 
 
 299. It has already been remarked that crystals of sodium 
 
 periodate carry forms which may be regarded as a hemimorphous 
 development of a tetarto-symmetrical type, and that the crystals, 
 like those of quartz, are right-handed and left-handed in their 
 rotation of the plane of polarisation. Twin-growths of this sub- 
 
Ortho-rhombic system. 331 
 
 stance have been described by Groth. The individuals of a com- 
 posite crystal are respectively right-handed and left-handed in 
 optical character, and are united in a face parallel to the pinakoid ; 
 they are not symmetrical with respect to this plane, but would 
 become so if one of the individuals were turned through 60, or 
 any odd multiple of 60, round the morphological axis (Fig. 244). 
 It will be seen from the figure that a half- re volution of one of the 
 individuals round the line ad would bring it into such a position 
 that the faces of the forms present would coincide with those of the 
 other individual, and that the two individuals would then be in 
 homologous positions. The line ad is parallel to the line be in 
 which a rhombohedron-face meets a plane parallel to the pinakoid ; 
 it is therefore perpendicular to a face of the hexagonal deutero- 
 prism. 
 
 SECTION IV. The Ortho-rhombic or Ortho-symmetric 
 System. 
 
 A. Holo-symmetrical Forms. 
 
 300. The characteristics of this system are that its holo-sym- 
 metrical forms are symmetrical to three perpendicular planes of 
 unconformable symmetry intersecting in three perpendicular mu- 
 tually incongruent axes of orthogonal symmetry. 
 
 These planes are, the proto-systematic plane S, the deutero- 
 systematic plane 2, and the trito-systematic plane C which is taken 
 as horizontal in position. 
 
 The systematic planes being taken for the axial planes, the three 
 systematic axes, which are also their normals, become the crystallo- 
 graphic axes ; the vertical axis \S 2] being taken for the axis 
 of Z, and the zone-lines [2C] and [SC] for the axes of X and 
 Y respectively. 
 
 The elements of the crystal are 
 
 f = T\ = C ~ 9 a\ b : c, 
 
 the parametral ratios having different values. The systematic 
 planes being mutually incongruent, and equally important in their 
 relation to the symmetry of the crystal, any one of them may 
 be selected as the plane 2, S, or C. It would have been more 
 
332 
 
 Ortho-rhombic system. 
 
 satisfactory if in the description of crystals it had been adopted as 
 a convention that the intercepts made on the axes OX, OY and 
 OZ by the parametral plane should be in descending order of 
 magnitude. Very often, however, that axis has been selected as 
 the direction OZ which presents itself as the axis of a well- 
 developed prismatic zone, or as the normal of a large pinakoid 
 face, independently of the magnitude of its parametral intercept; 
 the greater of the two remaining intercepts being then taken 
 for the axis OX : or again, the axes have been so selected as to 
 
 olo 
 
 Fig. 245. 
 
 suggest that the crystals are similar in developement to others 
 having an analogous chemical composition. 
 
 The systematic triangle is quadrantal and all its angles are right 
 angles, each such triangle being commensurate with one of the 
 octants of the system. 
 
 The general scalenohedron \hkl\ of the system is the scalene 
 octahedron ; and the remaining six forms, representing those of 
 which the poles lie on corresponding arcs or at the several angles 
 of the systematic triangle, are 
 
Holosymmetry. 333 
 
 The proto-dome { o k /}. 
 
 The deutero-dome { h o / } . 
 
 The prism \hko\. 
 
 The proto-pinakoid {100}. 
 
 The deutero-pinakoid {oio}. 
 
 The trito-pinakoid { oo i } . 
 
 The positions of the poles of such forms on the sphere of pro- 
 jection are illustrated in Fig. 245 for the case where 
 
 a : b : c = 1-894 : i : i'797> 
 
 as in topaz ; the axis of the well-developed prism of this mineral being 
 generally taken for the axis OZ, although the parametral intercept 
 on that axis is greater than the intercept on one of the other axes. 
 
 Fig. 246. Fig. 247. 
 
 301. The scalene octahedron {hkl}, Figs. 246-7, is the general 
 independent form of the system. Its faces are scalene triangles, 
 their symbols being 
 
 hkl hkl hkl hkl, 
 Ykl h~kl hkl ~hkl. 
 
 The three systematic planes pass through the twelve edges of 
 the form, the four similar edges in which the form is intersected 
 by each plane giving a rhombic section. The three rhombic sections 
 however in the different systematic planes are obviously not con- 
 gruent, the edges in one section being dissimilar from those in a 
 different section. 
 
 The six quoins are four-faced and ortho-symmetrical on the 
 respective axes, but the pair on one axis is not congruent with 
 that on a different axis. In Fig. 246 is represented the form 
 {213} corresponding to the above elements. 
 
 The parametral form {HI} is a scalene octahedron (Fig. 247). 
 
334 Ortho-rhombic system. 
 
 302. Prismatid forms. The designations of these open forms, 
 the horizontal prismatid forms as domes, the vertical one as a 
 prism, have already been given in article 109. They are the forms 
 of which the faces truncate the edges of a scalene octahedron, 
 each therefore consisting of a rhombic based prism, the rhombic 
 sections of which have their diagonals parallel to two of the axes 
 of the system. These diagonals are termed the proto- and deutero- 
 diagonals of the prism, and either of them associated with the 
 vertical axis [S 2] forms the pair of diagonals of one of the dome 
 forms. 
 
 i. The proto- dome {ok I}, Figs. 248-9, has its four edges parallel 
 to the proto-diagonal [2 C], the axis X ; the symbols of its faces 
 are 
 
 okl okl oYl okl-, 
 
 013 
 
 Fig. 248. Fig. 249. 
 
 they are intersected perpendicularly to their edges by the proto- 
 systematic plane S, which therefore divides them symmetrically, 
 the traces of the section being rhombs the diagonals of which 
 are the vertical axis and the deutero-diagonal. Fig. 248 represents 
 the proto-dome {013}, closed by the faces of the form {100}. In 
 the parametral dome {on} (Fig. 249) these diagonals are in the 
 
 . c 
 
 ratio - 
 o 
 
 ii. The deutero-dome {hoi}, Figs. 250-1, has its four edges 
 parallel to the deutero-diagonal [SC], the axis Y\ the symbols of 
 its faces are 
 
 hoi hoi hoi hoi] 
 
 they are euthy-symmetrical to the intersections with them of the 
 deutero-systematic plane 2, the diagonals of the rhombic section of 
 the dome being the vertical axis and the proto-diagonal. Fig. 250 
 
Holosymmetry. 
 
 335 
 
 represents the deutero-dome {203} closed by the faces of the 
 form {oio}. 
 
 The parametral deutero-dome {101} (Fig. 251) represents in its 
 
 diagonals the ratio - 
 
 Fig. 250. 
 
 Fig. 251. 
 
 iii. The four faces of the rhombic prism {hko\, Figs. 252-3, 
 have for their symbols 
 
 hko hkQ h~kv 
 
 they are euthy- symmetrically intersected by the plane C, which 
 is the plane of their zone, as S and 2 are of those of the proto- 
 and deutero-domes. Fig. 252 represents the rhombic prism {210} 
 
 210 
 
 JlO 
 
 Fig. 252. 
 
 Fig. 253. 
 
 corresponding to the above elements, closed by the faces of the 
 form {ooi}. 
 
 The parametral prism {no} (Fig. 253) gives in its diagonals 
 
 the ratio - 
 b 
 
 303. The faces parallel to each systematic plane constitute a 
 distinct pinakoid form. 
 
336 Ortho-rhombic system. 
 
 Those of the proto-pinakoid {100} are parallel to the proto- 
 systematic plane . They are 100, Too (Figs. 248, 249). 
 
 The faces of the deutero-pinakoid {oio}, namely oio and oTo, 
 are parallel to the deutero-systematic plane 2 (Figs. 250, 251): 
 and the faces of the trito-pinakoid or basal pinakoid {ooi}, 
 namely ooi and ooi, are parallel to the horizontal plane of sym- 
 metry C (Figs. 252, 253). 
 
 The poles of the pinakoids form the angular points of the 
 systematic triangles ; those of the prismatid forms lie on the arcs 
 of those eight triangles. 
 
 Ortho-rhombic System. B. Hero-symmetrical Forms. 
 
 304. In the Tetragonal and Hexagonal systems the mere-sym- 
 metrical forms admissible by the conditions of the systems were 
 remarkable for their variety, and except in the case of the rhom- 
 bohedral division hardly less so for the paucity and in some cases 
 the absence of known crystal species that exemplify them. 
 
 In the Ortho-rhombic system the simpler character of the system 
 is illustrated as well in the few and simple kinds of hemi-symmetry 
 it offers, as in the greater number of crystals by which these are 
 represented. 
 
 In the holo-systematic section of this system there are four 
 normals to the general form, the scalene octahedron {hkl} ; and 
 in the case of hemi-symmetry a face will be extant for each of 
 these normals. 
 
 There are two ways in which the suppression of the four absent 
 faces may occur. 
 
 305. I. The rhombic sphenoid a {hkl} or a {hkl}, Figs. 254 (a), 
 (c). In the first, the asymmetrical case, no systematic plane is sym- 
 metral ; but each of the three ortho-symmetral axes becomes an axis 
 of diagonal symmetry. Hence the alternate faces of the holohedral 
 form [h k 1} are absent or extant. The resulting form is tetrahedrid 
 in character, its four faces being similar scalene triangles. It is 
 therefore a scalene sphenoid. Its six edges are similar in pairs 
 only, a pair of similar edges being oppositely placed on the crystal 
 and having their directions symmetrically inclined in respect to one 
 of the systematic planes. The four quoins are three -faced and 
 
Merosymmetry. 
 
 337 
 
 similar, but present no symmetry. Figs. 254 (a) and 254 (c) 
 represent respectively the forms 0(213} and a {213}, which 
 together compose the scalene octahedron {213} of Fig. 254 
 
 Fig. 254 (a). 
 
 Fig. 254 
 
 Fig. 254 (<). 
 
 Sphenoidal hemi-symmetry of an ortho-symmetrical crystal can 
 only be represented in a distinct form by faces of the rhombic 
 sphenoid. Any other form of the system than the octahedrid 
 scalenohedron can only exhibit sphenoidal characters in the mero- 
 symmetrical features it acquires by association with sphenoidal 
 forms. 
 
 II. The second division of the holo-systematic section of this 
 system is one in which, two of the systematic planes being planes 
 of symmetry, the third fails of being so ; conditions resulting in 
 three kinds of hemimorphous forms. 
 
 i. The trito-hemioctahedron p{hkl\ or p{hkl}. Where the 
 proto- and deutero-systematic planes are symmetral, the form is 
 hemimorphous in respect to the plane C. and the form is a trito- 
 hemioctahedron p{hkl], the correlative form being p{hkl], the 
 former comprising the faces 
 
 hkl hkl hkl Yki, 
 the latter the faces 
 
 hkl hkl hkl hkl. 
 
 ii. The proto-hemioctahedron {hkl} or {h kl}. The form hemi- 
 morphous to the plane S is symmetrical to the planes 2 and C. 
 Its faces are 
 
 for the form {hkl}, 
 
 hkl hkl hkl hkl, 
 for the form {hkl}, 
 
 hkl hkl hkl hkl. 
 
338 Ortho-rhombic system. 
 
 iii. The deutero-hemioctahedron x {hkl} or x {hkl} is symme- 
 trical to the planes S and C. The faces of the form 
 
 x\hkl\ are hkl Jikl hkl ~hkl, 
 and of x{hkl} are hkl hkl hll hkl. 
 
 Since the forms are hemimorphous on the plane 2, in each of 
 them the sign of the k index is the same for every face. 
 
 In a hemimorphous developement of the scalene octahedron 
 {213} of Fig. 254 (3) the upper and lower halves would be 
 p {213} and ^{213}, the right and left halves {213} and 
 {"213}, and the front and back halves ^{213} and x {213} 
 respectively. 
 
 It is evident that all the forms may be hemimorphous to a 
 systematic plane with the exception of such as have their faces 
 perpendicular to the systematic plane in respect to which the 
 crystal is hemisymmetrical. Thus in the first case, where the 
 horizontal systematic plane C has its symmetral character in abey- 
 ance, besides the forms p{hkl} and p{hkl}, there may be a 
 hemi-protodome p {okl} or p {o//}, a hemi-deuterodome p{hol] 
 and /){^o/}, and a hemi-tritopinakoid p(ooi) or p(ooY); but the 
 proto- and deutero-pinakoids and the rhombic prism will only 
 betray a hemimorphous character in their association with the 
 other hemimorphous forms: and similarly for the forms f {o//}, 
 f{oio}, and f{ooi}, x{hol\, ^rjioo}, and x {001} in the 
 remaining groups of hemimorphous forms. 
 
 306. The question as to the existence of a hemi- systematic 
 section of the merohedral forms belonging to this system pre- 
 sents a certain ambiguity; for the hemi-symmetrical and tetarto- 
 symmetrical forms which might seem to accord with the law of 
 mero-symmetry will be found to present the symmetrical charac- 
 ters of the holo-symmetrical and hemi-symmetrical forms respect- 
 ively of the Clino-rhombic system. So that it is very ques- 
 tionable whether such forms could, in accordance with mero-sym- 
 metrical principles, hold a place in the system under consideration, 
 as not being impressed with any especial characteristics of that 
 system. Thus, if we consider the hemi-systematic diplohedral 
 form, it evidently presents four faces belonging to two normals. 
 
Mer asymmetry. 339 
 
 The four planes thus concurring will belong to the zone which 
 contains the two normals, and in the plane of which lies the zone- 
 axis to which these are diagonally symmetrical; the systematic 
 plane which is perpendicular to this axis will also belong to the 
 zone, and its trace on the zone-circle will, with the axis normal to 
 it, ortho-symmetrically divide the zone. 
 
 The form would be a rhombic prism, and, if a form of the 
 Ortho-rhombic system, would be represented by one or other of 
 the symbols it{hkl} or v{hkl} y ^{hkl} or ty{hkl}, ($>{hkl} 
 or < \hkl\. 
 
 Since no symmetral axes in the Ortho-rhombic system are 
 similar, each axis, as also each systematic plane, is independent of 
 the others in respect to the mero-symmetry it may thus present. 
 But this condition of a plane of symmetry having an axis of 
 diagonal symmetry for its normal is precisely that which charac- 
 terises the holo-symmetrical forms of the Clino-rhombic system. 
 A hemi-systematic ortho-rhombic form would thus differ from a 
 holo-systematic clino-rhombic form, not in the type of its symmetry, 
 but only in the ortho-symmetrical character of the one systematic 
 zone-circle to which it was symmetrical. 
 
 All the crystals to which this clino-rhombic habit in their hemi- 
 systematic forms had been attributed have been shown, by reference 
 to their optical and other physical characters, to belong either to 
 the Clino-rhombic or to the Anorthic system. 
 
 Ortho-rhombic System. C. Combinations of Forms. 
 
 307. (a) Holo-symmetrical Forms. Crystals belonging to the 
 Ortho-rhombic system present great diversity of aspect in their 
 combinations ; sometimes the predominance of an octahedrid or 
 pinakoid form imparts to the crystal a pyramidal or a tabular 
 character; or again, a more or less prismatid aspect results from 
 the presence of a conspicuous zone of prism- or dome-forms; 
 while frequently the different kinds of form are so balanced that 
 the crystal cannot be classed under either of these three types. 
 
 In Figs. 255-9 are illustrated various combinations of the forms 
 {in}, {no}, {on}, {101}, {100}, {o i o}, {oo i }, corresponding 
 to the elements given in Art. 300. In Fig. 255 the parametral 
 
 z 2 
 
340 
 
 Ortho-rhombic system. 
 
 octahedron {in} is predominant and the combination is pyra- 
 midal in aspect : four of the edges of the octahedron are truncated 
 by the faces of the proto-dome {on} and two of its quoins by the 
 faces of the proto-pinakoid {100}; the parallelism of the edges 
 formed by the faces (100), (in), (on), (In), (loo) points to the 
 tautozonality of these faces. The combination shown in Fig. 256 
 
 110 
 
 Fig- 255. 
 
 Fig. 256. 
 
 Fig. 257. 
 
 is tabular, owing to the size of the faces of the basal pinakoid 
 {001} : and that of Fig. 257 is prismatid through the predomi- 
 nance of the prism {no}, which is closed by smaller faces of the 
 basal pinakoid {001} and the deutero-dome {101}. In the com- 
 binations shown in Figs. 258, 259 more than one form is largely 
 developed: in Fig. 258 two edges of the dome {101} are trun- 
 
 m... 
 
 no 010 
 
 no ! 
 
 i 
 
 | 
 
 >H2I 
 
 
 
 \Jtt</ 
 
 Fig. 2 
 
 59- 
 
 Fig. 258. 
 
 cated by two large faces of the proto-pinakoid {100} perpendicular 
 to which are the faces of the deutero-pinakoid {oio}, while the 
 edges of intersection of the forms {010} {101} are replaced by 
 the faces of the parametral octahedron { 1 1 1 } ; on the other hand, 
 the crystal figured in Fig. 259 is prismatid in developement ; the 
 
Combinations. 
 
 prism {no} has two of its edges truncated by the faces of the 
 deutero-pinakoid {oio}, and is closed by those of the basal pina- 
 koid {001} ; the edges of intersection of the latter form with 
 {010} {no} respectively being replaced by faces of the forms 
 {011} {in}. 
 
 308. Fig. 260 represents a frequent combination exhibited by 
 crystals of native sulphur. The parametral octahedron {in} is 
 predominant, and imparts a distinctly octahedrid aspect to the 
 crystal. Four of its edges lying in the 2-symmetral plane are 
 truncated by faces of the dome {101}, and its summit-quoins by 
 the basal pinakoid {001} ; the edges formed by its faces with this 
 pinakoid are replaced by the faces of the octahedron {113}. Fig. 
 261 represents a crystal of witherite, and illustrates the pseudo- 
 
 Fig. 260. 
 
 Fig. 261. 
 
 hexagonal aspect of many crystals in this system: the angles 
 loo-no and iio-Tio are 59 15' and 6i3o' respectively, so that 
 the combination of the pinakoid {100} with the parametral prism 
 {no} approximates very nearly in angles and in aspect to the 
 regular hexagonal prism with normal-angles of 60. 
 
 Fig. 262 is that of a crystal of topaz (in the British Museum). 
 The crystals of this mineral are frequently embedded at one ex- 
 tremity in the rock, the faces on the other extremity being often 
 numerous, and those belonging to different forms very distinctly 
 recognisable by their physical features : the aspect is usually pris- 
 matid. In the figure the prisms {no}, {210} and {320} are 
 associated with the proto-domes {on} and {013}, the deutero- 
 domes {101} and {201}, the octahedrid forms {113}, {112}, and 
 (almost microscopically developed) {m}. 
 
342 
 
 Ortho-rhombic system. 
 
 309. Had the prism {210}, the angles of the adjacent faces 
 of which are 86 52' and 93 8', been adopted as the parametral 
 prism, topaz would have been a more evident illustration of the 
 approximation to tetragonal type seen in a large number of the 
 crystals that belong to the Ortho-rhombic system. Thus anda- 
 lusite, a mineral closely allied to topaz in chemical type, has 
 
 its parametral ratio | = 0-9856. Calco-uranite, a complex hy- 
 drated phosphate of uranium and calcium, has the ratio 
 - = 0-9876, while the analogous minerals in which copper 
 
 takes the place of calcium or arsenic takes the place of phosphorus 
 in the former compounds are actually tetragonal. In the enstatite 
 
 20 110 
 
 no s 
 
 \A 
 W 210 
 
 Fig. 262. 
 
 Fig. 263. 
 
 group of silicates the ratio = 1-03, and a similar approximation 
 
 to equality in parameters is seen both in the mono-symmetric or 
 augite group and the anorthic or pajsbergite group of minerals of 
 similar chemical type with enstatite. The natrolite group of silicates 
 offers analogous illustrations of this coincidence. The libethenite 
 group of phosphates and arsenates also presents parametral ratios 
 in the prism that differ little from unity; and in other crystals this 
 character is met with in one or other of the domes. In the group 
 of crystallised substances isomorphous with hepta-hydrated mag- 
 nesium sulphate the ratio =- varies from 0-9804 to 0-9901. 
 
 310. Fig. 263 illustrates the combination of a dome {104} with 
 a parametral prism {no} that is not rare in crystals of mispickel. 
 
Combinations. 
 
 343 
 
 In Fig. 264 a crystal of gothite of prismatid developement due to 
 the presence of the forms {100}, {no} and {120} is terminated 
 by the forms {in} and {101}. Fig. 265 shows a crystal of 
 aragonite on which the predominant forms are the proto-pinakoid 
 {100}, the parametral prism {no}, and the deutero-dome {101}, 
 the remaining forms, {ooi}, {in} and {211}, though subor- 
 dinate, being still well-developed. 
 
 2JJ. 
 
 Fig. 264. 
 
 And in Figs. 266 and 267 representations of barytes and bour- 
 nonite respectively illustrate tabular crystals of this system facetted 
 by a large number of forms. 
 
 Fig. 266. 
 
 Fig. 267. 
 
 311. (8) Mero-symmetrical Forms. The hemi-symmetrical de- 
 velopements of crystals in this system are numerous : illustrations 
 of crystals exhibiting combinations of hemi-symmetrical forms are 
 shown in Figs. 268 (/), 268 (r), 269, and 270. 
 
 Figs. 268 (/) and (r) represent crystals of sodium ammonium 
 laevo- and dextro-tartrate respectively, in which the forms {in} 
 and {211} are sphenoidally developed and are combined with the 
 
344 
 
 Ortho-rhombic system. 
 
 three pinakoids, the parametral domes and prism, as also the 
 prism {210}. If the mixed crystals deposited from a solution of 
 the corresponding racemate be separated into two sets according 
 as they present the developement shown in Fig. 268 (/) or that of 
 Fig. 268 (r), their solutions will rotate a plane polarised ray to the 
 left and right respectively, although the crystals are themselves 
 
 Fig. 268 (/). 
 
 Fig. 268 (r). 
 
 devoid of this property. A series of corresponding tartrates also 
 exhibits the same characters. 
 
 312. In Fig. 269 sphenoidal hemi-symmetry is seen in a crystal 
 of hepta-hydrated magnesium sulphate (MgSO 4 , 7H 2 O) in which 
 the sphenoid a{m} is combined with the prism a {no}, the 
 latter having all the faces of the holo-symmetrical prism {no}. 
 
 110 
 
 Fig. 269. 
 
 Fig. 270. 
 
 The correlative sphenoid a { 1 1 1 } is frequently associated with 
 a { 1 1 1 } on the crystals of this substance. 
 
 313. Fig. 270 of struvite and Fig. 271 of hemimorphite present 
 characteristic examples of hemimorphism in ortho-rhombic crystals. 
 The latter of these minerals is a conspicuous illustration of the 
 
Combinations. 
 
 145 
 
 significant concurrence of hemimorphism with pyro-electric proper- 
 ties already alluded to in Article 278 in the case of tourmaline. 
 
 314. The only variety of tetarto-symmetry of which an ortho- 
 rhombic crystal is susceptible is illustrated in crystals of milk- 
 sugar : Fig. 272 represents a crystal of this substance presenting 
 the form pa {nT} associated with other forms holo-symmetrically 
 developed. These are sphenoidal in their type, but are at the same 
 
 Fig. 271. 
 
 Fig. 272. 
 
 time hemimorphous in their developement. The solution of milk- 
 sugar rotates a polarised ray to the right, but the crystals them- 
 selves possess no gyratory action : in this respect they present 
 some analogy with the series of tartrates already mentioned. 
 
 Ortho-rhombic System. D. Twinned Forms. 
 
 315. The three ortho-symmetral axes in this system are by their 
 nature precluded from being twin-axes, except in the case where 
 the axis has been deprived of all symmetral character by mero- 
 symmetrical suppression. This is not the case with a sphenoidal 
 form, but is so where the crystal is hemimorphous in respect to 
 either of the systematic planes passing through the axis. Any 
 other face-normal of a holo-symmetrical crystal may, however, be 
 a twin-axis : hence either a dome- or prism-face, or again, a face 
 of an octahedrid form may be a twin-face : a pinakoid-face can 
 only be a twin-face when the crystal is hemimorphous in respect 
 to one of the remaining pinakoid-faces. 
 
146 
 
 Ortho-rhombic system. 
 
 I. Twin-plane a face of a scalene octahedron, 
 
 316. This mode of twinning is rare, and it is singular that, in 
 the two or three crystallised substances known to present it, the 
 
 \(ffoj 
 
 Fig. 273. 
 
 Fig. 274. 
 
 twin-plane is parallel to the face of a form not actually met with so 
 far on the crystals. Fig. 273 represents an interpenetrant twin- 
 crystal of staurolite twinned round the normal of a face with the 
 symbol (322) never observed on this mineral. The normal-angle 
 between a face of this form {322} and the adjacent pinakoid-face 
 of the form {001} is 60 37', so that the pinakoid-faces with 
 identical symbol ooi in the two individuals are inclined in the twin 
 at a normal-angle of 121 14'; the zone-plane of these faces is 
 inclined on the zone-plane [oio] at an angle of 5 4 3 7'. The 
 twin-axis is inclined on the normal of (100) or the zone-axis 
 [001-010] at an angle of 59 42', whence it follows that the normal- 
 angle of two pinakoid-faces with identical symbol TOO in the two 
 individuals will be 119 2 4'. 
 
 Copper glance, and stromeyerite isomorphous with that mineral, 
 also present twins of which the twin-plane is a face of an as yet 
 unobserved octahedrid form {112}. 
 
 II (a). Twin-plane a face of a dome. 
 
 317. Examples of this kind of twinning are afforded by stauro- 
 lite, the twin-plane being a face of the unobserved form {302} 
 (Fig. 274); by anhydrite, with a twin-face belonging to the form 
 {102}; and by copper glance, twinned on a face of the form 
 
Twinned forms. 
 
 347 
 
 {403}. In the staurolite twin, with twin-plane (302) (Fig. 274), 
 the angle between the Z-axes is near to 90 (91 36'), the normal- 
 angle (302-001) being 45 48'. 
 
 Fig. 275. 
 
 Fig. 276. 
 
 Fig. 275 represents a twin of manganite. The twin-plane is 
 (101): the normal-angle of the faces (001-101) being 2 8 3 5', the 
 Z-axes of the two crystals are inclined to each other at an angle 
 of 57 i o' (near to 60) and lie in the zone-plane [oio]. 
 
 Grouped twins of alexandrite, with faces of the form {301} for 
 twin-planes, have been alluded to in Article 165 as illustrating 
 a pseudo-hexagonal type of combination resulting from repeated 
 twinning on a dome-face. 
 
 Mispickel is sometimes twinned on a face of the parametral 
 proto-dome {on}, so that the Z-axes of the crystals lie in the 
 zone-plane [100] and are inclined to each other at an angle of 
 1 20 48' (near to 120), the angle between the faces oli-ooi being 
 60 24'. In Fig. 276 the twin-plane is (oil): a simple crystal is 
 shown as Fig. 263. 
 
 Twins of humite occur, twinned in two ways ; the one on faces 
 of a form {307}, the other on those of a form {107}, neither of 
 which forms has been actually observed on crystals of the sub- 
 stance : these twin-planes are inclined to the basal pinakoid at 
 normal-angles of nearly 60 and 30 respectively. 
 
 II (Z>). Twin-plane a face of a prism. 
 
 318. Aragonite offers a conspicuous example of the great variety 
 capable of being presented by the twinned crystals of this system. 
 
Ortho-rhombic system. 
 
 The twins of aragonite are frequently pseudo-hexagonal in their 
 character as a result of the prism-angle being not very remote from 
 60, and the twinning being repeated on the faces of the form 
 {no}. Usually the prism is associated with the pinakoid {100} 
 and the dome {101}, and forms belonging to the zone [100-101] 
 are of frequent concurrence with these. The octahedron {in} is 
 also a frequently prevalent form (Fig. 265). The normal-angle of 
 the faces iio-Tio of the prism {no} is 63 44', and the normal- 
 angle 100-110 is therefore 58 8'. 
 
 If now the crystal is twinned by the half-revolution of a parallel 
 individual round the normal of the face ilo, and this plane be 
 likewise the plane of junction of the two crystals, the re-entrant 
 normal-angle between the face 100 of the first and the face (Too) 
 of the second individual is (121 52' 58 8' or) 63 4 4', and the 
 
 ito 
 
 JJO 
 
 110 100 
 
 (wo 
 
 'too} 
 
 [770] 
 
 Fig. 277. 
 
 Fig. 278. 
 
 Fig. 279. 
 
 salient angle between the face no of the first and the face (no) of 
 the second individual is (116 i6 / 63 44' or) 5 2 32'. 
 
 319. The repetition of the twinning on the successive prism- 
 faces of the different individuals produces complex crystal-structures, 
 that vary in their aspect and angular relations according to the 
 faces on which the twinning occurs. 
 
 In Fig. 278 there are three individuals; while the front crystal 
 retains its position, the second crystal is twinned on the normal of 
 the face ilo of the first, and the third on the normal of the face 
 (ilo) of this intercalated second crystal. In Fig. 279 the three 
 individuals are successively twinned, the second as before on the 
 
Twinned forms. 349 
 
 i To face of the first and left-hand individual, but the third on the 
 (TTo) face of the second crystal. 
 
 320. At Herrengrund in Hungary and Girgenti in Sicily are 
 found very fine specimens of aragonite compounded of several 
 individuals, but which at first sight may be taken for simple hexa- 
 gonal prisms terminated by a basal pinakoid. It is seen, however, 
 that the striations on the basal pinakoid are in diverse directions, 
 and that obtuse re-entrant edges traverse some of the faces of the 
 pseudo-hexagonal prism in a direction parallel to its edges. The 
 structure of these crystals is illustrated by Figs. 280-2. 
 
 Fig. 280 is a projection of three individuals on the basal pina- 
 koid (ooi), each being a combination of the proto-pinakoid {100}, 
 the parametral prism {no}, and the basal pinakoid {ooi}, of 
 
 wo 
 
 - '' [ooi] p'zyJ! 110 
 
 . f " 
 
 Fig. 280. 
 
 which the latter is grooved with striations parallel to its inter- 
 sections with the faces of the proto-pinakoid {100}. The crystals 
 I and II are twinned respectively on the planes Yio and no of 
 the crystal A. The normal-angle iio-Tio being 63 44', the face 
 [TYo] of crystal I makes a small re-entrant angle of n 12' with 
 the face (iTo) of crystal II : the basal pinakoids ooi, [ooi], (ool) 
 are coincident in direction: the faces -ioo-(Too) and the faces 
 Too [100] form re-entrant normal-angles of 63 44', and the faces 
 [Too] (100) a re-entrant normal-angle of 52 32': hence the 
 striations of I and II make an angle of 116 16' with those of A, 
 and an angle of 127 2 8' with each other. 
 
 321. If the crystals of Fig. 280 are interpenetrant, the resulting 
 
350 
 
 Ortho-rhombic system. 
 
 compound crystal may present the aspect illustrated in Fig. 281. 
 The faces no, [no] and their parallels no, [no], of crystals A 
 aiad I are coincident in direction : similarly for the faces YYo, (TTo) 
 and their parallels no, (no), of crystals A and II: while the 
 
 faces (no), [no], and their parallels (no), [no], of crystals I 
 and II make re-entrant normal-angles of 11 12'. 
 
 322. If the crystals grow until the faces of the proto-pinakoid 
 disappear, the structure presents the aspect of a pseudo-hexagonal 
 prism, of which each edge has a normal-angle of 63 44', and of 
 
 Fig. 282. 
 
 Fig. 283. 
 
 which two opposite faces are traversed by an obtuse re-entrant edge 
 of 11 12'. In Fig. 282 are shown the striations to be observed 
 on the basal pinakoid of a compound crystal from Girgenti, now 
 in the British Museum, very similar in structure to the compound 
 
Twinned forms. 
 
 crystal of Fig. 281 : corresponding parts of the two figures are 
 distinguished by the same letters and numbers. In the building 
 up of this complex structure two additional individuals III and IV 
 have a part : the individual III is a twin of the individual I about 
 the plane [no], so that the faces [no], {no}, as also their 
 parallels [Ho], {I^o}, are co-planar; the minute individual IV 
 is similarlv a twin of the individual II about the plane (Ho). 
 
 Fig. 284. 
 
 323. Figs. 283 and 284 represent twinned crystals of cerussite, 
 the former being a simple twin on the face iTo, and the latter a 
 triple combination with faces of the form {no} for twin-planes. 
 
 Copper glance and stephanite again afford examples quite 
 analogous to the above in the character of the twins they form, 
 the twin-planes belonging in each case to the form {no}, and the 
 pseudo-hexagonal character being due to the approximation of the 
 prism-angles to 60 : in the case of copper glance it is so near that 
 value as 59 48'. 
 
 Crystals of cerussite twinned about a plane of the prism {310}, 
 the faces of which make an angle of 92 45' (nearly 90) with 
 those of the parametral prism {no}, have been observed. 
 
 III. Twin-plane a face of a pinakoid. 
 
 324. It has already been observed that a systematic axis can 
 only become a twin-axis in the event of its symmetral attributes 
 being in abeyance. This can only occur when the crystal is 
 hemimorphous with respect to the systematic plane in which the 
 
352 Mono-symmetric system. 
 
 zone-axis in question lies. Fig. 285 represents a crystal of struvite 
 twinned about the pinakoid (100), and having for plane of junction 
 the perpendicular face ooi with respect to which the crystal is 
 hemimorphous : crystals of hemimorphite and seignette salt (sodium 
 potassium tartrate with four equivalents of water) also afford 
 examples of this variety of twin. 
 
 SECTION V. The Mono-symmetric or Clino-rhombic 
 System. 
 
 A. Holo-symmetrical Forms. 
 
 325. The Mono-symmetric system is characterised by symmetry 
 to a single plane. The zone of which this is the zone-plane is 
 accordingly (see Art 91, p. 107) not symmetrical in respect to any 
 of its own planes, so that no faces in the zone can be permanently 
 perpendicular to each other or inclined successively at any crystallo- 
 metric angle. 
 
 Since the sphere of projection is divided into hemispheres by 
 a single symmetral plane, there is no systematic triangle. The 
 diplohedral form is however symmetrical to the zone-axis of the 
 systematic plane as an axis of diagonal symmetry, the general 
 form {hkl} presenting four faces symmetrical to the systematic 
 plane and to a plane perpendicular to it. The systematic plane 
 and two origin-planes perpendicular to it are taken as axial planes : 
 the lines in which they intersect each other are the crystallographic 
 axes, the one which is normal to the systematic plane being taken 
 as the axis F; the axes X and Zlie in the systematic plane and are 
 perpendicular to the axis Y. The obtuse angle 77 formed by these 
 two axes will be taken for the positive angle XOZ, so that OC, OA 
 the normals of the planes FOX, FOZ are contained within it. 
 
 The poles 100, Too and ooi, ooT of the axial planes will there- 
 fore lie on the arcs XZ and XZ the arc r[ between the poles 
 100 ooi or Too ooT being the supplement of the arc r/. In 
 stereographic projections the plane of symmetry is generally taken 
 as the plane of the drawing : the axis of Z being placed in a 
 
Holosymmetry. 
 
 .53 
 
 vertical position OA the normal to it in the systematic plane is 
 horizontal. The elements of a crystal in this system are 
 
 f =C= 9> "n > 9; a\b:c; 
 
 where the relative magnitudes of the parameters are not indicated 
 by the alphabetical order of the letters. 
 
 326. The varieties in the forms of the Mono-symmetric system 
 are: 
 
 I. Such as have their poles neither on the axis nor on the 
 circumference of the zone-circle [oio] ; 
 
 viz. (a) the general form, either a positive (prismatid form or) 
 
 prismatid {hkl}, or a negative prismatid \hkl\ ; 
 (b) particular cases of this form in which 
 
 i. the poles lie on a zone-circle passing through (100), 
 viz. the ortho-prism {hko}, including the para- 
 metral prism { 1 10} ; 
 
 ii. the poles lie on a zone-circle passing through (ooi), 
 viz. the ortho-dome {o//}, including the para- 
 metral dome {on}. 
 A a 
 
354 Mono-symmetric system. 
 
 II. Such as have their poles on the zone-circle [oio], 
 
 the positive and negative hemi-domes {hoi} and {-o/}, 
 
 including the parametral hemi-domes {101} and {Toi} ; 
 the ortho-pinakoid {100} ; 
 and the basal pinakoid {ooi}. 
 
 III. The systematic pinakoid { o i o } . 
 
 In this nomenclature the term dome is employed not in contra- 
 distinction to the term prism or prismatid^ but, like the latter term, 
 conventionally and merely to distinguish these forms from one 
 another: the dome, by analogy with the Orthorhombic system, 
 being a form with its poles on the zone-circles [oio ooi] and 
 [100 ooi]. 
 
 Fig. 286 represents in stereographic projection the positions of 
 the poles of these kinds of forms for the case where 
 
 r) = 106 i' and a : b : c 0-541 : i : 0-913, 
 
 as in diopside. 
 
 327. I. That the general mono-symmetric form {hkl} will be 
 a rhombic prism is evident from its having four poles lying on 
 an orthosymmetrical zone-circle passing through the pole of the 
 systematic plane (oio). If one of its poles lie in the octant formed 
 by the great circles passing through the poles 100, oio, ooi, the 
 form is termed a positive prismatid and its four faces are 
 
 hkl hkl hkl h~kl\ 
 
 those four of the octants formed by the great circles in question 
 which contain the poles of the positive form being similar. And 
 the remaining four octants will similarly contain the poles of the 
 negative prismatids, viz. 
 
 hkl III hkl hkl. 
 
 The symbol of the zone containing the four faces of the form 
 \hkl\ being [hkl, oio] or [7o^], the first and last index in the 
 symbol of each of the faces must have the same sign : and 
 similarly a face of a form {h k 1} must have different signs in its 
 first and third indices. Figs. 287 (a) and (b) represent respect- 
 ively for the above elements the positive prismatid {632} and 
 
Holosymmetry. 
 
 355 
 
 the negative prismatid {632} closed by the faces of the form 
 {100}. 
 
 Where the indices have the same magnitudes in each form, a 
 positive and a negative prismatid combine to produce a quasi- 
 
 63'J 
 
 C32 
 
 Fig. 287 (a). 
 
 Fig. 287 (b\ 
 
 Fig. 287 (c\ 
 
 octahedral form, as in Fig. 287 (). The faces however of the 
 positive and negative component prismatids are not similar. They 
 are scalene triangles of two kinds corresponding to the different 
 forms ; and the edge in which two adjacent faces of these forms 
 meet is incapable of undergoing truncation or bevilment. 
 
 V 
 
 iio 
 
 \ 
 
 Fig. 288. 
 
 Fig. 289 
 
 328. Among the varieties of prismatids, of which the poles 
 always lie in a zone perpendicular to the zone-circle of symmetry 
 [oio], two are especially noticeable. 
 
 One of these varieties includes the vertical or ortho-prism \hko}, 
 usually distinguished as the prism-form, the faces of which lie in 
 
 A a 2 
 
356 Mono-symmetric system. 
 
 the zone [100, oio]; in Fig. 288 is illustrated the ortho-prism 
 {210} closed by the form {ooi}. The parametral prism {no} 
 belongs to this type and is generally one of the most frequent and 
 important forms on crystals belonging to this system (Fig. 289); 
 in fact, the zone selected as that of the ortho-prism is usually 
 chosen on account of the pre-eminence of these faces. 
 
 The other especially important variety consists in the suite 
 of dome-forms with the general symbol {o//} (illustrated in Fig. 
 290 by the form {032} closed by the faces of the form {100}) 
 lying in the zone [oio, ooi] ; one of them is the parametral dome 
 {on } (Fig. 291). This variety of dome is termed the ortho-dome, 
 
 Fig. 290. Fig. 291. 
 
 because, like the ortho-prisms, it has an ortho-symmetrical 
 character, the zone containing it being symmetrical to the axis 
 [oio] and also to the normal of the face (ooi), perpendicular to 
 that axis. 
 
 Another important zone perpendicular to the plane of symmetry 
 is that passing through the parametral plane in. Its symbol is 
 evidently [ioT], so that for all planes belonging to it the first and 
 last index are equal and have the same sign. This zone is tauto- 
 hedral with the zone [oio] in the two faces 101 and lol of the 
 form {101}. The quasi-octahedrid form constituted by the union 
 of the positive prismatid {m} and the negative prismatid {In} 
 would correspond to the parametral octahedron of a rectangular- 
 axed crystal: Figs. 292 (a) and (c) represent the parametral posi- 
 tive and negative prismatids {m} and {In} respectively, each 
 closed by the faces of the form {100} ; in Fig. 292 (3) is illustrated 
 the quasi-octahedrid form produced by their union. The two faces 
 of the form {loi} are in the zone [oio, In]. 
 
Holosymmetry. 
 
 357 
 
 329. II. With regard to the forms the poles of which lie in the 
 zone [oio], which is symmetrical only to its centre, their faces 
 will of course consist in each case of a single parallel pair ; the 
 various special forms comprised under the general symbol \hol} 
 being in fact a series of pinakoids. And of this series the forms 
 will be positive when their poles lie on the arcs joining the poles 
 
 Fig. 292 O). Fig. 292 (). Fig. 292 (c\ 
 
 ooi, 100 and ooT, Too; negative when they lie on the arcs joining 
 ooi, Too or ooT, 100. 
 
 And if the indices in a positive and negative form irrespective 
 of sign are the same, the two pinakoids, though not of necessity 
 concurrent, may be conceived as uniting to produce a prismatid 
 
 Fig. 293. 
 
 Fig. 294. 
 
 or quasi-dome-form : in Fig. 293 is shown a combination of the 
 pinakoid {301} with the pinakoid {301}, closed by the faces of 
 the form {010} ; Fig. 294 represents the positive and negative 
 parametral pinakoids {101} and {Toi}, closed by the faces of the 
 form {oio}. Such a double form has been termed a clino-dome ; 
 and it will be convenient to retain the terms positive and negative 
 
Mono-symmetric system. 
 
 hemidome, in order to distinguish the forms that actually compose 
 it, the symbols of which are {hoi} and {hoi} respectively, from 
 the particular cases where one of the indices h or / in the symbol 
 {hoi} is zero; to which the term pinakoid will be restricted. 
 The positive hemidome {hoi} comprises the faces hoi and hoi', 
 the parametral hemidome being {101}; the negative hemidome 
 {hoi} comprises the faces liol, hoi, and includes the parametral 
 negative hemidome {Yoi}. 
 
 The ortho-pinakoid {100} is the form the faces of which are 
 parallel to the axial-plane YZ. Its faces are 100, loo, and are 
 tautozonal with those of the prism (the ortho-prism) (Figs. 287, 
 290-2). 
 
 The basal or clino-pinakoid is the form {ooi}, the faces of 
 which, namely ooi and ooT, are parallel to the axial plane YX and 
 are tautozonal with the faces of the ortho-dome (Figs. 288, 289). 
 
 III. Finally, of the systematic pinakoid {010} the two faces oio 
 and olo are parallel to the plane of symmetry (Figs. 293, 294). 
 
 Mono-symmetric System. B. Mero-symmetrical Forms. 
 
 330. The general form in the Mono-symmetric system, inclusive 
 of the ortho-prism and the ortho-dome, has four poles carried by 
 two normals and is symmetrical to the systematic plane S. Its 
 holo-systematic mero-symmetrical forms will therefore be of two 
 kinds. 
 
 I. One of these forms presents two poles symmetrical on the 
 plane S; the symbol of the correlative forms being ^therefore 
 s{hkl} and s{hkl} for a positive prismatid, and s{hkl} and 
 s{h~kl} for a negative prismatid. Thus the two extant faces of 
 the form 
 
 s{hkl} are hkl and hkl, 
 those of the form 
 
 s{hkl} are Jikl and Ykl 
 
 Of the prism-form {hko}, the correlative forms are 
 s { h k o } with the faces h k o and h k o, 
 and s{hlto} with the faces Ji~ko and liko. 
 
Merosymmetry . 359 
 
 And while the ortho-dome is resolved into the correlative forms 
 
 s { o k 1} with the faces o k I, ok?, 
 and s { o ~k 1 } with the faces o k /, o k I, 
 
 a form \hol\, the faces of which fall into the zone [oio], will be 
 capable of exhibiting the haplohedral hemi-symmetry in question 
 by one only of its two faces being extant, viz. h o / or hoi. And so 
 of the negative hemi-dome either Tivl or //o/ and of the pinakoids 
 100 or Too, ooi or ool will be the only extant faces, while the 
 systematic pinakoid {010} will have both its faces extant or absent 
 but without symmetry of form. 
 
 331. II. The other hemi-symmetrical variety of the holo- 
 systematic forms is that in which two extant poles corresponding 
 to two normals of the system lie on the same side of the systematic 
 plane; so that the form is hemimorphous on that plane. In 
 accordance with the symbolical notation we have adopted, since 
 the form is symmetrical to the diagonal axis of symmetry of the 
 system, though to no plane of it, the symbol for such a form 
 would be 
 
 a {hkl} with the faces hkl and hkl, 
 
 or a{Jikl,\ with the faces Jikl and hkl, 
 
 in the case of a positive form \hkl\\ a {hkl} or a {Tikl\ being 
 the symbols of the correlative semiforms of the negative form 
 \hkl\. The pinakoid {010} may present this hemimorphism by 
 the absence of one of its planes : but the forms the poles of 
 which lie in the zone-circle [oio] can only exhibit hemi-symmetry 
 of this kind in the distribution of their physical features and in 
 their association with the mero-symmetrical forms of this type. 
 
 332. The possibility of a hemi-systematic form, in which a single 
 normal is represented by both the faces belonging to it, involves 
 a question similar to that already considered under the Ortho- 
 rhombic system, Art. 306. For such a semiform would be repre- 
 sented in the case for instance of a positive prismatid form by two 
 faces, viz. hkl and Jikl, and the correlative form would present 
 the faces hkl, hkl. But such a semiform would differ from a 
 holo-symmetrical form of the Anorthic system only in its faces 
 belonging to an ortho-symmetrical zone or one potentially ortho- 
 
360 
 
 Mono-symmetric system. 
 
 symmetrical. But there is no sufficiently characteristic distinction 
 between such a form and an anorthic holo-symmetrical form to 
 satisfy the principle of mero-symmetry in the Mono- symmetric 
 system. 
 
 Mono-symmetric System. C. Combinations of Forms. 
 
 333. (a). Holo-symmetrical forms. The mono-symmetric cha- 
 racter of a crystal in this, the more symmetrical of the two oblique 
 systems, is generally recognisable by reason of the different 
 developement and diversity in the features of the forms it carries. 
 The antistrophic symmetry of the two halves of the crystal as seen 
 divided by an ideal plane of symmetry when it is looked at in the 
 direction of that plane, and the diagonal disposition of its homolo- 
 gous faces from any other point of view, but especially as seen in 
 the direction of its one axis of symmetry, are characters in general 
 easy of recognition. Where however the axial angle rj approxi- 
 mates to a right angle, or where the more prominent prism-forms 
 
 Fig. 295. Fig. 296. 
 
 have normal-angles approximating to 45 or to 60, and in particular 
 where the former is united with one or other of the latter specialities, 
 the crystal may occasionally wear the aspect of an ortho-rhombic 
 or even of a hexagonal crystal. In such case goniometrical 
 measurements must be had recourse to. In the not infrequent 
 case of one or more distinct cleavages being apparent, these will, 
 by the presence or absence of symmetrical repetition, generally 
 guide the judgment. 
 
 334. The habit of a mono-symmetric crystal is very often 
 distinctly prismatid ; the faces of a prism {no} or {hko} being 
 
Combinations. 
 
 361 
 
 then predominant: often, however, the edges hko, hko and 
 hko, liko are truncated by the faces of the ortho-pinakoid {100}, 
 or the other edges may be truncated by the faces of the systematic 
 pinakoid {oio}. The crystal of datolite in Fig. 295 exhibits the 
 two prism-forms {no} and {210}, the edges of the latter form 
 being truncated by the ortho-pinakoid {100}, while in Fig. 296 
 a crystal of gypsum is represented with prisms {no} and {120} 
 in which the edges of the latter form are truncated by the syste- 
 matic pinakoid {oio}. 
 
 The crystals of augite (Fig. 297) and of hornblende (Fig. 299) 
 exhibit prism-forms with both pairs of edges truncated by the faces 
 of the two pinakoids {100} and {oio}. The comparison of the 
 
 Fig. 297. 
 
 Fig. 298. 
 
 crystal of augite in Fig. 297 with that of diopside in Fig. 298, 
 belonging to the same mineral group with it, will serve to show 
 the different aspect assumed by a crystal according as the prismatid 
 forms preponderate over the prism- and pinakoid-forms, or as one 
 or both of the latter are the most developed. 
 
 335. In hornblende (Fig. 299) the normal-angles of ooi In 
 and ooi YYi are 34 25', while that of In TTi is 31 32': 
 further, the prism-angle no ilo is 55 30', and the angle 
 no oio is 62 15'. It will be seen, then, that a crystal of this 
 mineral carrying the above forms might readily be mistaken for 
 a rhombohedral crystal, the more so as the angle 100 ooi is 
 75 2 ', while the angle Too loi is 73 58'. 
 
 In some of its crystals, augite, on the other hand, presents 
 features, in a nearly square prism {no} of which the angle is 
 
362 
 
 Mono-symmetric system. 
 
 87 5', and in the presence of a face 102 inclined to 100 at an 
 angle of 89 20', that bring it near in aspect to an ortho-rhombic 
 crystal. 
 
 336. The crystal of sphene represented as Fig. 300 exhibits 
 a very oblique aspect due to a considerable prolongation of the 
 
 Fig. 300. 
 
 crystal in the direction of a negative prismatid {123}, and in 
 epidote as shown in Fig. 301 the preponderant forms are the 
 ortho-pinakoid {100} and basal pinakoid {001} with ortho-domes 
 and negative prismatids, which latter impart to epidote, by the 
 character of their developement and distribution, a characteristic 
 
 Fig. 301. 
 
 Fig. 302. 
 
 oblique appearance when looked at in the direction of the normal 
 to the ooi face. 
 
 Fig. 302 represents a crystal of the rare mineral turnerite in the 
 collection of the British Museum: its locality is Cornwall; it 
 serves to give an illustration of a mono-symmetric crystal with very 
 symmetrical developement of a large number of forms. 
 
Combinations. 
 
 363 
 
 337. (b\ Mero-symmetrical forms. The known mero-sym- 
 metrical crystals belonging to this system occur exclusively among 
 the products of the laboratory or of the organic world. In tartaric 
 acid an illustration is afforded of the mode in which correlative 
 forms occur on the crystals of a substance which assumes two 
 characters corresponding to a species of enantiomorphism in its 
 crystals. Racemic acid has the same percentage composition and 
 most of the characters of tartaric acid, but in solution has no 
 action on polarised light : if from the two varieties of sodium 
 ammonium racemate (laevo- and dextro-tartrate) described in Art. 
 311 the corresponding calcium salts be obtained by the addition 
 of chloride of calcium to their respective solutions, and the calcium 
 salts be then decomposed by sulphuric acid, solutions are obtained 
 which have opposite rotatory actions on plane polarised light : one 
 
 Fig. 303 
 
 of these solutions on evaporation yields crystals of ordinary or 
 dextro-tartaric acid, while the other yields crystals of the same 
 chemical composition which are termed laevo-tartaric acid: the 
 latter crystals have the form shown in Fig. 303 (/), the former that 
 of Fig. 303 (r). The crystals themselves do not possess this 
 rotatory character. It will be seen by the symbols* of the faces 
 present or absent in either case that the form {on} is hemi- 
 symmetrically developed and presents faces with positive indices 
 for the .iF-axis in the one case, and corresponding faces with 
 negative indices in the other. The crystals are thus hemimorphous 
 in developement. 
 
 338. The organic substance fichtelite or scheererite, a mineral 
 which occurs in fossil pine-wood in the peat of the Fichtelgebirge, 
 has been described as illustrating the variety of hemi-symmetry 
 
364 Mono-symmetric system. 
 
 in which four normals of a form {hkl} are represented by two 
 poles symmetrical to the systematic plane. 
 
 Mono-symmetric System. D. Twinned Forms. 
 
 339. The twinned forms in this system are numerous, and 
 present considerable variation in the laws they follow even for the 
 same substance. The analogy which is presented, at once in 
 chemical and in crystallographic type, by many minerals is not 
 confined to those which crystallise in one and the same system. 
 For instance, the ortho-rhombic minerals bronzite and enstatite are 
 homotypic in their composition with the minerals of the augite 
 group in the Mono-symmetric system, and in their crystallographic 
 constants present many points of close resemblance to the latter. 
 So the felspars, which form an important and extensive group of 
 minerals that crystallographically are anorthic, have at least one 
 representative mineral belonging to the Mono-symmetric system. 
 And this felspar, orthoclase, has many crystallographic features in 
 common with the numerous minerals that are grouped under 
 the general name of felspars in the Anorthic system. Among 
 these characters none are more important for comparison than 
 the methods of twinning exhibited by orthoclase, as compared with 
 the twin-structures of the anorthic felspars. In the discussion of 
 clino-rhombic twins this analogy has to be borne in mind. 
 
 340. In this system the twinned forms can always be explained 
 by taking for the twin-plane a face of the crystal: but in order to 
 keep in sight the analogy just alluded to, it is well to point out 
 that while a mono-symmetric twin can be most simply explained 
 as being twinned on a face-normal, it may happen that, by analogy 
 with a corresponding anorthic twin that cannot be so simply 
 explained, the mono-symmetric twin may also be described as 
 twinned round a zone- axis perpendicular to the face-normal. 
 
 341. The twin-plane for crystals in this system may be a face of 
 a form belonging to the zone [oio], and therefore perpendicular to 
 the plane of symmetry, or it may be a face oblique to this plane. 
 In holo-symmetrical crystals it cannot be the systematic plane 
 itself, as the normal of this plane is an axis of diagonal symmetry. 
 The following are the various twin-laws known in this system : 
 
Twinned forms. 
 
 365 
 
 A. Twin-plane perpendicular to the plane of 
 symmetry. 
 
 342. In the known twins falling under this law the twin-plane 
 is either a pinakoid-face or a face of the parametral hemidome 
 {101}. 
 
 Fig. 304- 
 
 Fig. 35- 
 
 i. I win-plane a face of the ortho-pinakoid {100}. Cases of this 
 kind of twin are exemplified in augite (Figs. 304, 305) and horn- 
 blende (Fig. 306). In these twinned forms the combination-plane 
 is parallel to the twin-plane, and the crystals are in apposition. 
 
 (5io) (Ho) i 
 
 Fig. 306. 
 
 The figure of hornblende (Fig. 306) illustrates the singularly 
 symmetrical aspect which a crystal of that mineral assumes, in 
 certain points of view, when thus twinned. The re-entrant angles 
 
3 66 
 
 Mono-symmetric system. 
 
 on the augite crystals (Figs. 304, 305) immediately betray the 
 twinned nature of their structure. 
 
 Fig. 58 (a) (p. 175) represents a twin of orthoclase from Carlsbad, 
 the kind designated as the Carlsbad twin : in this case the com- 
 bination-plane of the juxtaposed and slightly interpenetrant indi- 
 viduals is perpendicular to the twin-face, and is in fact parallel to 
 the plane of symmetry. The Carlsbad twin may equally be 
 represented as the result of twinning round the zone-axis [ooi]. 
 It is in this way described by Des Cloizeaux as indicating an 
 analogy between the felspars belonging to the two oblique systems : 
 the indices of the planes when the zone-axis [ooi] is assumed to 
 be the twin-axis are shown in Fig. 307. 
 
 343. 2. Twin-plane a face of the basal pinakoid {ooi}. This 
 mode of twinning is also known in orthoclase, and is well shown 
 in crystals from Silesia : the crystals are in juxtaposition with the 
 
 Fig. 308. 
 
 Fig. 309. 
 
 twin-plane as face of junction: Fig. 308 represents a specimen 
 in the British Museum. An analogous twin has been recognised 
 in the anorthic felspar, albite, but its occurrence is extremely rare. 
 
 The twin crystal of epidote represented in Fig. 309 is also an 
 example of this law. 
 
 In Fig. 310 is illustrated a twin-crystal of sphene in which the 
 basal pinakoid (ooi) is both the twin-plane and the face of junction : 
 often the individuals are interpenetrant and have a second plane of 
 junction normal to the first, as in Fig. 311. 
 
 344. Harmotome, long considered to be an ortho-rhombic 
 mineral by reason of the supposed perpendicularity of its axial 
 angles, has been shown by Des Cloizeaux to be mono-symmetric, 
 
Twinned forms. 
 
 367 
 
 and the twinned forms, in which alone the mineral occurs, are 
 recognised as falling, in part at least, under this law. 
 
 (00.0 
 
 The crossed twin Fig. 314 represents a characteristic and striking 
 variety of this mineral from Andreasberg. It is best explained as 
 the result of a double twinning: the first twinning is round the 
 normal to the face (ooi), which is likewise a face of union, as in 
 Fig. 312: such simple twins are not met with in nature, the indi- 
 viduals being always so intergrown that they cross over to opposite 
 sides of the first face of union and have a second plane of union 
 
 Fig. 312. 
 
 Fig- 3I3- 
 
 Fig- 3H- 
 
 perpendicular both to the first and to the plane of symmetry (Fig. 
 313) : the second plane of union is very nearly parallel to a face of 
 the hemidome {101}, the angle ooi . 101 being almost exactly 
 90. Such a twin has therefore three perpendicular planes of sym- 
 metry, and may be mistaken for a simple ortho-rhombic crystal. 
 The second twinning is about a plane of the ortho-dome {on}, 
 
3 68 
 
 Mono-symmetric system. 
 
 inclined to the basal pinakoid at an angle of 45 18': hence the 
 angles between the adjacent twinned pinakoids, which form the 
 re-entrant edges of this complex growth, are 90 36' and 89 24' 
 alternately. 
 
 345. 3. Twin-plane a face of the parametral hemidome {101}. 
 
 Gypsum affords an example of a twin following this law (Fig. 
 315): the twin-plane is also the face of junction. This twin is 
 
 Fig. 315- 
 
 Fig. 316. 
 
 of frequent occurrence, but the faces of the extant forms of the 
 crystals from Montmartre, near Paris, are usually much rounded, 
 as in Fig. 316. 
 
 B. Twin-plane oblique to the plane of symmetry. 
 
 346. In its second law, where the twin-plane is a face of the 
 ortho-dome {on}, harmotome has already supplied an illustration 
 
 Fig. 318. 
 
Twinned forms. 
 
 369 
 
 of this mode of twin-growth. The (orthoclase) felspar twin known 
 as the Baveno twin (from one of the localities in which this form 
 of felspar occurs) has for its twin-plane a face of the ortho-dome 
 {021}, which is also the plane of union. An analogous twin 
 occurs, but is very rare, in the anorthic felspar albite. Figs. 317 
 and 319 (a) represent simple twins according to this law: the 
 systematic and basal pinakoids of the two individuals form a prism 
 of which two normal-angles, namely ooY .010 and (ooi) . (olo), 
 are exactly right angles, and the remaining two, namely oio . (oYo) 
 and ooY . (ooi), are 90 6' and 89 54' respectively. Twins of 
 orthoclase, especially of the variety called adularia, that follow the 
 Baveno law, are often interpenetrant, and exhibit repetitions of the 
 twinning resulting in very composite structures : thus, in Fig. 319 (3) 
 is illustrated one end of a complex growth in which there is 
 
 001 
 
 Fig. 319 
 
 OOJ 
 
 Fig. 3 J 9 (0- 
 
 twinning about the faces (021) and (021) of the central individual; 
 Fig. 319 (c) shows one end of a similar growth in which a fourth 
 individual takes a part, being twinned on a face of the form [021] 
 of the left-hand individual ; in Fig. 3 1 8 is shown the latter growth 
 obliquely projected, the form {20!} being omitted for the sake of 
 simplicity. 
 
 As in the case of harmotome, complicated growths according 
 to two distinct twin-laws are met with in orthoclase, faces of the 
 basal pinakoid {001} and of the ortho-dome {021} being simul- 
 taneously present as planes of twinning. 
 
 Rare pseudomorphous forms after crystals of orthoclase twinned 
 about a face of the ortho-dome {051} have been described. 
 
 347. Fig. 320 is the representation of a twin of wolfram of 
 which the plane of twinning as well as of union is the face (023). 
 Like harmotome, wolfram has been removed from the Ortho- 
 
 Bb 
 
370 
 
 Anorthic system. 
 
 rhombic system, under which it had been previously classed in 
 a hemi-symmetrical category really incompatible with the symmetry 
 of that system. 
 
 348. Twinning about a face of a prismatid is very rare. In 
 Fig. 321 is shown an interpenetrant twin of orthoclase, belonging 
 
 Fig. 320. Fig. 321. 
 
 to the British Museum, in which the face (Tn) is the twin-plane. 
 Rare pseudomorphs after orthoclase crystals twinned about a face 
 (454) have also been described. 
 
 SECTION VI. The Anorthic System. 
 A. Holo-symmeirical Forms. 
 
 349. A form in the Anorthic system is constituted by two parallel 
 faces, the only symmetry which characterises the system being 
 that to a centre. 
 
 No zone on such a crystal can permanently present ortho- 
 symmetry, nor can it have three tautozonal faces inclined con- 
 secutively at crystallometric angles. The three edges to which the 
 axes are parallel will of necessity be inclined to each other at 
 angles of which no two can be right angles, and the elements of 
 the crystal are unfettered by any condition. Where more than 
 one of the axial angles may approximate to a right angle, it is 
 conceivable that changes of temperature inducing changes in the 
 axial elements might momentarily bring two or all the axes into 
 a position' of perpendicularity ; but it will be seen in a future 
 
Holosymmetry. 371 
 
 chapter that this is a condition transient with the temperature, 
 and is not a sufficient condition for the system to which the crystal 
 belongs to assume a different type of symmetry. 
 
 In the general case then the axes are to be assumed as oblique 
 to each other and the parameters as unequal; and in order to 
 assign a conventional orientation to such an axial system it will be 
 necessary to determine the ways in which a uniform principle may 
 be applicable to it. 
 
 Adopting the alphabetical order of the letters a, 5, c as that 
 of the descending magnitudes of the parameters assigned to the 
 axes X, Y, Z, respectively, we might further determine that in all 
 cases the axial angle 77 or ZX should be an obtuse angle, as in the 
 Mono-symmetric system. Now it will be found that under these 
 restrictions it is always possible to make the positive octant such 
 that the three axes which contain it shall be inclined either all at 
 obtuse angles, or so that two of the axial angles are obtuse and 
 one, for which f may be taken, is acute. The axial system would 
 then be represented by the expression 
 
 f> 9 o, T ? > 9 o, C^9> 
 a > b > c. 
 
 An orientation of this kind is not however suitable in cases 
 where analogies have to be kept in view between crystals chemi- 
 cally homotypic but belonging severally to the Monosymmetric 
 and Anorthic systems. An orientation corresponding to that em- 
 ployed for the monosymmetric crystal is in such cases adopted 
 in the Anorthic system. 
 
 350. The forms of the system being limited to pairs of faces, and 
 related to each other only by the bond imposed on them by the 
 crystalloid zone-law, a nomenclature distinguishing the forms which 
 lie in particular zones, as for instance the zones containing two of 
 the axial planes, possesses even less significance than is the case in 
 the Mono-symmetric system. The sort of octahedrid figure pro- 
 duced by the union of all the four forms in the symbols of which 
 the several indices have the same numerical values is one which 
 has no crystallographic meaning ; for the forms 
 
 {hkl}, \hkl\, {hkl}, 
 
37 2 Anorthic system. 
 
 are independent each of the other, and do not in fact concur with 
 any regularity or frequency on anorthic crystals. 
 
 Hemiprismatic forms would include a positive hemiprism {hko} 
 and a negative hemiprism {hko}, and of such united forms the 
 parametral hemiprisms {no} and {Tio} are more frequent in 
 their concurrence. So two concurrent hemidomes, consisting 
 of a positive pro-hemidome {hol\ and a negative pro-hemidome 
 {hoi}, would build a dome-form with its edges parallel to the 
 axis Y; while a positive para-hemidome {ok I}, with a corre- 
 sponding negative form {o//|, would constitute a dome-form 
 with edges parallel to the J^-axis. 
 
 Apart from the grounds on which particular faces are selected 
 for pinakoid or for parametral forms in this system, such pairs 
 of forms might be expected to be more frequent in their con- 
 currence as quasi-dome forms than in the case of the quasi- 
 octahedrid forms, inasmuch as in the one case only two hemidome 
 or hemiprism forms with numerically identical indices must concur, 
 while in the latter case four tetarto-octahedrid forms numerically 
 identical in their indices must of necessity be extant together on 
 the crystal. The three pinakoid forms parallel to the axial planes 
 may be termed 
 
 the para-pinakoid { 100}, parallel to the axes J^and Z\ 
 the pro-pinakoid {oio}, parallel to the axes X and Z\ 
 and the basal pinakoid {ooi}, parallel to the axes X and Y\ 
 the prefixes para and /r0-are suggested by the positions of the 
 pinakoids relative to an observer looking at the crystal in the 
 direction of the _F"-axis; the faces of the para-pinakoid being then 
 at the sides and those of the pro-pinakoid in front of the observer. 
 
 351. In selecting the faces that are to determine the axial elements 
 in an anorthic crystal, and assigning to particular faces the cha- 
 racter of axial pinakoids or prismatid or dome-forms, of course in 
 the first place importance has to be attached to the faces which by 
 the comparison of several crystals are recognisable as habitually 
 prominent on them. Sometimes also crystals of this system as 
 is conspicuously the case with the felspars present a close re- 
 semblance in the distribution of their forms with crystals of other 
 systems, and more particularly with crystals of the Clino-rhombic 
 
373 
 
 system, with which they are at the same time allied in the type 
 of their chemical composition. In such cases the orientation of 
 the axes is determined in subordination to the principle necessi- 
 tated by the symmetry of the system to which belong the crystals 
 with which a comparison has to be instituted: in Fig. 322 is shown 
 a stereographic projection of the more prominent forms exhibited 
 
 Fig. 322. 
 
 by the crystals of anorthite illustrated in Figs. 325 and 326, the 
 axial system of which would be represented by the expression 
 
 f=8647', 7? ="5 55', C=88' 4 8', 
 a \b\c\\ 0-635 : i : '55- 
 
 Anorthic System. B. Mero-symmetrical Forms. 
 
 352. Hemi-symmetry. The only conceivable case of mero- 
 symmetry in the Anorthic system would be that of a haplohedral 
 form, or forms represented each by a single face. 
 
 Anorthic System. C. Combinations of Forms. 
 
 353. In the previous systems the symmetrical repetition of faces 
 leads naturally to the grouping of forms in zones. In the Anorthic 
 system there is no such necessary tendency. Nevertheless the 
 aspect of an anorthic crystal very usually differs little in this respect 
 
374 
 
 Anorthic system. 
 
 from that of a mono-symmetric or ortho-symmetric crystal. The 
 symbols of the forms in the Anorthic system are usually of a very 
 simple kind, the indices rarely occurring with a higher numerical 
 
 Fig- 3 2 3- 
 
 Fig. 324. 
 
 value than four. Indeed the number of minerals crystallising in 
 the system is very small, though of organic products formed in the 
 laboratory this number is considerable. The forms, however, that 
 these present are usually few in number, and simple in their 
 symbols. 
 
 Fig. 325- 
 
 Fig. 326. 
 
 In Fig. 327 is illustrated a simple crystal of the felspar albite 
 with the forms {ooi}, (oio} 5 {no}, {ilo}, {iol}, and in Fig. 323 
 another crystal of the same mineral with forms additional to the 
 above : Fig. 324 shows the form of an ideal simple crystal of albite 
 elongated in the direction of the P'-axis, a variety to which the term 
 pericline is applied. Figs. 325 and 326 represent crystals of the 
 
Twinned forms. 375 
 
 felspar anorthite from Vesuvius (after vom Rath) remarkable from 
 the completeness of their developement. 
 
 Anorthic System. D. Twinned Forms. 
 
 354. In the Anorthic system the twin-growths are of two kinds ; 
 in the one the twin-axis is the normal of a face, in the other it is 
 parallel to a zone-axis ; and as in this system, from the principles of 
 symmetry, no face-normal can be a zone-axis, these two modes of 
 twin-growth are distinct. Another kind of twin-growth has indeed 
 been asserted to exist, in which the twin-axis is a line perpendicular 
 to an edge and lies in a face belonging to the zone to the axis of 
 which the edge is parallel ; and Neumann, Kayser and Rose gave 
 their great authority in favour of such a mode of twin-formation. 
 A line of this kind is not however a crystallographic line in the 
 Anorthic system, as neither the line itself nor the plane normal to 
 it would have a rational symbol : and the line has therefore no 
 significance other than what might be due to its supposed function 
 as a twin-axis. The twins of albite and of the pericline variety of 
 this mineral, for the explanation of which this mode of twinning 
 was virtually called into existence and maintained, can be ac- 
 counted for in a simpler manner, and will be found to be ex- 
 plained in a subsequent article by the mode of twin-growth round 
 an edge. In other cases in which this mode of twinning has been 
 called in, it will also be found that the two more simple kinds of 
 twinning are competent to explain the complicated growths to 
 which the former has been applied; and that where this more 
 complex principle would seem more simply to explain them, accu- 
 rate measurements, on which alone it could be established, are 
 wanting or were not possible by reason of the impracticability of 
 obtaining them. It is in fact only for complex structures con- 
 sisting of several individuals, explicable by one or both of the two 
 simpler methods of twin-growth, that this third mode of explanation 
 has been invoked, and in most cases so invoked to account for the 
 twin-relations of members of the group that are not in contact and 
 are only mutually related through the intervention, so to speak, of 
 one or more other individuals of the group. 
 
 355. The non-crystallographic line, round which the pericline- 
 
376 Anorthic system. 
 
 twin of two individuals was supposed by Kayser to be formed, is 
 perpendicular to the X-axis and is inclined on the JT-axis, the zone- 
 axis [oio], at so very small an angle that it is practically im- 
 possible, in the case of a mineral presenting the striated faces 
 common in pericline, to determine, by direct measurement of the 
 angles between the planes of the twin-growth, which of these 
 adjacent lines is the true twin-axis. The discussion of the argu- 
 ment in regard to these twins will be given later, when the 
 twinning round a zone-axis is treated of in Art. 362. 
 
 In the case of the twinned groups of albite, which were the first 
 that suggested to Neumann recourse to this irrational axis, the 
 crystals are twinned in pairs, each pair on a face-normal (oio), 
 and adjacent crystals of different pairs round a zone-axis [ooi]. 
 The explanation of the relative position of the first and third indi- 
 viduals offered by Neumann, purely as a geometrical explanation, 
 in which a line normal to a zone-axis and lying in a face is con- 
 ceived to be a twin-axis, introduces in fact a greater difficulty than 
 that of a double twin-law by giving to a line with an irrational 
 symbol a crystallographic function. 
 
 356. Besides the felspar-twins alluded to in the last article, the 
 only other cases adduced to support an irrational twin-axis are the 
 following : 
 
 1. A single specimen of labradorite 1 from Hafnefjord, con- 
 sisting only of a termination of a twin-crystal formed by a thin 
 lamina of only \\ millimetre in length, and yielding only ap- 
 proximate measurements made with a source of light very close to 
 the goniometer. Vom Rath described this crystal in 1871, but 
 did not again refer to it when in i879 2 he pointed out that the 
 evidence for the variety of twinning under discussion was not 
 sufficient to establish its recognition. 
 
 2. A complicated twin-growth of anorthite 3 , which did not admit 
 of accurate measurements and could be satisfactorily explained by 
 the ordinary rational twin-axes, as was pointed out by vom Rath 
 who first described it. 
 
 - Annal. vol. 144, 1871, p. 277. 
 
 2 Groth's Zeitschrift, vol. 6, 1879, ? I2 - 
 
 3 Pogg. Annal. vol. 147, 1872, p. 59. 
 
Twinned forms. 377 
 
 3. A single crystal of kyanite described by Bauer 1 . In this mineral 
 the twin-axis, if perpendicular to the ^-axis as supposed by Bauer, 
 is inclined to a second edge at a small angle which has been 
 determined by Bauer himself as 23', by Brooke and Miller as 15', 
 and by vom Rath on one specimen as 5', and on another as zero. 
 It is therefore more probable that in the specimen described by 
 Bauer this edge, and not the perpendicular to the ^-axis, is really 
 the twin-axis ; and in fact in two other very similar specimens, also 
 described by Bauer, the 2T-axes of the two individuals, instead of 
 being quite parallel, were distinctly inclined to each other. 
 
 It is quite evident that no new mode of twin-growth could be 
 established by any of these specimens, and that, had not the above 
 mode been considered to be well-established by means of the 
 specimens of pericline, a simpler explanation would have been 
 suggested in each instance. 
 
 357. As the characters of the twin-growths observed in other 
 anorthic crystals, as for instance those of brochantite and kyanite, 
 are not essentially distinct from the characters presented by the 
 anorthic felspars, the illustrations will be selected from the last- 
 mentioned and more plentiful minerals. 
 
 I. Twin-plane a face; the face of union parallel 
 to the twin-plane. 
 
 Since in this system there is no symmetral plane and the 
 crystallographic axes are consequently selected more or less arbi- 
 trarily from the zone-axes, no essential difference in the twin- 
 growths can be associated with a mere difference in the symbol of 
 the twin-plane : as, however, the lines chosen for axes are usually 
 the intersections with each other of planes either of cleavage or of 
 prominent development, a numerical difference in the symbol will 
 correspond to a distinction in the aspect of the twin-growth. 
 
 358. Twin-face the pro-pinakoid {oio}. 
 
 This mode of twin-growth is very common in all the anorthic 
 
 felspars. Fig. 327 represents a simple crystal of albite; two 
 
 nearly perfect cleavages are parallel to the faces (oio) and (ooi), 
 
 and correspond to analogous faces of the mono-symmetric felspar 
 
 1 Zeitsch. d. deutsch. geol. Gesell., vol. 30, 1878, p. 306. 
 
378 
 
 Anorthic system. 
 
 orthoclase. If the crystal be bisected by a plane passing through 
 its centre and parallel to the pro-pinakoid {oio}, the section 
 will have the form defghk and will be centro-symmetrical : hence 
 the section will be congruent with itself after a rotation of 180 
 round the normal to its plane. If, therefore, the front half of the 
 crystal of Fig. 327 be turned through two right angles round the 
 normal to the face (oio), the sections will again coincide and the 
 resulting compound crystal will have the form shown in Fig. 328, 
 and be symmetrical to the plane of union. No face being per- 
 pendicular to the twin-plane, with the exception of the faces of 
 the pro-pinakoid {oio}, no face of one individual is parallel to a 
 face of the other. The normal-angles made by the planes ooi, 
 Yoi with the twin-plane olo are 86 24' and 86 21' respectively: 
 
 no 
 
 oio 
 
 oio 
 
 3 
 
 Fig. 327- 
 
 3 
 
 Fig. 328. 
 
 Fig. 329. 
 
 hence there will be two nearly equal re-entrant angles, 7 1 2' and 
 7 1 8', at the upper end of the twin, and two salient angles of the 
 same magnitudes at the lower end : the angle no. olo being 
 119 40', the angle no . (TTo) will be salient and equal to 59 20'. 
 
 359. If the twinning be repeated the growth has sometimes the 
 form shown in Fig. 329, and may then be otherwise regarded as 
 a single crystal enclosing a twin-lamina : in most cases the twin- 
 ning is repeated several times, as already illustrated in Fig. 59 and 
 in some cases the twin-growth consists of a hundred or more 
 laminae, of which adjacent individuals are in twin- and alternate 
 ones in parallel positions. 
 
 In the mono-symmetric felspar orthoclase the normal to the 
 corresponding face (oio) is an axis of diagonal symmetry, and 
 is impossible as a twin-plane, for a rotation of the crystal through 
 
Twinned forms. 
 
 379 
 
 two right angles round such a line would bring it into a position 
 not crystallographically distinct from the first : hence the presence 
 of twinning about a face of the pro-pinakoid is sufficient evidence 
 that a specimen of felspar belongs to the anorthic series. 
 
 360. Sometimes, as in the fine crystals of albite from the dolo- 
 mite of Roc-tourne in Savoy, described by Gustav Rose 1 , the 
 individuals are interpenetrant. The crystals from this locality are 
 quite tabular through the prominent developement of the pro- 
 pinakoid, and have the habit shown in Fig. 330 : these twin- 
 crystals are remarkable in that the two angles formed at opposite 
 ends by the basal pinakoids of the two individuals, instead of 
 being one salient and the other re-entrant, are both re-entrant, while 
 
 Fig. 33. 
 
 Fig. 33 1 - 
 
 the angles formed at opposite sides by the faces of the form {201} 
 are both salient : further, a vertical furrow bounded by faces 
 belonging to the form {130} traverses each face of the pro- 
 pinakoid {oio}. If the twin-crystal be broken across so as to 
 show the basal cleavages of the two individuals (Fig. 331), it is 
 seen that on the broken part at one side of the furrow the cleavages 
 form a re-entrant angle, and on the part at the other side of the 
 furrow a salient angle, and that each of the cleavage-faces on one 
 side of the furrow is parallel to the diagonally opposite face on the 
 other side of the furrow. The individuals thus appear to cross 
 over to opposite sides of the junction-plane at its intersection with 
 the furrow. 
 
 361. Other faces presenting themselves as planes of twinning in 
 
 1 Pogg. Annal. vol. 125, 1865, p. 460. 
 
380 Anorthic system. 
 
 the anorthic felspars are comparatively rare. LeVy 1 and also 
 Schrauf 2 have described specimens of albite twinned about the 
 basal pinakoid (ooi) and similar in aspect to the orthoclase-twin 
 of Fig. 308; growths of albite already twinned about the zone- 
 axis [oio] are likewise known in which there is a further twinning 
 about the basal pinakoid (ooi), as shown in Fig. 340: Amazon- 
 stone from Pike's Peak, a green variety of microcline, is sometimes 
 twinned similarly to the so-called Baveno-twins of orthoclase, 
 namely about a plane (021); an albite crystal has been described 
 by Neumann 3 , and one also by Brezina 4 , twinned about a plane 
 having the same indices. 
 
 In kyanite, crystals enclosing laminae of which the position is 
 due to twinning about a plane (308), not observed as a face of the 
 crystals, have been described by vom Rath 5 . 
 
 II. Twin-axis an edge: plane of union perpendicular 
 to the twin-plane. 
 
 362. i. Twin-axis the Y-axis [oio]. 
 
 The most important twin-growths assigned to this law are the 
 common twins of pericline. Their most characteristic feature is 
 the presence of an obtuse re-entrant edge of very small angle 
 traversing a face of the pro-pinakoid {010} in a direction more 
 or less nearly parallel to its intersection with the basal pinakoid 
 {ooi}. The basal pinakoids of the two individuals being parallel, 
 the twin-plane must be either parallel or perpendicular to these 
 faces ; that the latter is the true explanation is at once evident from 
 the fact that corresponding faces of the two crystals, instead of 
 being symmetrically disposed with respect to a plane parallel to the 
 basal pinakoids, are diagonally symmetrical to a line lying in that 
 plane : the exact position of this line, the twin-axis, has been long 
 a matter of doubt. 
 
 363. The growth was first mentioned in 1824 by Mohs 6 , whose 
 
 1 Catalogue des mineraux de M. Turner, 1837, vol. 2, p. 194. 
 
 2 Atlas der Kry stall -formen, 1864. 
 
 3 Abhandl. Ak. Berlin, vol. 19, 1830, p. 218. 
 * Tschermak's Min. Mittheil. 1873, p. 19. 
 
 5 Groth's Zeitsch. vol. 3, 1879, p. 9. 
 
 6 Grundriss der Mineralogie, 1824, vol. 2, p. 295. 
 
n^ ^ forms. 
 
 381 
 
 statement of the law as twin-axis [oio] was accompanied by a 
 figure analogous to Fig. 332, in which the two individuals are 
 represented as intersecting in a plane parallel to the faces of the 
 basal pinakoid; thus the edge X, formed by the intersection of 
 olo . ooi and therefore parallel to the J-axis, is represented as 
 parallel both to the corresponding edge X 1 and to the edge R 
 formed by the intersection of olo . (oYo), pro-pinakoid faces 
 belonging to the different individuals. In 1835 Kayser 1 pointed 
 out that as the line [oio], the .F"-axis, assumed as twin-axis by 
 Mohs, is not at right angles to the J^-axis but is inclined to it at an 
 angle of 8 9 13^' (according to Breithaupt), the edge X 1 instead 
 of being parallel to X ought to be inclined to it at an angle of 
 I 33i / j an d further that the edges X and R ought to have a 
 mutual inclination of 13 n J', as shown in Fig. 333. The question 
 
 Fig. 332. 
 
 Fig. 333- 
 
 Fig. 334- 
 
 thus arose as to whether the statement of the law or the figure 
 given by Mohs was most nearly in accordance with the features 
 of the specimens themselves. Kayser, and later Rose 2 , after 
 examination of numerous specimens came to the conclusion that 
 all deviations from parallelism presented by the edges X, R, X l 
 were due to the striated and imperfect character of the faces, and 
 that a line exactly perpendicular to the J^-axis, and not the .XT-axis 
 which makes an angle of 89 13^' with that line, must be regarded 
 as the twin-axis : Fig. 334 illustrates the form of the ideal twin 
 corresponding to this view. Kayser's explanation of the twins of 
 pericline became very generally accepted, although three eminent 
 
 1 Pogg. Annal. vol. 34, 1835, p. 108. 
 
 2 Ibid. vol. 129, 1866, p. i. 
 
382 
 
 A nor thi^sy stem. 
 
 observers, Miller, Des Cloizeaux and Schrauf, adhered to the older 
 statement given by Mohs. 
 
 By the publication of an elaborate memoir of Vom Rath in 
 1876 1 the accuracy of the older view may be regarded as now 
 fully established. Vom Rath has pointed out that the approxi- 
 mate parallelism of the edges X, R, X l is only exceptional, and is 
 then due to irregularity in the growth of the individuals beyond 
 their plane of junction, and that the angle between the edges X 
 and R changes considerably for a very slight change in the angles 
 of the individuals, which in albite and its variety pericline vary 
 somewhat with the locality of the specimens : he further states that 
 Kayser's conviction of the parallelism of the above edges was 
 arrived at chiefly from consideration of some excellent twin- 
 growths not of pericline but of oligoclase, a felspar of which the 
 angles are such that the J^-axis [oio] is itself almost precisely 
 normal to the edge X, so that the two modes of explaining the 
 twin become in this particular case practically indistinguishable. 
 
 Fig. 335- 
 
 Fig. 336. 
 
 364. The relative positions of two individuals of an anorthic 
 felspar after the rotation of one of them through 1 80 round the 
 axis [oio] may be determined as follows: 
 
 In Fig .335, <2, , c are the points of intersection of the crystallo- 
 
 graphic axes OX, OF, OZ with the edge formed by the faces 
 
 ilo .no, with the face olo and with the face ooi respectively: 
 
 aOa is thus parallel to the edge X and cO to the edges of inter- 
 
 1 Monatsber. d. Ak. Berlin, 1876, p. 147. 
 
Twinned- forms. 383 
 
 section of the prism-faces. Since the normals from a point in a 
 twin-axis to two corresponding faces are co-planar with the twin- 
 axis, the edge of intersection of the faces is perpendicular to that line : 
 hence the edge of intersection of the face olo with the corresponding 
 face of the other individual must be the line in the face oio which 
 is perpendicular to the axis bOb. Hence if the line fbg be in the 
 face oio and be at right angles to bO~b, and eOh, dbk be lines 
 parallel to fbg, the figure defghk will be the intersection of 
 corresponding faces of the two individuals. The position of the 
 \mefg, or of its parallel eh, may be thus calculated : 
 
 Let the spherical triangle abc (Fig. 336) be obtained from the 
 intersections of the lines Oa, Ob, Oc with a sphere having its 
 centre at : e, the point of intersection of Oe with the sphere, will 
 fall in the great circle passing through c and a, for the line Oe is 
 in the plane aOc, parallel to oTo : further, the arc eb is a quadrant, 
 since Oe is at right angles to the axis Ob : whence, by the 
 ordinary formula 
 
 cos eb = cos ea cos # + sin ea sin ab cos eab, 
 we have 
 
 cot ea = tan ab cos eab : 
 
 ea is the angle which Oe or fg makes with the line Oa, or the 
 edge X, and may be denoted by 0; eab is the angle between the 
 planes aOc, a Ob, or the supplement of the angle </> between the 
 normals to the faces oYo, ooi ; ab is the supplement of the angle 
 between Oa and Ob : 
 
 hence cot = cos (/> tan f. 
 
 365. In the case of albite from Schmirn = 86 30' and 
 = 92 8', whence = + 3i 23': in albite crystals measured by 
 Breithaupt < = 86 41' and C = 90 47', whence 9 = + 13 18': in 
 anorthite < = 85 50' and f = 88 48', whence 16 5'. 
 
 Thus while in the case of albite the edge of junction R of the 
 pro-pinakoids of the two individuals would meet the edge X in 
 front, in that of anorthite the two lines would meet if produced 
 backwards, and in the one felspar the inclination may be as high 
 as 31 or as low as 13 in one direction, and in the other felspar is 
 1 6 in the opposite direction. For the remaining felspars with 
 
384 A north^c system. 
 
 intermediate parameters the line R will have an intermediate 
 direction, and in oligoclase is almost exactly parallel to the edge 
 X : hence vom Rath points out that the direction of the edge of 
 junction on the pro-pinakoid face may be of occasional service 
 for the determination of the kind of anorthic felspar to which 
 the individuals of such a twin-growth belong. 
 
 366. The above section defghk determines the directions of 
 the lines in which corresponding prism and pinakoid faces of the 
 two individuals would meet, and is, so far at least, merely a 
 geometrical fiction and analogous to the twin-axis as an axis of 
 revolution : it is a constructional plane useful for the determination 
 of the directions of the edges in which the corresponding faces 
 would meet, if they met at all, and even in the latter case the whole 
 surface of the figure might not be common to the two crystals. 
 In fact Mohs regarded the plane parallel to the basal pinakoid as 
 being in every case the plane of junction, while Kayser, who first 
 pointed out that the edges of the basal pinakoid could not take up 
 a congruent position on rotation through two right angles about 
 any line in its plane, still considered that this plane was the plane 
 of junction, holding that the frequent intersection in edges was due 
 to the growth of parts which lay outside the boundaries of the 
 surface common to both the individuals. 
 
 367. Similar growths to the twins of pericline are met with in 
 anorthite, but in the latter felspar have a more precise character 
 owing to the greater constancy of the angles of the individual crys- 
 tals, and have thus been always and unhesitatingly assigned to this 
 twin-law; in anorthite, too, the faces of the zone [oio] are more 
 numerous and are more prominent in their developement, so that 
 the exact parallelism of the faces of this zone, corresponding to 
 each other on the two individuals, is capable of easy demon- 
 stration. Vom Rath, to whom we are indebted for a careful 
 description of these twin-growths l , has described crystals enclosing- 
 numerous twin-laminse of which the faces are parallel to the above 
 section, and has thence inferred that, notwithstanding its irrational 
 indices, it has generally a physical existence and is the plane of 
 union, whenever corresponding faces of the two individuals actually 
 
 1 Pogg. Annal. vol. 147, 1872, p. 42. 
 
Twinned forms. 
 
 385 
 
 intersect, as in Fig. 337. Sometimes, however, the individuals 
 have their basal pinakoids in contact, and present incongruent 
 edges as illustrated in Fig. 338, which represents an actual twin- 
 growth described by vom Rath. 
 
 Fig. 337- 
 
 368. In the case of pericline itself the surface of junction is 
 frequently irregular and is rarely quite plane : sometimes the indi- 
 viduals have their basal pinakoids in contact, and have incongruent 
 edges similar to the anorthite-twin of Fig. 338 : at other times they 
 are more or less interpenetrant, and if the twin be broken parallel 
 to the pro-pinakoid cleavages it presents a zigzag line of 
 
 Fig. 338. 
 
 separation of the individuals, one side of the zigzag being deter- 
 mined by the above section and the other by the vertical axis : in 
 two exceptionally good specimens the above section seemed to be 
 actually the face of junction. 
 
 369. Simple twins of pericline have not been observed: the 
 crystals always interpenetrate and appear to cross over to opposite 
 sides of the junction-plane, as in the crystals of albite from 
 
 c c 
 
;86 
 
 Anorthic system. 
 
 Roc-tourne' described in Art. 360, which are twinned about the 
 pro-pinakoid (oio). The edge traversing the pro-pinakoid face 
 thus comes to be re-entrant at both sides of the twin, instead 
 of being re-entrant at one side and salient at the other : such a 
 twin is represented in Fig. 339. 
 
 Fig. 339- 
 
 In Fig. 340 is illustrated a specimen of pericline, described by 
 vom Rath, in which two twins of the kind just mentioned are 
 united by their basal pinakoids and are symmetrical to their face of 
 union, the two innermost individuals on the right or left sides of 
 the growth being therefore twinned about the face (ooi): the 
 equality of the inclinations of the edges R, R l to the edge X l in 
 this specimen furnished a convincing proof that the obliquity is a 
 fundamental, not an accidental, character of the pericline-twins. 
 
 Fig. 340. 
 
 Fig. 341- 
 
 370. 2. Twin-axis the X-axis [100]. 
 
 This mode of twinning is described by Des Cloizeaux as occur- 
 ring, though very rarely, in pericline from Tyrol: in Fig. 341 is 
 illustrated a simple twin figured by Schrauf. The plane of union 
 
Twinned forms. 
 
 387 
 
 is nearly parallel to the basal pinakoid and intersects the pro- 
 pinakoids in an edge T parallel to the edges X and X 1 : the 
 twin-axis being parallel to the intersection of the basal pinakoid 
 with the pro-pinakoid, a pro-pinakoid face of one individual will 
 be co-planar with a pro-pinakoid 
 face of the other, and the basal 
 pinakoids of both will be parallel. 
 The law is more often met with in 
 complicated twin-growths of eight 
 individuals, twinned in addition about 
 the face (ooi), in the albite of Ala 
 in Piedmont. 
 
 371. 3. Twin-axis the Z-axis 
 [ooi]. 
 
 This law of twinning, which is 
 more rarely met with than the or- 
 dinary pericline law, corresponds 
 to the Carlsbad law of the mono- 
 symmetric felspar orthoclase, and 
 the growths are similar in aspect. 
 As before, the plane of junction is perpendicular to the twin- 
 plane and intersects the crystal in a section which, though sym- 
 metrical to its centre, is not symmetrical to the twin-axis lying 
 in it, whence the edges of the section are not congruent in the 
 two positions. The twins are dissimilar in character from the 
 pericline twins, for there is no growth of the individuals beyond 
 the incongruent edges, nor do the individuals cross to opposite 
 sides of the plane of junction : in Fig. 342 is illustrated such a 
 twin-crystal of anorthite from Vesuvius, described by vom Rath. 
 
 Fig. 342. 
 
 c c 2 
 
CHAPTER VIII. 
 
 THE MEASUREMENT AND CALCULATION OF THE 
 ANGLES OF CRYSTALS. 
 
 SECTION I. The Goniometer. 
 
 372. Two methods have been employed for measuring the angle 
 between the faces which form an edge on a crystal. The one is a 
 mechanical contrivance by which the angle contained between two 
 steel bars brought into contact with the faces of the crystal is read 
 off on a graduated arc ; the other consists in making the crystal 
 revolve round the edge to be measured, and determining by the 
 aid of a ray of light fixed in direction the angle through which 
 it must be turned in order that each of the two faces in succession 
 may be in a position to reflect the light in an identical direction. 
 By the former method only approximately correct measurements 
 can be obtained even from faces of exceptional smoothness and 
 considerable size; while the accuracy of the latter method need 
 only be limited by the precision with which the measuring instru- 
 ment is graduated and the perfection as reflecting planes of the 
 faces to be measured. The instrument employed in either case is 
 termed a 'goniometer'. 
 
 373. The contact- or hand-goniometer (Fig. 343). As first employed 
 by Rome* de lisle, and subsequently modified by Haiiy, it has the 
 form of a flat semicircular arc of brass or silver graduated from o 
 to 1 80 and subtended by a flat steel bar, which, for convenience 
 in manipulation where the crystal is entangled with associated 
 mineral, is able to move in the direction of its length, and is 
 constrained so to move by two screw-pins which pass through 
 two long slots cut in the bar parallel to its sides and clamp the 
 bar to a corresponding bar which is fixed in the position of a 
 radius to the graduated arc : one of these screw-pins is fixed at 
 
The contact-goniometer. 389 
 
 the centre of the arc, the other near its circumference. The 
 edges of this bar are parallel to the diameter which passes through 
 the zero of the graduation. A second moveable bar corresponding 
 to the first has only one slot extending to about half its length, 
 and being held by the central pin only, is capable of being rotated 
 round it as a pivot with any desired amount of friction, while for 
 convenience of adjustment it also can be moved in the direction 
 of its own length. Of the two bars or arms of the instrument 
 thus mounted, the parts of the narrower sides or edges that are to 
 be brought into contact with the faces of the crystal are worked 
 
 Fig. 343- 
 
 into perfectly squared and plane surfaces. The limb of the rotating 
 arm which moves over the graduated face of the goniometer is cut 
 away to one half its width and worked to a straight chisel-edge, 
 the continuation of which would thus pass through the centre 
 of the graduated semicircle ; this edge therefore indicates the arc 
 subtended by the two arms, and thus when the arms have been 
 applied to two faces of a crystal indicates the angle between them : 
 the instrument is more convenient when graduated in the opposite 
 direction to that shown in the figure ; the indicated angle will then 
 be that between the poles of the faces. 
 
 The instrument is usually constructed with a hinge on the 
 graduated arc, whereby one quadrant can be doubled back behind 
 the other, so as not to impede the measurement of an embedded 
 
3QO The reflection-goniometer 
 
 crystal; and to this purpose also the sliding motion of the arms 
 subserves (Fig. 343). 
 
 In using the hand-goniometer the points to be attended to are 
 the perpendicularity of the plane of the instrument and of the 
 broader faces of the steel arms to the edge to be measured, and 
 the complete simultaneous contact of both the straight nar- 
 rower surfaces of the arms with the crystal-faces, as tested by 
 examination between the eye and the light. And during the 
 operation, and especially when removing the instrument from 
 the crystal, it is well to maintain a slight pressure of the crystal 
 on the arm that does not rotate, and none, or the least possible, on 
 the crystal-face in contact with the rotating arm which indicates 
 the arc on the graduated circle; the crystal being drawn away 
 from the measured edge along and with a slight pressure against 
 the fixed arm. With every precaution, and where the faces to be 
 measured are well adapted by size, smoothness, and accessibility, 
 the successive readings of the measured arc may accord within the 
 limit of a quarter of a degree. Such cases are however very rare, 
 and it is evident that even this is not a sufficient degree of accuracy 
 for the purposes of exact crystallography. 
 
 The instrument described is often employed in a simpler form ; 
 the arms being separate from the arc but capable of being applied 
 to it by inserting the pivot of rotation of the two arms in the centre 
 of the metal arc, which then presents the character of an ordinary 
 semicircular protractor. 
 
 374. The reflection- goniometer of Wollaston, in its original form, 
 consists of a circular metallic disc four to eight inches in diameter, 
 with a flanged rim graduated with degrees and subdivisions of a 
 degree (Fig. 344). An axle at the centre of the disc revolves in a 
 collar of metal fixed to the support of the instrument, and is firmly 
 attached to a flat circular handle with a milled edge behind the 
 disc, whereby the disc with its graduated arc is carried round. The 
 graduations are read by the aid of fixed verniers. The axle of the 
 disc is hollow, and a second and solid axle or core passing through 
 it and ground accurately into it can be turned independently of 
 the disc by a smaller handle similar to the one which turns the disc 
 itself and parallel to but beyond it: the extremity of this axle, which 
 
of Wollaston. 
 
 projects beyond the face of the goniometer, bears a small and 
 simple apparatus for the support and adjustment of the crystal 
 to be measured. 
 
 In the adjustment of a crystal this independent rotation of the 
 axle of the instrument is convenient, as it enables the crystal and 
 the apparatus holding it to be turned without moving the graduated 
 circle. 
 
 For use the goniometer is planted in front of a window, a bar 
 of which may be taken as the object to be reflected in the crystal- 
 faces ; or, preferably, the object or signal may be a slit in a dark 
 screen illuminated by the 
 sky or by artificial light ; 
 or a beam of sunlight may 
 be directed on the crystal 
 by a heliostat and seen 
 through a protecting glass 
 held before the eye. 
 
 The crystal is now at- 
 tached to the holder by a 
 little plastic material formed 
 by fusing together beeswax 
 and pitch in suitable pro- 
 portions ; this may be kept 
 at hand conveniently if 
 rolled out into a pencil- 
 form. 
 
 The holder is a small 
 rectangle of thin sheet-brass which is gripped in a slit at the end 
 of a rod, which rod serves to turn and to give a traversing motion 
 to the holder by its play in a collar attached to a curved brass 
 support, which again has a turning motion round a pivot in a 
 second similar support which is fastened to the axle of the gonio- 
 meter; see Fig. 344. 
 
 The variety of motions that can by these means be given to the 
 holder enables the observer with a little practice to adjust the 
 crystal in any position. For the purpose of measuring, however, 
 especially for following the measurements of a zone, the position 
 
 Fig- 344- 
 
392 The reflection-goniometer. 
 
 of the crystal usually presented by books on crystallography is 
 impracticable. The crystal must in fact project clear of the ap- 
 paratus and beyond the different portions of the support ; the eye 
 can then uninterruptedly observe the faces of the zone as they 
 in succession come round and, if properly adjusted, reflect to it 
 the signal, each in turn. 
 
 Assuming then that the crystal has been or is readily capable of 
 being thus adjusted since the precautions to be taken in the 
 centering and adjustment will have to be discussed presently in 
 detail we may proceed now to the other conditions that have to 
 be fulfilled if a correct measurement of the normal-angle of two of 
 the faces of an adjusted zone is to be made. This can only be 
 attained when the rotation of the crystal round the axis of the 
 goniometer through this normal-angle shall have brought the two 
 faces in succession into a position in which each reflects the axis 
 of a pencil of rays, coming from the centre of the signal, in the 
 same direction, a condition which necessitates the position into 
 which the second face is brought being parallel to and, for exact 
 measurement, coplanar with that previously occupied by the first 
 face. 
 
 The direction of vision must evidently be unchanged during the 
 measurement. Wollaston achieved this by placing in the line or 
 vision of the reflected signal a second signal similar to it but more 
 faintly illuminated. Then viewing this second signal by direct 
 vision he would bring the image of the first signal, as seen by 
 reflection from one crystal-face, into contact with it, and when 
 the one signal, seen so through a portion of the pupil of the eye, 
 exactly covered the other as seen directly through the remaining 
 portion of the pupil, he arrested the turning motion. A second 
 crystal-face being treated in the same way, the difference of the 
 readings on the circle gave the normal-angle between the faces. 
 Kuprfer substituted for the second signal the image of the first 
 signal as seen by reflection in a mirror of black glass. 
 
 375. But the most important modification that Wollaston's 
 goniometer has received is that which, in the hands of Mitscherlich, 
 has made it a new instrument by endowing it with a telescope 
 fitted with cross-wires, by the aid of which the direction of the 
 
Improved forms. 393 
 
 rays reflected from successive crystal-faces was fixed with great 
 precision. 
 
 The greater rigidity of direction thus imparted to the axis of 
 vision involved the necessity of a motion either of the telescope or 
 of the crystal in a direction parallel to the axis of the instrument, 
 so that the faces of the zone to be measured might be brought with 
 precision into the field of view. This is usually effected by giving 
 the telescope a sliding movement parallel to itself on a bar that 
 supports it. But Professor Viktor von Lang introduced an im- 
 provement on this method by fixing the ' head* (or part of the 
 goniometer which carries the crystal and the screws for its adjust- 
 ment) on the end of a hollow cylinder, which fits on and can be 
 clamped immoveably to an accurately turned solid cylinder of 
 steel, forming a continuous piece with the solid core of the axle of 
 the goniometer : the hollow cylinder can thus slide on the solid 
 one and be fixed by the clamp. This method offers the advantage 
 that the goniometer being adjusted and fixed in position the tele- 
 scope may be also adjusted once for all to the centre of the re- 
 flected signal ; any variations in situation of the edge to be mea- 
 sured, whether from the magnitude of the crystal or from the con- 
 ditions under which it is mounted, being compensated by the 
 motion not of the telescope but of the 'head' and of the crystal 
 carried by it. 
 
 If a collimator be used, the signal introduced by Websky is very 
 convenient : it is limited by two brass discs of which the distance 
 from each other can be altered by means of a screw, and by two 
 straight edges parallel to the line joining the centres of the discs. 
 
 In Wollaston's and Mitscherlich's goniometers the axis is hori- 
 zontal and the plane of the graduated circle is vertical. Professor 
 Miller, however, adopted a form of the instrument in which the 
 axis is vertical and the disc horizontal: a form very simple in 
 adjustment, and furthermore one especially adapted for measuring 
 such crystals as are either too heavy of themselves, or are attached 
 to masses of mineral and rock too considerable, to be carried by 
 a vertical goniometer without causing strains that must conduce to 
 flexure in the support and to very appreciable inaccuracy in the 
 results obtained. 
 
394 
 
 The reflection-goniometer. 
 
 376. We shall now proceed to discuss the process of putting the 
 goniometer into proper adjustment for measuring a crystal, and 
 for this purpose we shall have recourse to methods independent 
 of the use of a mounted crystal. Certain qualities have of course 
 to be presupposed as being presented by the instrument itself. 
 These include accurate graduation involving a power of reading 
 by the aid of the verniers v (Fig. 345) to 15"; the perpendicularity 
 
 v 
 
 Fig. 345- 
 
 of the axle to the face of the goniometer disc ; a minimum of 
 * backlash ' in all the screws that impart motions ; and a firm 
 fastening of the handles to the axles which they turn, so that any 
 motion of either handle is instantly responded to by the graduated 
 circle or by the mounted crystal, as the case may be. 
 
 In order to proceed in a defined manner we shall assume the 
 goniometer, in the first place, to be one of the vertical construction 
 with the telescope permanent in its position and the head move- 
 able (Fig. 345). The foot on which the telescope is mounted has 
 
Its adjustment. 395 
 
 adjustment-screws pp, to regulate in azimuth the position of the 
 plane of observation relatively to that of the goniometer-disc D. 
 There are also screws yy for regulating the horizontally of the 
 cradles or F's in which rest the trunnions xx on which the 
 telescope T turns. 
 
 Now the following are the ends to be attained in planting the 
 goniometer and adjusting the telescope : 
 
 a. The goniometer-axis is to be horizontal. 
 
 b. The plane of observation which passes through the centre of 
 the signal and the optic axis of the telescope is to be vertical, and 
 is to be parallel to that of the goniometer-disc. 
 
 c. And it is convenient that it should be perpendicular to the 
 plane of the signal. 
 
 d. The optic axis of the telescope is to intersect the axis of the 
 goniometer. 
 
 In order to effect these adjustments, and to do so in a certain 
 order, the following adjuncts to the signals employed are of the 
 greatest use : 
 
 A long wire of substantial thickness, swinging freely and as 
 close to the plane of the signal as is consistent with its carrying a 
 suspended plumb-weight : it must exactly intersect the centre of the 
 signal. A second and similarly weighted wire (for which however 
 may be substituted a distinctly seen vertical line on the wall or 
 window) placed at a distance to the right of the former wire equal 
 to twice the distance of the goniometer-face from the intersection of 
 the cross-wires of the telescope. The latter will be termed the adjust- 
 ment-wire, the former the signal-wire. A flat dish or tray con- 
 taining clean mercury. A metal pin w (Fig. 347), one end of which 
 fits into the ' head ' of the instrument while the other terminates 
 in an accurately turned point. A lens / (Fig. 345) of such focal 
 length that when applied in front of the object-glass of the telescope 
 the crystal can be observed as by a microscope of low power. This 
 lens turning on a pivot can be applied or removed at will : the 
 telescope when used with the lens will be termed the micro-tele- 
 scope. Where a collimalor is not employed it is important that 
 the signal should be at a considerable distance about 20 feet to. 
 
396 The reflection-goniometer. 
 
 30 feet from the telescope is sufficient from the goniometer, 
 and, where this instrument is one of the vertical construction, 
 it is desirable that the signal should be at such a height that 
 the angle of incidence and reflection from a crystal-face may 
 be about 45: and the slit or bar employed as a signal should 
 be quite horizontal. 
 
 377. In the first place, the goniometer is planted on a firm 
 table, in such a position that the plane of observation is ap- 
 proximately perpendicular to the window or plane of the signal. 
 A method will be given for obtaining this perpendicularity with 
 accuracy, but a first approximation attained by any simple geo- 
 metrical method that may suggest itself according to circumstances 
 is desirable as simplifying the processes. 
 
 a. Horizontality of axis. The goniometer is to be turned 
 round upon the table until the image of the signal -wire can 
 be seen in the lacquered face of the disc D, and the turning move- 
 ment continued until this image is seen to be in contact at the 
 edge of the disc with the adjustment-wire as seen directly. If the 
 adjustment-wire and the image of the signal-wire are not in con- 
 tact throughout their length but are seen to be inclined to each 
 other, the levelling screws -S* are set in motion to bring them into 
 parallelism when a little separated; after a slight turning of the whole 
 instrument the one wire will then be seen to eclipse the other. A 
 small spirit-level with a stirrup mounting is also convenient for 
 securing the horizontality of the steel core of the instrument. 
 
 Two objects will now have been effected: the axis of the 
 goniometer will be horizontal and the disc will lie in a vertical 
 plane which bisects the space between the signal- and the ad- 
 justment-wires, so that if the telescope is in correct adjustment, 
 on directing it to the signal, the signal-wire should be seen passing 
 through the cross-point of the telescope-wires in every position of 
 the telescope as it turns on its trunnions. 
 
 b. Adjustment of plane of observation. But if the centre of the 
 signal lie on one side of this point when the telescope is turned 
 to it, the adjustment-screws pp are brought into use to turn the 
 telescope in azimuth until the cross-wires and the signal-centre 
 are seen as if in contact. The effect is of course to bring the optic 
 
Its adjustment. 397 
 
 axis of the telescope, when it is in this one position, into parallelism 
 with the plane of the disc. 
 
 The mercurial trough is now placed in such a position on the 
 floor of the room that the signal as reflected in the mercurial sur- 
 face can be seen in the telescope ; and if the centre of the reflected 
 signal is found to fall on the cross-wire, there is nothing more 
 needed in this part of the adjustment: the telescope obviously 
 must have been moved in a plane perpendicular to the surface 
 of the mercury and therefore containing the signal-wire ; and if 
 the vertical wire of the telescope be correctly placed to cover any 
 part of this signal wire it will continue to do so when the tele- 
 scope is directed to any other part of that wire and its reflected 
 continuation. 
 
 But if these events do not take place and the signal-centre 
 is on the cross-wire when seen directly and to the side of it 
 when seen reflected in the mercurial surface, the adjustment 
 screws yy that regulate the relative elevation of the trunnions of 
 the telescope are set in action to bring the axis on which the 
 telescope turns into parallelism with that of the goniometer. 
 In fact the correction has to be made to half the extent of the 
 error as shown in the mercurial mirror. If this and the previous 
 operation have been correctly performed, and have in fact been 
 repeated to ensure greater exactitude, the telescope will now be 
 in accurate adjustment as regards the parallelism of the plane 
 of observation with that of the disc of the goniometer and there- 
 fore with the plane containing the graduated circle. 
 
 c. Plane of observation and plane of signal made perpendicular. 
 So far we have been content with only an approximately perpen- 
 dicular position for the plane of observation in respect to the 
 plane containing the signal. 
 
 The cross-wires in a telescope are rarely strictly perpendicular 
 to each other; it is not always that they intersect exactly in the 
 centre of the field of view. Adjustments to the true centre can 
 however be effected by turning the eye-piece through 180, and 
 taking for the ' cross-wire' the point on the horizontal wire to which 
 the two positions of the actual wire-crossing are symmetrical. 
 
 In order to make the plane of observation perpendicular to 
 
398 The reflection-goniometer. 
 
 the plane of the signal, let the horizontal wire be first turned 
 into a position of exact parallelism with the sides of the signal, 
 which is assumed to have been previously mounted in a horizontal 
 position by the aid of a good spirit-level. On observing the image 
 of the slit from the mercurial surface, if the plane of observation 
 is not strictly perpendicular to that of the signal, the image will 
 now be seen inclined at a certain angle to the horizontal wire of 
 the telescope. Let therefore the eye-piece be turned till the hori- 
 zontal wire bisects this angle. 
 
 If now the goniometer be moved in such a manner that on 
 the one hand the adjustment-wire is kept in apparent contact 
 with the image of the signal-wire as in the adjustment (a), and 
 so that, on the other hand, the image of the signal-slit seen 
 on the mercury is brought into parallelism with the horizontal 
 wire, the adjustment of the instrument in respect to the signal 
 will be found to be complete. 
 
 d. Direction of telescope-axis. We have only, in fine, to be careful 
 that the optic axis of the telescope accurately intersects the axis 
 of the goniometer : and this is readily effected by first mounting 
 the pointed metal pin w (Fig. 347), to which allusion has been 
 made, so centiically that when the goniometer -disc is turned the 
 point makes no excursion from the axis; and then by means 
 of the adjustment-screw (not visible in the figure) which regulates 
 the inclination of the telescope the horizontal wire is brought 
 accurately to cover the point of the metal pin as seen in the 
 micro-telescope. Or if, first, one of the traversing screws be 
 placed in a vertical position, and the point of the metal pin be 
 brought into apparent contact with the horizontal wire of the 
 micro-telescope by turning the traversing screw ; and if, secondly, 
 on bringing the second traversing screw into the vertical posi- 
 tion the point has not moved away; or, if it has moved away 
 and after being brought back to the wire by the second tra- 
 versing screw it .remains permanently in contact with the wire 
 as the goniometer-disc is turned, the adjustment is complete. If, 
 however, in the last event the position of the point is not per- 
 manent, the point should be turned by means of the axle into the 
 position of its maximum excursion from the .centre, that namely 
 
Its adjustment. 399 
 
 in which the goniometer has revolved through 180 from the last 
 position. The declination-screw of the telescope should now be 
 turned until the point is moved through one half the distance 
 separating it from the horizontal wire, and the point itself should 
 be brought down by the second traversing screw into contact 
 with the wire. The adjustment will be found to be now complete. 
 The best assurance that all the corrections have been accurately 
 made will be obtained when a zone, preferably an ortho-symmetrical 
 zone, has been adjusted on the goniometer, and the images from 
 the successive faces have been found to pass through the field of 
 view, each moving symmetrically in respect to the vertical wire and 
 in parallelism to the horizontal. Any deviation from this symmetry 
 or parallelism should be at once traced to its source in one or 
 other of the causes that have been discussed. 
 
 378. Where a collimator is used, it is advantageous to have a 
 small triangular prism fixed to it, just above the objective, in such 
 a position that the image of the signal given by a face may be 
 viewed after total reflection in the prism. A pencil of rays from 
 any point of the signal is first reflected from the face in a direction 
 nearly perpendicular to it, then totally reflected in the prism, and 
 next enters the telescope, which in this case is so arranged that 
 its axis can be directed to intersect the axis of the goniometer 
 either directly or after total reflection in the prism. This arrange- 
 ment is occasionally convenient when the faces are much striated 
 or the crystal is attached to its matrix. 
 
 379. Before passing from the adjustment of the goniometer to 
 the mounting of a crystal on the adjusted instrument it should be 
 remarked that it is very desirable either to have both the gonio- 
 meter and the signal fixed permanently in position, or to have in 
 front of the permanent signal a fixed stand to which the goniometer 
 can be transferred without involving the necessity of readjustment. 
 
 The three-armed groove proposed by Sir W. Thomson forms an 
 admirable stand for this purpose. The top of the goniometer-table 
 or a slab Zf that can be fixed to it is formed of hard wood or slate, 
 and has three grooves ggg (Fig. 345) cut in its surface which meet 
 at angles of 120: their section is a right angle, and one of the 
 three cylindrical feet of the goniometer is placed in each groove. 
 
400 
 
 The reflection-goniometer. 
 
 In planting the goniometer in position before the signal and in 
 levelling it, this stand has to be shifted until the adjustment is 
 complete and is then fixed in the position thus determined. The 
 goniometer may now be removed and replaced on its stand in 
 identically the same position as often as is requisite. 
 
 380. It has already been indicated that, in order to ensure 
 accuracy in the process of measuring the angles of a crystal, the 
 precautions to be taken relate not only to those preparatory ad- 
 justments of the goniometer and the telescope which have been 
 
 . Fig- 346- 
 
 discussed in detail, but also to the proper mounting of the crystal 
 itself; but before defining these latter precautions it will be well 
 to discuss the optical principles involved in the method of measure- 
 ment by a reflection-goniometer, certain of which have been in 
 fact already assumed by anticipation during the discussion of the 
 adjustments of the instrument. 
 
 Let S, Fig. 346, be the centre of the signal, B> a point in the 
 axis of the goniometer, and EL the optic axis of the observing 
 telescope of the instrument of Mitscherlich, or the line of sight 
 of the naked eye as in the method of Wollaston, and so directed 
 that if continued it passes through B. If then -5", the centre of the 
 slit, is seen reflected by a crystal -face so as to coincide with 
 
Error due to imperfect centering. 401 
 
 the intersections of the cross-wire in the telescope, the following 
 statements follow from the elementary principles of optics : 
 
 (1) Every image of the centre of the signal S that is to be seen 
 placed at the intersection of the cross-wires must be due to rays 
 of which the axis is reflected in the direction ^BE. 
 
 (2) If 2 be the image of S, S and 2 will be symmetrically 
 situate in regard to the reflecting plane. So that if M be the 
 intersection of the line S2 with the plane of the crystal-face, 
 SM= 2JJ/and -5*2 is perpendicular to the reflecting plane ; and 
 this plane forms an oblique section with the cone of rays of which 
 2 is the apex and It the object-glass of the telescope or the pupil 
 of the eye is the base : the brightness of the image will therefore 
 increase as more of this oblique section is a reflecting surface, and 
 will therefore increase with the magnitude of the crystal-face so 
 long as this is not greater than that of the section. 
 
 (3) If the crystal-face lie in a plane BM passing through the 
 axis of the goniometer, its position must be at or near B and within 
 the limits of the cone of rays 2//'; the direction of MB being 
 determined by the angle SB E, since the angle SBM is the 
 complement of \ SB E. 
 
 In the case of the crystal-face lying in a plane not passing 
 through the axis of the goniometer (though necessarily parallel to 
 it), if the plane be parallel to the plane B ' M "the axis of the reflected 
 rays will not intersect the horizontal wire of the telescope, for 
 the rays will appear to proceed from a point not lying in the 
 line ^BE. 
 
 In order that the image of the slit may be reflected on to the 
 horizontal wire of the telescope by the crystal-face, this face must 
 be turned into such a position that the image of will lie on the 
 line .#2 though it will be at a different point from 2 ; say at 2'. 
 Let KM' be the trace of this plane perpendicular to and bisecting 
 6*2'. The crystal-face will now lie in some part of this plane, 
 as at If, and will have been turned from its previous position, 
 round an edge perpendicular to the plane of the figure (and there- 
 fore parallel to the axis of the goniometer), through an angle which 
 may be denoted by 0. Let BM and B'M' intersect in R. 
 
 Then S and 2' will be symmetrical in respect to 'M', and 
 
 ' Dd 
 
402 The reflection-goniometer. 
 
 B'R will bisect S^f perpendicularly in M f . From B draw per- 
 pendiculars BN on S2' and BD on B'R. Let M be supposed 
 joined to M' and N. 
 
 Let .#> = AW= d, SB = r, and SBR = o> ; 
 
 then 6 = BRD = 2 S2'= MBN. 
 
 Since the angles at M and A^are right angles, SMNB are points 
 on a circle, and MNM'= MBS o>. 
 
 Also since -flf and M' bisect S2 and S3', MM' is parallel to 
 22', and 
 
 2JOTjy=#2W =25^252'=: ---a> + 0, 
 
 2 
 
 and M'MN = -n-MNM'-MM'N = - -0. 
 
 2 
 
 Also 
 
 Therefore MM'= sin co 
 
 COS0 
 TVTJVT' c' T\,T^\7\/T' 
 
 Also 
 
 sin MM' S cos(co 0) 
 
 Therefore MM'= sin co rS . m m . 
 
 cos (w 0) 
 
 Equating these values, we have 
 
 d rsmO 
 
 cos ~~ cos (to 0) ' 
 whatever be the value of 0. 
 
 This angle is however necessarily very small, while, in relation 
 to d, r is always very large ; whence, assuming 
 sin = and cos = i, 
 
 we have d = r or = - cos co. 
 
 cos co r 
 
 T) T~\ 
 
 It may be observed that, since sin = -^-^ , we have within the 
 assumed degree of approximation, = 
 
Error due to imperfect centering. 403 
 
 and, substituting this value of 6 in the above expression, we have 
 
 SB 
 
 = cos o> 
 
 whence J5SR = 90, to the same degree of approximation. 
 
 Hence if d, the distance from the axis of the goniometer of 
 a plane passing through the crystal-face that gives an image of S 
 on the horizontal wire of the telescope, be given, we can find 
 the direction of the plane by drawing a line through S at right 
 angles to S and meeting BM in R, and drawing from R a line 
 RD tangent to the circle described with radius d round B as a 
 centre: the position of the reflecting crystal-face in this plane 
 RD will be determined by the intersection of this plane with the 
 cone S'/r. 
 
 381. It will be seen then that the position of a reflecting face 
 as determined by the reflection-goniometer is different according 
 as the plane of the face contains the axis of the instrument or is 
 only parallel to it : and the correction for the error an error of 6 
 if we estimate the angle and of d if we consider the position of 
 the plane must be treated as positive if it lie on one side of 
 BM, negative if on the opposite side of that plane. 
 
 If then we are measuring the angle between two faces on a 
 crystal, it is clear (i) that their edge must be parallel to the axis of 
 the instrument, (2) that it should be coincident with this axis. 
 Where however it is not coincident the correction for the errors of 
 angle will be given by 
 
 = cos cu for the one plane, 
 
 r 
 
 Q'= cos co for the other. 
 r 
 
 And as we can always in practice so centre the crystal as that the 
 axis may fall within the angle formed by the two faces, the correction 
 
 can be put into the form - cos co ; and this can always be made 
 less than - cos co, where / is the greatest thickness of the crystal ; a 
 fortiori less than - , or the maximum angle which the crystal when 
 placed at B subtends at S. 
 
 D d 2 
 
404 The reflection-goniometer. 
 
 And, further, the amount of error will depend on the magnitude 
 of w, becoming smaller in proportion as the directions of the ray 
 from the signal and of the axis of the telescope approach to 
 coincidence with the normal to the reflecting face; and also on 
 the magnitude of r, vanishing when r is infinite and the rays from 
 the signal S become parallel. 
 
 The latter case, where the rays from -S" are parallel, is a con- 
 dition which can be induced by the use of a collimator ; the illu- 
 minated slit being placed in the geometrical focus of the collimator- 
 lens, and the focus of the telescope adjusted for parallel rays. 
 
 382. In the discussion of the adjustments needed for the gonio- 
 meter, the processes employed for detecting and correcting error 
 did not necessarily involve the employment of a mounted crystal : 
 and the method for attaching and properly mounting a crystal 
 remains to be considered. It will have been seen from the result 
 of the theoretical discussion of the last articles that the conditions 
 to be fulfilled are that the edge to be measured should coincide 
 as nearly as may be with the axis of the goniometer, and that the 
 part of that edge selected for measurement should be intersected 
 perpendicularly by the plane of observation. 
 
 The instrumental means at our disposal for effecting the dif- 
 ferent motions of the crystal are the following, and they belong 
 entirely to the head or moveable apparatus for adjusting and cen-. 
 taring the crystal. 
 
 (1) Two circular motions cc in planes perpendicular to each 
 other may be imparted to the crystal by reason of its being placed, 
 when mounted, at or near the common centre of two toothed 
 circular arcs worked by tangent screws VV'\ see Fig. 347. The 
 crystal can thus receive a partial rotation, approximately round its 
 own centre, in either or both of two perpendicular planes which are 
 parallel to the axis. 
 
 (2) Two traversing motions tt' perpendicular to each other and 
 to the axis of the instrument. These are worked by two screws, 
 UU', parallel to the tangent screws that impart the rotatory move- 
 ments, and they operate in the directions of these tangent screws. 
 
 (3) A motion already alluded to whereby the head can be 
 moved along the cylindrical core A of the goniometer away from or 
 
Adjustment of a crystal. 
 
 405 
 
 towards the disc ; and a means b of clamping the head at any point 
 firmly to the steel core (Figs. 345, 347). 
 
 (4) A supply of small brass crystal-holders in the form of cubes 
 or flattened prisms, each with a projecting peg fitting into a 
 corresponding socket drilled into the ' head ' in such a manner 
 that the axis of the peg is in the axis of the instrument when 
 the traversing and tangent screws are in their normal position: 
 
 17. 
 
 with the crystal it carries the holder is subject to the motions im- 
 parted by these screws. 
 
 The forms hh in Fig. 347 are the most useful; the one 
 when a wax cement is the adhesive material employed, the 
 other when the crystal is attached to a platinum wire. The latter 
 is one of the most convenient modes of mounting a small crystal, 
 as then the wire can be screwed firmly to the support and 
 can be bent into any position required for bringing different zones 
 of the crystal into adjustment. When a wire is thus employed 
 
406 The reflection-goniometer. 
 
 the best cement wherewith to effect the fastening is formed of 
 gelatine and acetic acid, just sufficiently viscous to become hard 
 in the course of two or three hours. Or, as recommended by 
 Klein, gum-arabic thickened with milk-sugar, kept dried and 
 moistened when needed, will answer the purpose. 
 
 383. Let it be supposed, as an example, that we have to 
 measure the angles of a zone in which two good reflecting faces 
 P and Q meet in an edge. In the first place, the goniometer itself 
 (Fig. 345) having been adjusted, the apparatus forming the head is 
 brought into the normal position, that namely in which the circular 
 arcs cc worked by the tangent screws are both symmetrically divided 
 by the axis, so that when the crystal-holder is in its place its edges 
 are parallel to the axis ; further, the head is turned round until the 
 traversing screws /'/ are respectively vertical and horizontal, the 
 former being on the upper side and the latter on the side away 
 from the observer. 
 
 The crystal o is so adjusted on the brass holder A, that the face 
 P is as near as may be parallel to the flat side of the holder, and 
 that the edge to be measured is parallel, to its axis; and care is 
 afterwards taken, when fixing the crystal-holder in its socket, that 
 the flat side is as near as may be horizontal, and therefore perpen- 
 dicular to a definite tangent screw. The head is next clamped to 
 the steel core A in a position in which the crystal can be seen in 
 the micro-telescope T. By means of the traversing screws the edge 
 to be measured is brought as near as possible to the horizontal 
 cross-wire and in focus ; one method of quickly effecting this is 
 as follows: the inner axle is turned until one of the traversing 
 screws is perpendicular to the axis of the telescope, and by means 
 of this screw the edge is brought to the horizontal wire ; the inner 
 axle is then turned until the other traversing screw is perpendicular 
 to the axis of the telescope and the edge is again brought to the 
 horizontal wire in the same way as before : the edge of the crystal 
 is now close to the axis of the goniometer, but is only approximately 
 parallel to it. 
 
 384. The inner axle is next turned round until the face P is as 
 nearly as may be in position to reflect light from the signal into the 
 telescope : the face P having been arranged to be approximately 
 
Adjustment of a crystal. 407 
 
 perpendicular to a definite tangent screw, the requisite position of 
 the head is a constant of the instrument and becomes familiar to 
 the observer. By means of the tangent screw perpendicular to P, 
 the edge to be measured is now brought into parallelism with the 
 horizontal cross-wire, and next into apparent concidence with it by 
 the opposite traversing screw : on turning aside the lens / from the 
 object-glass an image of the signal should now be visible, and its 
 centre may be brought on to the vertical cross-wire by a further 
 rotation of the same tangent screw : the edge will require to be 
 once more brought to the horizontal wire by means of the opposite 
 traversing screw. The inner axle is now turned round until the 
 face Q is seen in the micro-telescope to be illuminated with light 
 from the signal : by a rotation of the second tangent screw the 
 position of Q is altered until the face is at its brightest, the lens / is 
 again turned away from the object-glass, and the centre of the 
 image of the signal then visible is brought on to the vertical cross- 
 wire by means of the same tangent screw, and the edge as seen in 
 the micro- telescope is again brought to the horizontal wire by 
 means of the opposite traversing screw. Since, in the preliminary 
 arrangement of the crystal, the face P was made approximately 
 parallel to the plane of the circular motion produced by the second 
 tangent screw, the adjustment of the face Q by means of this 
 screw will little affect the direction of the face P. 
 
 385. These operations must be repeated until, when the gonio- 
 meter disc makes a complete rotation (i) the crystal is seen to 
 turn round the edge as if hinged on the horizontal wire, and (2) the 
 images of the signal reflected from the faces P and Q, and indeed 
 from all faces of the zone, pass in succession through the field of 
 view symmetrically to the vertical wire, and are ortho-symmetrically 
 divided by the two wires when at the centre of the field. 
 
 The nearer the angle between the faces P and Q approaches to 
 a right angle, the more quickly is the accurate adjustment of the 
 zone arrived at : for, by the preliminary arrangement of the crystal, 
 the face Q will in such cases become more nearly parallel to a 
 tangent screw, and thus be less affected by the adjustment of the 
 other face. 
 
 386. The chief advantages of an instrument of the above con- 
 
408 The reflection-goniometer 
 
 struction over one with a horizontal disc are the following : (i) the 
 graduation being on the edge of the disc is more easily read ; (2) 
 owing to the inclined position of the telescope the work is less 
 fatiguing to the observer ; (3) the crystal, the traversing and adjust- 
 ment screws on the head, and the milled heads of the axles are all 
 more easily accessible than when they are above or beneath the 
 instrument ; (4) the sliding of the hollow cylinder which carries the 
 'head' renders the adjustment more easy and speedy; (5) the 
 correctness of adjustment of the goniometer itself is easily verified 
 or restored. On the other hand, a goniometer with a horizontal disc 
 can be used for the measurement of larger crystals, and the adjust- 
 ment of the crystal is less liable to disturbance, through the yielding 
 of the wax support during the measurement : also the motion of 
 the telescope renders it possible to alter, when desirable, the angle 
 of incidence of the light upon the reflecting faces; further, the instru- 
 ment can be used for the determination of indices of refraction. 
 
 387. In the vertical goniometer (Fig. 348) the ' head ' is fixed to 
 a hollow cylinder which can be clamped to a solid core continuous 
 with the inner axle, but in the horizontal goniometer such an 
 arrangement is impracticable : in the latter instrument the head is 
 fixed to a solid steel cylinder which passes concentrically through 
 the inner axle, and is terminated by a screw : it can be raised or 
 lowered by the rotation of a milled head through which this screw 
 passes. 
 
 When the crystal has been, brought to the height of the horizontal 
 cross-wire, the cylinder is clamped to the inner axle by tightening 
 a collar situated just above the graduated disc. During the measure- 
 ment of a zone, the inner and outer axles are fixed together by the 
 pressure of a screw. A slow motion of the graduated circle can 
 be produced by an ordinary tangent screw. Finally, the telescope 
 and also the verniers of the graduated circle are fixed to another 
 concentric hollow axle, external to the other two ; this moves in 
 a hollow cone fixed to the three legs of the instrument : hence, if 
 the graduated circle be clamped to the tripod, the angle of rotation 
 of the telescope can be measured : the telescope can be fixed in 
 any of its positions and is provided with a tangent screw for slow 
 motions. The collimator is fixed to the tripod. 
 
with vertical axis. 
 
 409 
 
 Corresponding parts of the vertical and horizontal goniometers 
 (Figs. 345 and 348) are indicated by identical letters. 
 
 Fig. 348. 
 
 388. In the horizontal goniometer, as made by Fuess, the tele- 
 scope, instead of being capable of adjustment, is screwed fast to 
 its carrier : the instrument is so carefully constructed that adjust- 
 ment can be made precise by a simple motion of the cross-wires, 
 and is therefore less easily disturbed. The process of adjustment 
 of the instrument, recommended by Websky, is the following : 
 
 (1) One of the eyepieces having been adjusted so that its 
 cross-wires are distinctly visible, is inserted in the tube of the 
 telescope until a distant object is in focus. 
 
 (2) A needle is adjusted in the axis of the instrument, and the 
 eyepiece is turned round until one of its cross-wires is as near as 
 may be parallel to the needle, as seen in the micro-telescope : the 
 cross-wires are then moved horizontally until this wire is in 
 apparent coincidence with the axis of the needle. 
 
4i o The reflection-goniometer. 
 
 (3) A plate of ' parallel glass ' is adjusted in approximate 
 parallelism to the axis of the goniometer, and the axle is rotated 
 until the plate is as near as may be perpendicular to the axis of the 
 telescope. 
 
 (4) A small plate of glass is now fastened in front of the 
 eyepiece at an inclination of about 45 to the axis of the telescope: 
 a lamp having been so placed that light is reflected from this plate 
 into the telescope, the cross-wires of the eyepiece are seen both 
 directly by dispersed light and by reflection from the parallel glass : 
 if the latter be adjusted by means of the circular motions so that 
 the images of the cross-wires, due to reflection from opposite sides 
 of the parallel glass, take in turn exactly the same position in the 
 field, the plate will be exactly parallel to the axis of the goniometer. 
 By raising or lowering the cross-wires their centre may be brought 
 into coincidence with its image as seen after reflection : the axis 
 of the telescope is then exactly normal to the plate and therefore 
 to the axis of the instrument. 
 
 (5) When the outer-axle is rotated, the image of the intersec- 
 tion of the cross-wires will move across the field in a direction 
 exactly perpendicular to the axis of the goniometer : if this 
 direction be inclined to the * horizontal ' wire, the eyepiece must 
 be turned until parallelism is produced : coincidence may be 
 obtained by a vertical translation of the cross-wires. The ' hori- 
 zontal ' wire and the axis of the telescope are now exactly normal 
 to the axis of the goniometer. The glass plate in front of the 
 eyepiece is now removed. 
 
 (6) Four tubes with signals are generally supplied with the 
 instrument; the signals being respectively, a cross-wire, a small 
 circular hole, a straight-edge slit, and a Websky's slit. The second 
 of these is first inserted in the collimator : the telescope is then 
 turned until the image is seen either directly or after reflec- 
 tion from the parallel glass : the signal is - moved about in the 
 tube of the collimator until the image is in focus : the screws which 
 fasten the pillar of the collimator to the tripod are adjusted so that 
 the centre of the image is seen on the horizontal cross-wire. The 
 straight-edge slit and the Websky's slit are next focussed. Finally, 
 the cross-wire signal is inserted and focussed, and then by means 
 
Error due to imperfect adjustment. 4 1 1 
 
 of their traversing screws the cross-wires of the collimator are 
 brought into apparent coincidence with those of the telescope. 
 The remaining eyepieces of different magnifying powers are now 
 adjusted by means of the cross-wire signal : in fact, the chief use of 
 the latter is for the verification of the adjustment, since few crystals 
 present faces so perfect as to give images of such a signal. All 
 the eyepieces and signals are provided with sliding collars with a 
 projecting piece which fits into a corresponding notch in the collar 
 of the tube of the telescope or collimator : if this collar be tightened 
 when the tube is in its final position, the eyepiece or signal may 
 be taken away and afterwards replaced in exact adjustment without 
 necessitating the removal of an adjusted crystal. 
 
 In performing the actual measurements of the successive angles 
 between the faces of a zone, each edge has to be separately 
 adjusted by means of the traversing screws; unless the crystal 
 is very minute' in its dimensions, in which case the crystal, after 
 one of the edges of the zone has been adjusted, may be brought 
 by the traversing screws into a position in which the zone to 
 be measured is seen always symmetrical to the horizontal wire. In 
 this case the resulting error is too minute to be estimated. If the 
 edge itself cannot be seen, from injury or other cause, the centering 
 may be effected by making the profile of each face coincide with 
 the cross-wire parallel to the axis of the goniometer. 
 
 389. Error of adjustment. It has been seen that, if A and B 
 be two faces of a zone, their edge will be seen in the field of the 
 micro-telescope to be precisely parallel to the horizontal wire, and 
 can be brought into apparent contact with it when the zone is 
 correctly adjusted ; and the images of the signal reflected from 
 the two faces will each be seen in the telescope to pass over 
 the field of view symmetrically to the vertical wire, and to be 
 ortho-symmetrically divided by both wires when the centre of the 
 image has reached the horizontal wire. 
 
 Neither of these statements holds good when the axis of the 
 zone \AE\ is not strictly parallel to the axis of the goniometer. 
 Let the angle of inclination of the zone-axis to the axis of the 
 goniometer be a ; and if the face A be adjusted so that the image 
 reflected from A has its centre at the crossing of the wires in the 
 
412 
 
 The reflection-goniometer. 
 
 telescope, let 8 be the distance from this crossing of the centre 
 of the image reflected from B when moved through the field till it 
 is in contact with the horizontal wire. The value of 8 in any 
 particular case can be determined by finding out the angle which a 
 responding distance on the vertical wire represents when an image 
 from a correctly adjusted crystal-face traverses that distance ; 
 i.e. finding the difference of the two readings on the goniometer 
 when the image is central and when its centre is on the vertical 
 wire at the distance 6 from the crossing ; b is twice this difference. 
 Let T, Fig. 349, be the projection of the telescope-axis perpen- 
 dicularly to the plane of the 
 figure, ZJ'the axis of the goni- 
 ometer, ST the direction of the 
 axis of the collimator, or of the 
 ray from the centre of the signal 
 which intersects the goniome- 
 ter-axis in the plane of observa- 
 tion. 
 
 For simplicity, let the axis of 
 the telescope be assumed to be 
 perpendicular to ST; hence 
 Z, T, and -S 1 may be three 
 J'ig- 349- points on a sphere separated 
 
 by quadrants. 
 
 If now A and B are in correct adjustment, and the letters 
 A v B l , &c. in the figure represent the projections on this sphere 
 of the poles of the planes in question, it is evident that for the 
 image reflected from the first face A to be central we must have 
 
 And similarly for jB lt since the pole of B must be at the point A 1 
 when the image from it is central, if the zone be in correct ad- 
 justment. Where however the zone-axis is inclined to the gonio- 
 meter-axis at an angle a, let P 1 be the pole of B when the centre 
 of the image from A is on the horizontal wire. 
 
 Let the angle of the edge \AB\ be y ; then, since the angle 
 between the zone-circles A l S and A l P l will be the same angle a as 
 
Error diie to imperfect adjustment. 413 
 
 that of the inclination of their zone-axes, and A 1 JB 1 = A 1 P 1 = y, 
 if the position of the pole A 1 is given anywhere on the great circle 
 ST the corresponding position of PI can be determined. When 
 the image from A is on the horizontal wire the pole of A is at A l 
 and the pole of B at P lf and when the image from B is at T' on 
 the horizontal wire the pole of B is at P 2 . Then we have 
 
 P 2 S = P 2 T'= 45 and TT- 8. 
 And if the great circles -LP l and LP^ intersect ST'mp^ and/> 2 , 
 
 LP 1 = LP,, 
 and sin P 2 p 2 = sin P 1 p 1 = smA l P l sin P l A l p l 
 
 = sin y sin a ; 
 also sin P 2 p z = sin SP Z sin P 2 Sp 2 
 
 sin 8 
 
 = sin 45 sin o = ; 
 
 V 2 
 
 sin 8 , , 
 
 whence sin a = ......................... (1) 
 
 V2 sin y 
 
 The angle, as measured on the instrument, between A and B 
 
 w 
 
 ill be 
 
 __ AS, __ yi 
 
 A A A 
 
 further we have 
 
 tan p 1 A l = tan P 1 A l cos ^ ^j^ = tan y cos a, 
 
 tan p 2 S = tan P 2 6" cos P z Sp 2 = cos 8. 
 
 Let E be the error in the measurement of the angle, i.e. the 
 true angle the measured angle ; then 
 
 tany tan^, p. tany(i cos a) 
 
 Now tan (y A. A) = \ l = ^-5- 
 
 i + tany tauA l p l i+tan 2 ycosa 
 
 . a 
 tan y sin 2 - 
 
 i + tan" y cos a 
 
 i cos 8 8 
 
 and tan (45 Sp 2 ) = - - = tan--, 
 i + cos 8 2 
 
 a and 8 are small quantities and their cubes may be neglected; 
 then a 2 
 
 T tany a 2 
 tan (y A l pj = = sin 2 y, 
 
414 The re/lection-goniometer. 
 
 The product of these two tangents will thus be of the fourth order, 
 and may be neglected. 
 
 a 2 8 2 
 
 = sin 2 y H 5 
 
 4 4 
 
 or to the same degree of approximation, 
 _ a 2 sin 2 y + 6 2 
 
 4 
 
 From (1) sin 2 8 = 2 sin 2 y sin 2 a, 
 
 and 8 2 = 2 a 2 sin 2 y. 
 
 2 a 2 sin y cos y + 2 a 2 sin 2 y 
 Whence E = 
 
 a 
 
 7= sinysin(y + 45), ................ (2) 
 
 V2 
 
 
 -v/2 sin y 
 
 jg 1 , a, and 8 are in circular measure, and if E^ a lt ^ be the corre- 
 sponding angles in degrees, we shall have 
 
 2 
 
 EI = ifo 5" sin y sin ( y + 45 ) 
 
 ^ = * V_sin(y+45). 
 
 1 1 80 2 <v/2 Sm 7 
 
 This is an error clearly small in quantity and of the second order. 
 If a be given we may determine the value of y for which E is a 
 maximum or minimum. 
 From (2) we have 
 
 E= - a ^ [cos45-cos(2 y + 45)]- 
 
 2V 2 
 
 For a given value of a, E will be a maximum when 
 cos (2 y + 45) = i, or when y = 67 J, 
 
 and a minimum when 
 
 cos (2 y + 45)= i, or when y=i57j; 
 
 that is to say, with a given error of adjustment a, the error in the 
 
General observations. 415 
 
 angle measured increases with the angle until that angle is 
 and diminishes until the angle is 157 J, &c., &c. 
 
 390. Before proceeding to measure any particular zones on 
 a crystal it is best to examine the crystal carefully and to assign 
 letters to prominent and distinctly recognisable faces, after having 
 made a few small free-hand sketches of it in its different aspects. 
 Thus, in subsequently noting the results, one has in the letters 
 assigned to these faces a means of provisionally designating the 
 various zones. It is well in entering measurements in a note- 
 book to head the page with this symbol of the zone, and to 
 make a sketch of each face measured, as seen in the micro- 
 telescope, noting any peculiarities it may exhibit. Opposite the 
 entry of the arc measured, an abbreviation (im., m. 1., &c.) should 
 indicate that the observation was made from an image, max- 
 imum light, striae, bands, a broad belt, &c. ; a good image being 
 indicated by one or by two straight lines scored under it, an 
 indifferent or bad one by a waved or more or less jagged line; 
 and similarly for the quality of the readings. Every fresh adjust- 
 ment of an edge should be noted in the margin. 
 
 Finally, a certain relative weight is assigned to the readings 
 for each edge in the zone while the observations are still fresh 
 and the crystal is under the eye and in adjustment, and an 
 average is taken of the different measurements thus weighted of 
 any symmetrically repeated edges. 
 
 It cannot however be too decidedly dwelt on that one fine 
 measurement with distinct images from the faces of a well-ad- 
 justed edge has an altogether different intrinsic value from that 
 of any average obtained from less reliable measurements made 
 on faces that, from whatever cause, give imperfect or confused 
 images. 
 
 In noting the nature of the observation, other terms than an 
 1 image ' have been alluded to ; and in fact one of the great 
 difficulties that baffles the crystallographer is the comparative 
 rarity with which he is able to record an observation from two 
 consecutive faces in a zone with the note that they gave perfect 
 images. Sometimes in lieu of a single image two or three are 
 seen, of which perhaps some lie slightly out of the zone to left 
 
4i 6 The reflection-goniometer. 
 
 or right, the images in fact overlapping, but with their centres 
 not on the vertical wire. And not rarely these take the form of 
 a long ribboned band made up of an indefinite number of such 
 images, images in fact reflected from a legion of striations, or 
 of the numerous pseudo-parallel tesserae of a tesselated face. 
 In other cases, again, they form a broad band of light, the result 
 either of the massing together of striae, in which case they may 
 be resolved by narrowing the slit, or else of the confused character 
 of the image reflected from a face which presents unevenness of 
 reflecting surface. 
 
 391. It is in dealing with such difficulties that the qualities 
 of the good observer come into play. 
 
 A whole zone should be considered in determining which images 
 may indicate the faces that are most to be relied on for giving direc- 
 tion to the zone-axis on the goniometer. Sometimes one half of the 
 zone can be determined consistently from one group of images, 
 while no good images can be found from faces parallel to them 
 in the other half of the zone ; and in this other half again a 
 different set of images may give results concordant with the 
 former, although representing faces which are slightly inclined 
 to those which gave the former images ; indicating in fact a 
 small inclination of the two hemi-zone axes on each other. This 
 is sometimes traceable to a twinning of two forms, but oftener to 
 a composite structure in which the numerous crystal-individuals are 
 in approximately but not completely parallel union. 
 
 392. When there are numerous separate images it may be 
 hopeless to identify those belonging to two faces which corre- 
 spond to each other. Sometimes however by patience and 
 scrutiny such correspondence can be found, evidenced by the 
 repetition of an identical angle measured over other symmetrically 
 repeated edges in the zone; and such corresponding striae are 
 by no means as a rule those of equal brightness or breadth. 
 They are however generally to be found in the same zone except 
 in the case alluded to in the last article. 
 
 But often we get no image at all from a face, and are driven 
 to the unsatisfactory expedient of estimating the position in which 
 the face under measurement gives in the micro-telescope the 
 
Crystallographic calculation. 417 
 
 brightest illumination. For this purpose the slit should be narrowed 
 to the smallest width that will give any appreciable illumination of 
 the face : and in measuring from a face that gives an image to 
 one that does not, this method of maximum illumination should 
 be employed for both faces. In every case the measurement 
 should be repeated several times with the object of obtaining 
 an average result; and, even in determining an angle where the 
 faces both give perfect images, it is always well to take the mean 
 of three measurements on successive portions of the graduated 
 circle. 
 
 SECTION II. Crystallographic Calculation. 
 
 General Problems. 
 
 393. From the instrumental means of measuring the angles 
 between the faces of a crystal, we pass to the consideration of 
 the Crystallographic problems by which the results of these 
 measurements are to be interpreted. 
 
 These problems are of two kinds; involving, first, the deter- 
 mination of the elements of the crystal and there with its symmetry; 
 and secondly, the assignment to each form and face of the crystal 
 of its proper symbol and position. 
 
 In proportion as the data obtained by the aid of the goniometer 
 are exact and numerous are these problems simplified ; and where, 
 from the imperfection or the absence of faces, the observations 
 taken are few or are wanting in precision, the results arrived at are 
 necessarily incomplete. 
 
 The crystals which belong to the more complete types of 
 symmetry are so far the more simply dealt with that the elements 
 to be determined are few, there being none in the case of the 
 Cubic system, while from the symmetrical recurrence of the same 
 angles the opportunities of obtaining good measurements of any 
 angle are multiplied. Crystals presenting a lower order of sym- 
 metry require a greater number of independent angles to be 
 measured, at the same time that the calculations become more 
 complex, involving the more frequent determination of oblique 
 triangles and purely trigonometrical processes. 
 
 We have then in the investigation of a crystal to employ 
 
 E e 
 
4i 8 Crystallographic calculation. 
 
 methods which deal with the general relations of the planes of 
 a crystalloid system to each other and to the axial planes, 
 combined with special methods which belong especially to each 
 particular type of symmetry. Those of the former class have 
 been already to a large extent considered : they fall under the 
 following heads : 
 
 (1) The general equations (A) connecting the symbols of the 
 different faces belonging to a crystalloid system (Art. 17). 
 
 (2) The expressions connecting the symbol of a zone with 
 those of faces belonging to it and of a face with those of zones 
 containing it. 
 
 (3) The relations connecting three faces in a zone. 
 
 (4) The problems relating to four tautozonal faces and their 
 anharmonic ratios. 
 
 (5) The expressions involved in the transformation of an axial 
 system . 
 
 (6) Those dealing with the faces of twinned crystals. 
 
 And certain other general problems 
 still remain to be considered before we 
 pass to the discussion of problems 
 peculiar to each different system. 
 
 394. In the first of these general 
 problems it is sought to represent the 
 angles between the normal of a face 
 and the several axes as a general func- 
 tion of the parameters of the system, 
 the indices of the face, and of the 
 angles between the axes ; or in the 
 equivalent form, as a function of the 
 
 parameters and indices, and of the angles between the normals of 
 
 the axial-planes. 
 
 I. To establish the relation connecting the arcs PX, PF, PZ, 
 
 rz, zx, xr (Fig. 350 ). 
 
 YPZ = 2V-FPX-XPZ, 
 
 cosFPZ = cos(FPX + XPZ), 
 and = cos YPX cos XPZ- sin YPX sin XPZ, 
 
 (cosrPZ-cosFPX cos XPZ) 2 = (i-cos 2 rPX) (i - cos 2 XPZ). 
 
General problems. 4 1 9 
 
 Whence 
 
 i = cos 2 YPZ + cos 2 ZPX + cos 2 XP Y- 2 cos YPZ cos ZPX cos XP Y, (i) 
 a condition which holds true whatever the position of P. 
 
 Also we have 
 
 vn cos f - cos /T'cos PZ 
 
 cosFPZ = \ prr . , 
 
 smPYsmPZ 
 
 and the other expressions symmetrical therewith. 
 
 Substituting these values in (i), transforming the sines into 
 cosines, and reducing, we get for the required relation, 
 i cos 2 cos 2 77 cos 2 + 2 cos COST; cos 
 
 = cos 2 />Jr+cos 2 />F+cos 2 />Z-cos 2 cos 2 PX- cos 2 77 cos 2 P Y- cos 2 C cos 2 /"Z 
 - 2 (cos cos P Y cos />Z + cos 77 cos />Z cos PX + cos C cos PX cos P F) 
 
 an expression which is symmetrical in respect to PX, PY : PZ, 
 and also to f, r?, . 
 
 395. To express cos PX, cos PY, and cos PZ in terms of h, /, I, 
 
 From the general equations 
 
 b PV c 
 
 ~ ~k cos ~ 7 
 
 we have 
 
 h k I 
 
 a b c 
 
 Substituting these values in (ii) we get, to determine /, 
 
 i cos 2 f cos 2 77 cos 2 + 2 cos f cos 17 cos f 
 
 i 2 T] H ^ sin 2 f 2 - (cos f cos r] cos f ) 
 2 (COST/ cos f cos f ) 2 (cos f cos f COST?) >; (iii) 
 
 whence, if /$, ^, /, f , TJ, f, fl, 3, <: be given, / is known, and there- 
 fore also cosPX, cosPF, cosPZ. 
 
 If A, B, C, Fig. 351, be the poles of the axial planes YZ, ZX, 
 XY, and therefore ABC, XYZ be polar triangles, we have, if BC, 
 CA, AB be a, /3, y, respectively, 
 
 a = TT JY", /3 = TTY, y = irZ, 
 
 ACT) /"* /* 
 
 -4 = 7T , JD = 7T ?/, C = 77 (,. 
 
 E 6 2 
 
420 Crystallographic calculation. 
 
 cos X + cos Y cos Z 
 Since cos f = - : ^r^ ^ 
 sin J^ sm Z 
 
 i - cos 2 J - cos 2 Y cos 2 Z- 2 cos .Y cos .F cos Z 
 sm * = - o T ^ ;~T ^ 
 
 sm 2 Y sin" Z 
 
 Let 4 A' 2 = i cos 2 X cos 2 Y cos 2 Z 2 cos X cos .F cos Z, 
 
 = i cos 2 a cos 2 /3 cos 2 y + 2 cos a cos/3 cosy; 
 then 
 
 f- 2N 2N 
 
 ~ sin J^sin Z ~" sin /3 sin y 
 
 2 TV 2 A" 
 
 Similarly, sin 17 = -^ , sin f = - 
 sin y sin a 
 
 X 
 
 Fig. 351. Fig- 35 2 - 
 
 4 A 72 cos a 
 
 Also cos cos -T] cos = sm 77 sm cos A = r^- 
 
 sin 2 a sin (3 sin y 
 and other expressions symmetrical therewith. 
 
 Substituting in (iii) we get 
 ., _ T cos 2 A cos 2 B cos 2 C 2 cos ^4 cos ^ cos C 
 
 ~~ 
 
 
 a 2 sin 2 /3 sin 2 y be sin 2 a sin sin y 
 
 _ (r cosM cos 2 ^ cos 2 C 2cos^4 cos,5'cosC)sin 2 asin 2 ^sin 2 y _ ,. 
 
 !ffi 2 kl i 
 
 2 sin 2 ^ + 2 cos a sin /3 sin y > 
 a* be } 
 
 where the series indicated by 2 sin 2 # is 
 
 / 2 
 
 -fiin z fl4- 
 
 3 2 
 
 sin 2 a + sin 2 /3 + sin 2 y ; 
 
General problems. 421 
 
 2kl 
 
 and that by 2 -= cos a sin /3 sin y is 
 
 2kl 2lk 2kk 
 
 -= cos a sin /3 sin y -\ cos /3 sm y sin a H cos y sin a sin (3. 
 
 be ca ab 
 
 But 
 
 ! _ CO sM cos 2 ^ cos 2 C 2 cos^t cos -5 cos C = sin 2 a sin 2 ^ sin 2 C ; 
 
 and since 
 
 4 ^ 2 4 ^ 2 
 
 siir/? = 5 and sin-C = . 
 
 sm 2 y sm- a sm- a sin 2 /3 
 
 4^V 2 
 
 we have /> = r- 2 TJ 
 
 2 sin 2 a + 2 cosa sin/3 siny 
 a 2 be 
 
 Substituting in the original expression, we have 
 
 -" Vs^i 
 
 kl . _ . 
 
 i cos a sm p sm y 
 
 / _ 4^; 
 * V 2 ^ sin2a + 2 ^ 
 
 " 1 
 
 cos a sin /3 sin y 
 
 /Z . o -2KL . . 
 
 -5 sm^ a + 2/ - cos a sm p sin y 
 a" 1 be ) 
 
 396. In the second problem we have to find the angle between any 
 two normals of a crystal in terms of the parameters, the normal-angles 
 of the axial planes, and the indices of the two faces. 
 
 If P (h k /) and P' (Ji k' I') be two poles, the axial points being 
 as before X, Y, Z (Fig. 352) ; then 
 
 XPY= XPP f - 
 
 and cos XPY = cos XPP' cos P f Pr+ sin XPP f sin P'PY. . . ( vi) 
 
 cos C cos PX cos PY 
 But also cos*/^: , mPXsinPr ' ........ (-) 
 
 In the triangles XPP', YP*P, 
 
 cosP'X-cosPXcosPP' 
 
 COS XPP = - - ; - . - - - 5 
 
422 Crystallographic calculation. 
 
 V sin 2 PX sin 2 PP' - (cos P f X - cos PX cos PPJ 
 = - - : nv , ^-^ 
 
 sm PX sin PP 
 
 cosP'r-cosPrcosPP* 
 
 COS YPP = - : - ptr . - 5^7 -- > 
 
 smPFsmPP 
 
 2 TV"- (cos P'Y- cos P.F 
 
 sin PY sin /Y* 
 
 By substituting these values and that in equation (vii) in equa 
 tion (vi), we obtain a quadratic equation in cos PP', the co- 
 efficients of which are the cosines and squares of the cosines of 
 PX, PY, P'X, P'Y and f . This equation may be solved in the 
 ordinary way : the solution, however, is more readily arrived at by 
 performing operations identical with the above for the triangles 
 XPZ, XP'Z, whereby a second quadratic equation in cos PP f is 
 obtained, having for coefficients the cosines and squares of cosines 
 of PX, PZ, P'X, P'Z and r? ; from the two quadratics the term 
 cos 2 PP f may be eliminated, and there will remain a simple equa- 
 tion giving cos PP' in terms of the cosines and squares of cosines 
 of PX, PY, PZ, P'X, P'Y, FZ, <n and f . 
 
 Also, since the arcs f, 77, f represent the angles of inclination 
 of the axes, and the arcs a, /3, y are supplementary to those 
 representing the inclinations of the axial planes, we have the 
 equations 
 
 , cos /3 cos y cos a cos 77 cos f cos f 
 
 cosf = - : - - ; cos a = - : - : -z - 
 sin /3 sin y sin 77 sin f 
 
 cos y cos a cos B cos C cos f cos 17 
 
 COST? = -- : - : - I COS p = - : z : ^ - 
 
 sin y sin a sin sin 
 
 cos a cos /3 cosy cos COST? cos C 
 
 COSC= - : - 47: - -; COSy= - ^ - pr-^ - - 
 
 sin a sin p sm sin 77 
 
 If now in the above simple equation in cos PP / , the values of 
 cosPX, cos /IF", cosPZ, cos P'X, cosP'F, cos P'Z given in equa- 
 tions (v) and the values of cos 77 and cos f given in equations (viii), 
 are substituted, we ultimately obtain on reducing the expressions 
 the equation 
 
General problems. 423 
 
 r>p 
 
 g sin/3 siny 
 
 " v y tan a 
 
 
 / 2 fcsin 
 
 a V7,7 fl 
 
 / VA/2 ^sina o2/-7 x 
 
 a 
 tana 
 
 ^ 0sin/3siny tana'\ 
 in which 
 v fc-sina , fcsina ( AJ/ 
 
 a sin j8 siny 
 f a sin /3 ,,, <^3 sin y 
 
 2/ nn ; - ; fin 
 a sm p sin y 
 
 v^ 2 ^sina __^ 2 
 
 ^ sin p sin y 
 
 ^r sin a 
 
 _i_ ^-2 
 
 <5 sin y sin a <: sin a sin 
 <:# sin (B ^ n ab sin y 
 
 - ) 
 
 a sin (3 sin y 
 fcsin* 
 
 ^ sin /3 sin y <5 
 
 - 1 Z>'2 
 
 sin y sin a ^ sin a sin /3 ' 
 
 r# sin ^3 7/ <2(5 sin y 
 
 - +/ = ; 
 
 g sin /3 sin y ^ sin /3 sin y 
 
 tan a tan a tan /3 tan y 
 
 tan a tan a tan /3 tan y 
 
 The equation may also be put into the form 
 
 ^hh' sin 2 f- 2 (&T + /^) a (cos - cos TJ cos C) 
 cos^= ^ - ^^^^ - , Ox.) 
 
 in which 
 
 S=?.tf sin 2 22 Ida (cos -cos 17 cos Q 
 
 y = 2^ 2 sin 2 -22 If I a (cos - cos ij cos Q- 
 
 
 397. The equations (ix) or (ix a] represent in the most general 
 form, though not a form directly adapted for logarithmic com- 
 putation, the arc-distance of any two poles whatever of a crystal ; 
 and in the most general case, that namely of the Anorthic system, 
 in order to determine the five elements of a crystal there are 
 required five such equations connecting the elements with the 
 indices of the faces between the poles of which five independent 
 arc-distances have been measured, the symbols of the faces being 
 of course supposed to be known. 
 
424 Crystallographic calculation. 
 
 The use of spherical trigonometry, however, combined with the 
 application of the rules of zones and the methods that will be 
 presented in the discussion of the various systems, render the 
 problem of finding the elements of an anorthic crystal less 
 laborious than it would be if the only method adopted were 
 that of dealing with a series of such equations. Where however 
 the data obtained by the goniometer are not sufficiently exact, 
 and the determination of the elements from these data with the 
 closest attainable precision has to be sought in the employment 
 of the method of least squares, the application of the equation 
 (ix) to the determination of a large series of angles will be 
 found the most direct, since, in the form in which it is here 
 given, the aid of logarithms in determining the recurring constants 
 can be advantageously brought into the calculations of these 
 angles. 
 
 For other systems than the Anorthic the general equation (ix) 
 assumes forms which become simpler in proportion as the arcs 
 a, /3, y become two or three of them quadrants, and as the para- 
 metral ratios become reduced to a single ratio or to equality. 
 
 Thus, in the Mono-symmetric system, where the arcs a and y 
 are quadrants and the arc (3 is the arc-distance A C of the poles 
 100, ooi, the equation is 
 
 +(hl'+ lh'} b cos/3 
 
 + &c. 
 
 where the expression under the second square root is identical 
 with that under the first, but with h'k't substituted for hkl. 
 In the Ortho-rhombic system the equation becomes 
 
 ''V + ^ 2 7 +//2 7 
 
 and in the Tetragonal and Cubic systems the expression for cosPP' 
 
General problems. 425 
 
 becomes further simplified as a, &, and c are two of them or are all 
 three of them equal. 
 
 398. It is often necessary in crystallographic problems to 
 assign symbols to particular faces of a crystal the relative positions 
 of which have been already determined by measurement and 
 projection, and to proceed to deal with the remaining faces or 
 with the determination of the crystallographic elements by methods 
 which involve the symbols and positions established for these 
 particular faces. For this purpose certain definitions and pro- 
 positions are of general application, and receive special forms in 
 the different systems. 
 
 399. A pole will be spoken of as being given in position 
 when its arc-distance from a certain given pole has been ascer- 
 tained, together with the angle contained by two zone-circles, 
 of which one traverses the two poles and the other passes through 
 the given pole, but is only known in symbol; and where this 
 angle is not a right angle it may always be taken as the acute 
 angle in which the zones meet, care being had as to the positive 
 or negative character of the angles as ascertained for different 
 faces referred to a common face and 
 
 zone; positive angles being always 
 measured in one direction from the 
 zone of reference. 
 
 In discussing each system methods 
 will be sought for finding the symbol 
 of a face the pole of which is given 
 in position in regard to some deter- 
 mined pole and zone -circle of the 
 crystal. The following propositions will 
 therefore deal with the general methods Fig. 353. 
 
 for determining the position of a pole 
 
 or poles : the elements of the crystal are in each case supposed 
 to have been previously determined. 
 
 I. Given the elements of a crystal and the symbols of two of 
 its poles R and S; given, further, the symbols of two poles P 
 and P' : it is required to find the magnitude of the arc PP'. 
 
 Instead of calculating PP f directly by means of equation (ix), it 
 
426 Crystallographic calculation. 
 
 will often be more convenient to calculate, by means of that equa- 
 tion or otherwise, the magnitudes of RP, PS, SR, RP', P'S, 
 Fi g- 353 and thence to deduce the angles PRS, P'XS, and the 
 arc Pf by means of spherical trigonometry. 
 
 In any system but the Anorthic a pole can be found to represent 
 R, which is also the pole of a zone-circle identical with it in the 
 indices of its symbol. The zone-circle [/'/''] will in such case 
 intersect this zone-circle say in a pole Q so that, taking R 
 and \RQ\ as the pole and zone-circle of reference for deter- 
 mining the positions of P and P f 1 the arcs PQ and P' Q are each 
 a side of a triangle, of which another side RQ is a quadrant, and 
 the arc PP f is the sum or the difference of these two arcs. 
 
 II. T and T' being two poles of which the symbols are known, 
 R, \RS~\ being respectively a pole and zone given in symbols 
 and position; let P be a pole of which the position relative to 
 
 W 
 
 the pole R and the zone \RS\ is to be determined, the arcs PT 
 and PT' being the data as regards P. 
 
 Here (Fig. 354) RT, SRT t RT', SRT' t and TT f are to be de- 
 termined from the symbols of T and T f , and the known elements 
 of the crystal by means of equation (ix) or otherwise, and by 
 spherical trigonometry RTF, RT'T, PTT', PT'T, and so RTP 
 can be computed, and therefore also the arc RP and the angle 
 TRP in the triangle RPT. Hence SRP and the arc RP which 
 give the position of P are known. 
 
Anorthic system. 427 
 
 The symbol of P can now be determined by the methods 
 special to the system. 
 
 III. Let T be a pole of which the symbol is known, and let 
 the arc-distance of T from a pole P be given : the symbol of 
 [uvw] a zone traversing P is given, R and RW being given 
 as the pole and zone of reference. The position of P is required 
 with a view to the determination of its symbol. 
 
 Let [uvw] intersect the zone-circles [RT] in Q and \RW} 
 in (Fig. 355). The symbols of Q, R, S and T 7 being known, the 
 arcs RS, RQ, QT&nd TS can be found from equation (ix), and 
 then the angles SRT and RQS or PQT can be calculated by 
 spherical trigonometry. In the triangle PQT the arcs PT, TQ 
 and the angle PQT being known, the angle QTP or RTP can 
 be calculated: and finally, from the arcs RT, TP and the angle 
 RTP of the triangle RTP, the arc RP and the angle PRT czn 
 be found. The position of P being thus established in reference 
 to R and [R W\ its symbol may be determined by the methods 
 special to the system. 
 
 SECTION III. Crystallographic Calculation. 
 The Oblique Systems. 
 I. The Anorthic System. 
 
 400. In the Anorthic system the five axial elements have to 
 be determined irrespectively of symmetry, for the only symmetry 
 in a crystal belonging to this system is that to a centre. 
 
 A, B, C, Fig. 356, being the positive poles of the axial planes 
 YZ, ZX, XY } their symbols are 100, oio, ooi, respectively; and 
 the arcs a, /3, y, or BC, CA, AB, form a triangle polar to that 
 formed by the arcs f, rj, f, or YZ, ZX, XY, traversing the axial 
 points X, Y and Z, arcs which measure the axial angles of the 
 crystal. Thus the angle 
 
 A = i8o- B = i8o-Tf, C = i8o-C; (i) 
 
 and for the sides of the triangles XYZ and ABC we have, from 
 spherical trigonometry, 
 
428 
 
 Crystallographic calculation. 
 
 cot 
 
 f /sin(S- 
 
 - = A / -- ^ o 
 2 V sin o 
 
 o /o \ 
 in o sin (6 a) 
 
 ?] /sin ( y) sin (S a) 
 
 2 = V sinSsin(S-/3) 
 
 where 
 and 
 
 C / sin (S- a) sin (S- 13) 
 
 COt = A / - ; - p; : 77^ 
 
 2 V sin o sin (o 
 
 a _ /sin ( 
 
 2 "" A/ ' si 
 
 y) 
 
 2 -77) sin (2-C) 
 
 sin 2 sin (2 
 fl /sin (2-Q sin (S 
 
 COL = A / ; ^ : 7^ 
 
 2 V sm-lsin(2/ 
 
 i,) 
 
 cot- = 
 
 where 
 
 401. Let P, Fig. 356, be the pole of hkl, and let the zone- 
 circles [AP] t [JBP], [CP] intersect [JBC], [CA], [AJB] in the 
 
 Fig. 356. 
 
 poles H, K, L. Then H is o k /, ^T is ^ o /, and Z is ^ /^ o, and 
 from the fundamental equation, 
 
 -r T _ - TT 
 
 r cos LY - cos LX, 
 k n 
 
 cos XX = cos JCZ. 
 
Anorthic system. 429 
 
 Further, from the triangles HAY, HAZ, since 
 
 AY = AZ = - = CY = CAY = BZ = BAZ, 
 
 2 
 
 we have 
 
 cos HY = sin HA cos HAY = sin HA sin JZ4 C, 
 cos HZ = sin JK4 cos HAZ = sin Zf,4 sin HAB ; 
 
 and by substitution from these and similar expressions in the 
 
 preceding equations, 
 
 b I _ sin HAB < c h _s\nKBC ^ a k _ sinLCA 
 7~k~ smHAC' ~a 1 ~~ sin A"A4 ' ^"sirTZOZT ^ m 
 
 For logarithmic computation these expressions may be put into 
 the form 
 
 tan (HAB- \ CAB] = tan | CAB tan (45- 0), ^ 
 
 tan - 
 
 (ivfl) 
 
 al b h 
 
 - - ? tan \1/ = - - ; 
 c h a k' 
 
 or also into the equivalent form 
 
 c k a I b h 
 
 where tan Q - - , tan d> = - - > tan \lc = - T ; 
 ^ / c h a k' 
 
 The arcs HB, KC, LA can now be readily determined. For 
 H t for instance, we know the angles NBA and HAB and the 
 arc AB] and similarly for KC and LA. 
 
 402. From the six triangles into which the triangle ABC is 
 divided by the arcs APff, BPK, CPL, we obtain the equations 
 
 sin AB sin BAP = sin BH sin BHP, 
 sin C^4 sin CAP sin C# sin CHP, 
 sinC sin C^P = sin CK sin C^P, 
 
 sin^4^P = sin^f^ smAKP, 
 
 smACP = s'mAL smALP, 
 
 smBCP sin^Z sin^ZP; 
 where 
 
 , smALP=sinLP. 
 
430 
 
 Crystallographic calculation. 
 
 Hence, from equations (iii), 
 
 sin AB sin CH sin CAP c k 
 
 b /' 
 
 sin AB sin CK 
 sin CA sin #Z 
 
 sn 
 
 sin BCP 
 
 sin 
 
 sin 
 
 (v) 
 
 Fig- 357- 
 
 and smIfsmCJsmAL = s'mCH sinAK smL; .. .. (vi) 
 
 an equation which is true for every 
 point P of the sphere, and is there- 
 fore true for every pole P of the 
 crystal. 
 
 If any five of the arcs it involves 
 are given, the sixth is known. 
 
 403. Let O be the parametral 
 face in, and the poles correspond- 
 ing to H, K, L be D, E, F- } so 
 that (Fig. 357), 
 
 D is on, Z'is 101, and Fis no. 
 The axial angles may be obtained 
 in terms of the arcs AB,BC, CA by means of equations (iio;), and 
 the parametral ratios in terms of the arcs AB, BC, CA, and of 
 the segments of these arcs as formed by the poles D, Z, F, by 
 means of equations (v). Thus 
 
 a _ sin BC smAF _ smACO , 
 
 6 ~ sin CA sin BF ~~ sin BCO 
 
 b sin CA sin BD sin 
 
 c sin AB sin CD 
 
 c _ sin AB sin CE 
 a ~ 
 
 The segmental arcs AF, FB, BD, DC, CE, EA may thus be 
 taken as the arc-elements of the crystal, any five of them serving 
 to give the five axial elements, independently of the sixth segment. 
 404. If the two segments of one of the three arcs a, /3, y are 
 unknown, and the other four segments are given, the two unknown 
 
 sin CBO 
 
Anorthic system. 431 
 
 segments can be readily found; for, from (vi), if the segments of 
 AB and CA are given, 
 
 sin BH sin AK sin BL 
 sin CH ~ sin CK sin AL 
 
 which may be written as 
 
 whence 
 
 = tan i ,(7 tan (4 5 -0); 
 
 an equation which gives CH, and therefore BH, since BC is 
 known. 
 
 405. The position of a pole H or (ok I), K or (hot], L or (hko) 
 in a zone containing two of the poles A, B, C is determinable by 
 the equations (v), supposing the elements of the crystal and the 
 symbol of the pole to be given. And if the position of a pole on 
 one of these zone-circles is known, the same equations give the 
 symbol of the pole. 
 
 When the elements of a crystal are given, and it is required 
 to determine the position of a pole P of known symbol not lying 
 on one of the arcs a, /3, y, instead of making use of the general 
 equation (ix) of Art. 396 it will usually be more convenient 
 to project the great circles APH, BPK, CPL, and find the 
 arc-distances from P of the poles A, B, C, and the angles 
 which the zone-circles PA, PB, PC make with the arcs a, /3, y. 
 These values are in fact deducible by spherical trigonometry 
 from any two of the angles PAB, PBC, PC A, or any two 
 of the arcs PA, PB, PC, or from one of the angles together 
 with the corresponding arc ; so that if any pair of these data be 
 given, the position of the pole is determined : but the equations 
 (iv<5), and the values assigned in them to tan 0, tan (/>, and tan a//-, 
 enable us to determine the angles PAB, PBC, PCA at once 
 from the symbol of the pole. Hence when this symbol can be 
 assigned in consequence of the known distances of the pole from 
 each of three other poles of known symbols in one zone-circle, 
 or from other data, the position of the pole in question relative 
 to the poles A, B, C can be found by any two of the equations 
 
432 Crystallographic calculation. 
 
 (iv). The inverse problem of finding the symbol of a pole 
 given in position, the elements of the crystal being known, differs 
 from the last in that here the position of the pole may be given 
 in reference to some two poles or to a pole and zone of the 
 crystal not belonging to the elementary triangle ABC, though 
 known themselves in position relatively to the sides and angles 
 of that triangle. This in fact is the problem for finding the 
 position of a pole in respect to a given pole and zone-circle 
 which has already been considered in Article 399. 
 
 Taking in place of the pole R and zone of reference [^S'], 
 in Article 399, a pole and a side of the elementary triangle, for 
 instance C and [^C*]; and, in place of T and T', or T and 
 [uv w], taking the poles or pole and zone to which the position 
 of P has been referred, we have to proceed, as in that article, 
 to find the position of P in respect to all the poles and zone- 
 circles of the elementary triangle. And the symbol of P can then 
 be calculated by means of equations (v). 
 
 406. To find the arc-distance of two poles with known symbols, the 
 elements of the crystal being given. 
 
 Instead of calculating 
 the arc-distance by help of 
 the general equation (ix) 
 of Art. 396, we may pro- 
 ceed thus : 
 
 The poles P and P' can 
 be severally determined in 
 position in regard to a 
 pole and adjoining side of 
 the elementary triangle, by 
 equations (iv#). 
 
 Thus, for instance, PAC 
 and ^^C as well as PA 
 
 and P'A being known, the arc PP f is found by spherical trigo- 
 nometry. 
 
 Or, also, Fig. 358, [-W] intersects the zone-circles a, /3, y in 
 poles S, T, U to which the symbols can be assigned. The arcs 
 SC, S3, TA, TC, UB, UA can be computed by equations (v) : 
 
Anorthic system. 
 
 433 
 
 and the arcs ST'and TU from any two of the triangles STC, TA U, 
 SBU. We have then two groups of four tautozonal faces UTPS 
 and UTP'S from which the arcs UP, UP' and therefore PP r can 
 be determined : see also Art. 408. 
 
 407. In the previous articles the position of a pole has been 
 determined relatively to a pole and side of the elemental triangle, 
 the elements of the crystal being given. The problem however of 
 determining the position of a pole 
 when its symbol is given, or of 
 finding its symbol when its posi- 
 tion is given relatively to four 
 other known poles, may be put 
 into a completely general form. 
 
 Let U, V, PFand T(Fig. 359) be 
 four heterozonal poles, the sym- 
 bols of the zones [VW], [WU], 
 
 wj 
 
 Fig. 359- 
 
 \UV\ being [u^wj, [u 2 v 2 wj, 
 [u 3 v 3 w 3 ] severally, and (efg) 
 being the symbol of T. Then, by spherical trigonometry, if five of 
 the six arcs joining the four poles are given, the sixth can be found, 
 as well as the angles at U, V, W and the angles which [777], 
 [TV}, [TW] make with the arcs joining U Fand W. 
 Let P be a fifth pole with the symbol (hkl). Then, if 
 
 ~ sm(VUT-VUW) ' 
 
 sin UVT 
 
 sm(UVT-UVW)' 
 
 we have, by articles 48 and 51, 
 
 \an(VUP-\VUW) = tan J 
 
 whence the angles which \PU"\ and [PV] make with [^7K], and 
 so the arcs PU and PV, and if needed PW } can be found, and 
 the position of P in respect to any pole and adjacent arc of the 
 triangle UVWis determined. 
 
 408. If P' or tik't be a sixth pole of the system, it can also 
 be found in position by an identical method ; and the arc-distance 
 
 Ff 
 
434 Crystallographic calculation. 
 
 PP' between two poles of known symbol can then be directly 
 found by the methods of spherical trigonometry. 
 
 409. The symbol of a pole P can be found if its arc-distances 
 from four poles of given positions and symbols are known. For if 
 [UT] and [ VT], and \UP] and [ VP] (Fig. 360), intersect the zone- 
 circles [VW] and [UW] in poles R, R', Q, and Q respectively, 
 the positions of these poles on UW and VW can be determined 
 
 by spherical trigonometry, and there- 
 with the symbols of Q and (X, since 
 the anharmonic ratios of \U Q' R' W] 
 and [VQR W] are then known: the 
 symbol of P, in which the zones 
 [UQ] and [VQ'] of known symbols 
 are tautohedral, is then straightway 
 deduced. 
 
 41O. To determine the elements. 
 Since in the Anorthic system the 
 elements of a crystal are five in- 
 dependent quantities, the requisite data for determining them are 
 five measured angles between poles of which the symbols are 
 known but of which none are deducible from the rest. 
 
 For this purpose therefore three poles forming the angles of a 
 spherical triangle, together with a fourth pole heterozonal to them, 
 and therefore not on a great circle containing a side of the triangle, 
 are available ; since five out of the six angles they form are 
 quite independent : or, the three poles forming the angles and 
 two other poles each lying on a great circle containing a different 
 side of the triangle may be employed, the arcs joining all the 
 five poles being known. 
 
 If however the faces thus to be employed are initially to have 
 known symbols, it will need a preliminary investigation of the 
 crystal with a view to selecting faces to which it may be possible to 
 assign such symbols, and of which some at least may be employed 
 in determining the axial system, either by taking the axial planes 
 parallel to them or by treating them as parametral faces. 
 
 For such a selection, the measurement and projection of the 
 zones containing the faces which are the most prominent or 
 
Anorthic system. 435 
 
 which give the best images, will be undertaken first: and, after 
 independent symbols have been assigned to certain of the faces, 
 it is generally possible, by the rules of zones and of the tauto- 
 hedral law, or by the application of the problem of four planes, 
 to determine the symbols for other faces of which the poles are 
 projected. Usually this assignment of symbols leads directly to 
 the determination of certain arc-elements for the crystal. But 
 sometimes it is not possible to assign at once the poles most 
 suitable for the elementary triangle, or those which offer the best 
 data for determining the elements ; in which case these elementary 
 planes have, in the first place, to be, as it were, tentatively selected 
 and, even then perhaps only approximately, determined. Never- 
 theless, when they have been so determined, the symbols assigned 
 by inspection and the application of the zone-laws to the different 
 poles of the crystal, or calculated for them by aid of the provisional 
 elements, will generally have validity; and ultimately, if a different 
 axial system be adopted, the application of the formulae for the 
 transformation of all or of part of the elements of an axial system 
 will readily convert the symbols of these faces to the new symbols 
 representing them in the transformed system. 
 
 If however the elementary triangle and the arc-elements cannot 
 all, or some of them, be thus directly determined, they may be 
 deduced from a series of measurements between faces heterozonal 
 to the elementary triangle, to which the symbols have been 
 tentatively assigned consistently with the zone-laws. It is how- 
 ever a case of most improbable occurrence that no one of the zones 
 a, /3, y, and at the same time no one of the poles A, JB, C, can 
 be identified with zones or faces determinable by the goniometer. 
 
 The simplest method to be followed in such case is that 
 indicated in Articles 407-8, the expressions in which are inde- 
 pendent of the elements of the crystal. 
 
 Thus, where five of the six arcs joining the four poles U, V, 
 W, T having known symbols are given by measurement, but none 
 of these poles are identified with A, B or C, the arc AC between 
 the two poles A, C having the known symbols 100, ooi can be 
 calculated ; and so on, for the remaining arc-elements. Usually 
 however V t or U and F, can be taken to represent one or two 
 
 Ff 2 
 
436 
 
 Crystallographic calculation. 
 
 of the poles A, B, C, and not unfrequently also one of the quasi- 
 octahedral parametral forms can be represented by one of the 
 remaining poles in the figure : and the process for determining the 
 elements is proportionately simplified. 
 
 The Oblique Systems. 
 
 II. The Mono-symmetric System. 
 
 411. In this system, the normal of the plane of symmetry is 
 taken for the axis Y. The axial points Z and X lie, as do the 
 poles C, ooi, and A, 100, of the axial planes XY and YZ, in 
 
 the plane of symmetry ZX, while 
 the pole B, or oio, of that plane 
 lies on the axis Y', so that in 
 trigonometrical problems recourse 
 can frequently be had to quad- 
 rantal and right-angled triangles. 
 The normal -angle AC, i.e. 
 (100.001), or /3, is the supple- 
 ment of the axial angle 17, and 
 AZ = CX = 90. 
 
 To determine the position on 
 the sphere of a pole P or hkl, 
 we have from the quadrantal 
 
 triangles PBZ and PBX (Fig. 
 Fig. 361. 
 
 X 
 
 cos PZ = sin BP cos PBZ = sin BP sin PBA, 
 
 cos PX = sin BP cos PBX sin BP sin (A C-PBA ), 
 
 ^cosPZ; 
 
 7 
 
 But - cos PX = - cos PY - 
 
 k k I 
 
 whence - sin (PBA 
 
 and : - 
 
 sin Jr^oA a I 
 
 Let be the pole of in and let zone-circles through BO 
 
Mono-symmetric system. 437 
 
 and BP meet the zone-circle [AC] in E and K respectively. 
 Then E is 101, K is h o /, OB A = AE, arid PBA = AK. 
 For 0, sin (r? -M^) = sin (i8o-^C) = sin EC, 
 and for P, s'm^ + AK) = sin(i8o-^C) = sin^C, 
 and equation (i) becomes 
 for O, a sin^C = I cot OB = c sin ^, \ 
 
 forP - iuKC-- oiPB-- ' AK\ ^ 
 
 h sin AE sin ^fC ^ sin yl^' tan -50 
 
 whence - = -^ = - - -^ and - = - 
 
 The arcs EC, AE, OB are parametral, and may be employed 
 as the arc-elements of the crystal in place of the axial elements 
 
 a c 
 
 yj T' and T). 
 
 o o 
 
 412. If 
 
 then tan (PBA + J r/) = tan J ry tan (135-^), 
 and 
 
 T y sin/^^c = T ysin.r^'^, ^..x 
 
 ^ <5 / f- .. (m) 
 
 cot 
 
 J 
 
 From the triangles PBA, PBC, 
 
 cos PA = sinPB cos PBA = sinPB cosAK, ^ 
 . cos PC = sinPB cos PBC = smPB cosKC, i .. ..(iv) 
 = sin PB cos (AK-}- 77). ) 
 
 413. The position of a pole K of a hemidome {hol\ is given 
 on the zone-circle [oio] by equations (iii). PB is a quadrant 
 since P coincides with K, 
 
 h c 
 
 where tan = - - ; 
 
 / a 
 
438 Crystallographic calculation. 
 
 or, where AK is known, the symbol is given by the expression 
 
 from (ii), 
 
 h a sinJfC s'mA-E sin 
 
 / c sin AK sin EC sin AK 
 
 414. For a face L of the prism [hko], P in the foregoing 
 formulae becomes Z and PBA vanishes, while 
 
 k a , k cot OB . 
 
 and cot LB - - sin A C - - =^ sin ^4 C, 
 
 h b h s 
 
 cosZC = sinZ.Z? 
 Also, if/ 7 is the pole of the parametral face no, 
 a h tan AL tan AF 
 
 
 b ~~k sin A C 
 h a s'mAC 
 
 k ~~ b tan AL t&nAL 
 415. For a face H of an ortho-dome {okl}, BH+HC 90, 
 
 ^ . 
 
 cotHB = -smAC= - 
 
 I b I smAJ? 
 
 cos HA = sin Z^5 cos A C, 
 
 . ., _ 
 
 and T = j- ^ , if D is on; 
 
 b 
 
 also we have 
 
 k _b cotBH _ 
 I c sin^4C 
 
 / _ tan CD 
 
 416. The poles of a form \hkl\ are symmetrical in pairs, where 
 the signs of their k indices are the same, to the axis [oio], and 
 where the signs of their h and / indices are the same, to the 
 systematic plane (oio). Hence if the symbol of P be h k I, the angle 
 between the faces 
 
 and the angle between the faces 
 hkl.hkl = 
 
Mono-symmetric system. 439 
 
 The equations ( iii ) may be written 
 
 h sin EC I sin AE 
 
 tan B0~~k 
 
 (v) 
 
 where A K and KC are obtained as in Article 413 ; the arcs PA 
 and PC being given by equation (iv). 
 
 417. If P' or h f K I' be a pole in a zone passing through (oio) 
 and P or (hkt), we have by equating values for cot BP and 
 cot BP', from the third equation in (iii), 
 
 h k' _ k' I ^ 
 ~h' ~k ~~ I?' 
 
 from which, P being given in position, the position of P' can 
 be found if its symbol is known, or its symbol can be found if its 
 distance from B is given. 
 
 418. If K and K' be any two poles in the zone [oio], we have, 
 from Article 413, 
 
 / sm(AC-AK) _ l f s' 
 
 i sin AX ~ 
 whence 
 
 7 7/ 77 / _ 7 7/ 
 
 {cotA-l 7 cotA' = i,, coUC; 
 n fi hh 
 
 an expression which is often convenient in computing such angles 
 when a table of natural cotangents can be employed. 
 
 For the parametral hemidome faces E, 101 and E' ', Toi it 
 becomes, if A' be Too, 
 
 cot AE- 2 cot AC = 
 
 and for a plane K^ h o / of a single hemidome form 
 
 cot AE - cQiAK = -^ cQiAC. 
 h h 
 
 419. The elements of the crystal being given and the symbols of two 
 poles ) to find the arc joining these poles. 
 
 P (hkt) and P' (h'k'T) being the two poles, their positions 
 may be determined in reference to B and \BA\ by the equations 
 (iii). Thus, PB, P'B, PBA, and P'BA are known, and the 
 triangle PBP f can be solved. 
 
440 
 
 Crystallographic calcidation. 
 
 420. To find the arc-elements of a mono- symmetric crystal. Since 
 the elements in this system involve three unknown magnitudes, 
 three equations of the form of equation (ix), Art. 396, would 
 be required in the most general case; that, namely, where the 
 poles, of which the arc-distances are given, are heterozonal and 
 have no zero in their indices. 
 
 In fact however the problem always presents itself in a more 
 simple form than this, since the preliminary investigation of the 
 crystal in deciding the character of the symmetry will have always 
 
 given measurements between poles 
 which, by their symmetrical repe- 
 tition, or by the position of one 
 or more of them in the plane of 
 symmetry, or, again, by the identi- 
 fication of one of them with the pole 
 of that plane, lead to the direct ap- 
 plication of the equations (ii) or (iii) 
 (Arts. 411-12). And where the ap- 
 plication is not so simple it is fre- 
 quently possible to obtain the arcs 
 between poles in the zone [100, 
 
 ooi] by aid of the 'problem of four planes' or of the expressions 
 in Article 396 ; and where the arc-distance of a pole from the pole 
 of symmetry oio is not obtained by direct measurement (as for 
 instance in the absence of this pole), it can frequently be deduced 
 ' from a measured angle between the normals of two symmetrically 
 repeated faces. 
 
 Let the distances of four poles p l , p 2 , ft , ft be given ; and let 
 Aft = Aft and A ft = Aft Fi g- 362, 
 Let [Aft] [Aft] intersect in S, 
 
 [A ft] [Aft] in ^> 
 [A A] [ft ft] in#, 
 and [XS] intersect [p L p 2 ] [ft ft] in K and Z. 
 Then \KS\ lies in a plane of symmetry to the crystal, and B is its 
 pole and is (oio). 
 
 Fig. 362. 
 
 Also 
 
 A K = i A A and ft z = 4 ft 
 
Rectangular-axed systems. 441 
 
 Project the poles on the zone-plane [^T-S 1 ]. BK and BL are 
 quadrants (Fig. 363) ; hence B p 1 and B q^ are known, and from 
 the triangle p l q^B, the angles p l J3q l (and therefore arc KL] and 
 Bp\ $1 ( Kp\ S) mav t> e computed. Hence also in the right- 
 angled triangle Kp^ S, KS can be determined. 
 
 Let p be in and q^ be Tn. Then K is 101, L is Yoi, R is 
 ooi, and S is 100 ; and from the four poles S, K, R, L known in 
 symbol and known in distance as regards SK and KL the angle 
 KR can be computed; whence the arc-elements SK t KR, and 
 
 Bpi are found. If the relative positions of p^ and ^ had led to 
 a different appropriation of symbols to those poles, the poles S, K, 
 R, and L would also have different symbols, but the method of 
 procedure would be the same. 
 
 For instance, if p l be assumed to be in and ^ 021, L becomes 
 ooi ; R, 203 ; K, 101 ; S, 201 ; and the distance of a pole A, 
 or 100, from K has to be found. 
 
 SECTION IV. Crystallographic Calculation. 
 The Rectangular-axed Systems. 
 
 421. The Ortho-symmetric or Ortho-rhombic system represents 
 the most general case of a crystallographic plane-system referred 
 to rectangular axes ; the case, namely, in which the parameters are 
 all different, and the three axes are axes of ortho-symmetry. 
 
 The three systems which have been referred in this treatise to a 
 rectangular axial system have the directions of their axes deter- 
 mined by considerations of physical as well as morphological 
 
442 
 
 Crystallographic calculation. 
 
 
 symmetry, though in no system are the three directions so 
 emphatically fixed by physical conditions as in the Ortho-rhombic 
 system ; since all the physical characters which distinguish a crystal 
 in this system are ortho-symmetrical to the three perpendicular 
 axial directions to which the forms of the crystal are referred, and 
 to these alone. 
 
 The characters which are common to the different systems of 
 
 planes referred to a rectangular 
 axial system may be considered 
 as introductory to those which 
 more particularly bear on the crys- 
 tallography of the Ortho-symmetric 
 system itself. 
 
 422. In a system in which the 
 faces of a crystal are referred to 
 three perpendicular axes the direc- 
 tions of the axes X, F, Z coincide 
 with those of the normals to the 
 axial planes, and the axial points 
 X, Y, Z fall on the poles A, B, C of the axial planes ; which 
 poles are therefore those of the faces belonging to the forms 
 {100}, {oio}, and {ooi}. 
 
 Hence the fundamental equation (A) becomes 
 
 (a] 
 
 From the triangles PCA, PCB, (Fig. 364) 
 cos PA = sin PC cos PC A, j 
 cos PB = sin PC sin PC A, $ 
 cos PC = sin PA cos PA C, 
 
 
 and 
 
 whence, 
 
 = ^smPCcosPCA, 
 ch. 
 
 - 
 ck 
 
 cot PC = cos PC A = sin PC A, 
 
 ch ck 
 
Rectangular-axed systems. 443 
 
 and similarly, we obtain 
 
 tan = 7k ' 
 
 and tan PBC = . 
 
 al 
 
 Squaring equations (3) and adding, 
 
 cos 2 PA + cos 2 PB = sin 2 PC = i - cos 2 PC, ) 
 and cos 2 PA + cos 2 P+ cos 2 PC = i. \"' 
 
 From (c), and substituting values from (3) and (a), we have 
 
 cot PC = ^ 
 
 and similarly, 
 
 = "~ 
 
 and cot = ~ 
 
 b 
 
 423. If ^", K, L be the poles ok I, hoi, hko they will be the 
 poles in which the zones AP, BP, CP intersect the arcs C, CA, 
 AB respectively; and being the pole in, and Z>, E, /'being 
 the poles on, 101, no in which the zone-circles OA, OB, OC 
 intersect the arcs BC, CA, AB, we have from (a) 
 
 abc 
 
 7^ ' 
 
 hbc 
 
 cos AP = sin PH = - 
 VS 
 
444 Crystallographic calculation. 
 
 For the parametral plane 0, 
 
 An nn be 
 cos AO = sin UD = ? \ 
 
 Vs 
 
 ca 
 
 ~7s 
 
 ab 
 
 cos CO = sin OF = 
 
 These equations may be put into the form 
 
 a:l\c h cosPPcosPC : k cosPC cosPA ilcosPA cosPP, (g) 
 = cos OB cos OC: cosOCcosOA: cos OA cos 03. (/) 
 
 424. If P and P' be the poles of two faces hkl and h'k'l', we 
 have, from the spherical triangle PCP r formed by the arcs CP, 
 CP', PP', and by substitution from (3) and from (/), 
 
 cos PP f cos PC cos P f C + sin PC sin P'C cos (P'CA - PC A ) 
 
 = cos PC cos P / C+ sin PC cos PC A sin P'C cos P'CA 
 
 + sin PC sin PC A sin P'C sin P'CA 
 
 = cos /M cos P'A + cos PP cos P'B + cos PC cos P'C 
 
 (h) 
 
 an equation identical with that to which the general equation (ix), 
 Article 396, becomes reduced by the condition of the axes being 
 perpendicular and the arcs a, /3, y, which also measure the angles 
 f, 77, f, being quadrants. The equation (h], though not directly 
 adapted for logarithmic computation, involves only terms which 
 are readily so computed. 
 
 The equations in the last article are evidently directly dedncible 
 from the above form (ti) of the general equation by substituting in 
 it the different values of the indices of P and P'. Evidently also 
 the following expressions will hold good : 
 
Rectangular-axed systems. 445 
 
 If P' coincide with in, i.e. with 0, P being any pole hkl, 
 the expression (ti) becomes 
 
 and if P' be one of the poles on, 101, or no, i.e. D, E, or F, 
 we have 
 
 COS P^ = 
 COS P/^ = 
 
 and if also P be one of the poles D, E, or F, 
 
 be 
 
 cos EF = 
 
 ab 
 
 425. Since, by equation (d\ in any system referred to rect- 
 angular axes, 
 
 PA =-c 
 
 -cos(Vc+P^)cos( J PC'-P^), [ ...... (A) 
 
 cos 2 PC = - cos (PA + P) cos (PA -PS};} 
 and since 
 
 a:d:c cosOjB cos OC : cos 6>C cos OA : cos (9^4 cos OB, 
 by substituting in any two of the last ratios, 
 
 for cos OA, </-cos(O+OC)cos(0-OC), 
 for cos OB, </-cos(OC+OA)cos(OC-OA), 
 for cos OC, V-cos (OA + OB) cos (OA - 0), 
 
446 Crystallographic calculation. 
 
 we can obtain the parametral ratios for a crystal whenever two of 
 the arcs OA, OB, OC have been determined. 
 
 Similarly, by substituting the square roots of the quantities in 
 (A) in the expression 
 a : b : c = h cos PB cos PC : k cos PC cos PA : I cos PA cos PB, 
 
 we can find the parametral ratios from two measured arcs between 
 P and A, B, or C, if the symbol of P be given. 
 
 And the parametral ratios can equally be found if a pole P 
 or hkl be given in position in reference to one of the poles 
 A, B, or C, and to an arc BC, CA or AB for instance if PC and 
 AL (L being hko) are given since, by equations (b}, the arc- 
 distance of P from all the poles A, B, and C can then be de- 
 termined. 
 
 So if two of the arcs PH, PK, PL be given, since they are the 
 complements of the arcs PA, PB, PC, the parameters may be 
 found. 
 
 The Rectangular-axed Systems. 
 
 I. The Ortho-rhombic System. 
 
 426. In considering the application of these general results to 
 the particular problems arising in the investigation of an ortho- 
 rhombic crystal, it will be seen that the equations (c] give the 
 position of any pole in respect to the pole C and the arc CA , 
 when the elements of the crystal and the symbol of P are known ; 
 and that by employing corresponding expressions the position of 
 the pole may be determined in regard to A and \AB\ or to B 
 and \BC\ 
 
 427. Prism- and dome-forms. For a pole H belonging to the 
 proto-dome {okl} the angle PAB becomes HAB or HB\ hence 
 from equation (c] of Art. 422 
 
 tan^^=-|. . . .. (i) 
 
 c k 
 
 Similarly, if K and L belong respectively to the deutero-dome 
 {hoi} and the prism {hko} we have 
 
 c h . ,. a k /-'\ 
 
 - T5 tan^4Z = TT ( n ) 
 
 a I b h 
 
Ortho-rhombic system. 447 
 
 428. being as before a pole of the parametral scalene 
 octahedron {m}, and D, E, F adjacent poles of the parametral 
 domes and prism {on}, {101}, {no}, we have 
 
 cosOA cos OB 
 cosOF= - =-:- = =-:-; .......... (ma) 
 
 cos FA sin. FA 
 
 hence 
 
 cos OB a 
 
 
 = tan FA - from equations (n). 
 
 cos OA b 
 
 Similar!y 
 
 cos 
 
 also fan. FA tznDB fan EC = i. 
 
 Any two of the arcs FA, DB, EC may thus be taken to determine 
 the parameters and serve for the arc-elements of the crystal. 
 Further, from (i) and (ii), 
 
 , ?, tan CK = - fan EC. (iv] 
 
 ,,. k I ^ ' 
 
 429. The parameters of the crystal being given, let P and P f 
 be the poles of two faces (hkl) and (h'tff). 
 
 If now the zone-circle [./V] traverse one of the poles A, 
 B, or C and let it for example pass through C then, from 
 equation (c), 
 
 cot CP _ lh' _ Ik' 
 hl'~ kl'' 
 
 and similar expressions hold where A or B lie in the zone 
 Whence 
 
 /IN r r^ DD /i h'^AP k' l f 
 
 (1) for a zone \APP 1, - - - = =:-- 
 
 k 
 
 1D/1 tffan.BP' I' 
 
 (2) for a zone \BPP>\ 
 
 _ / I'fanCP' h' 
 (3) for 
 
 430. If, again, the symbol has been determined of a second 
 pole P f t representing a face (h'k'f\ where PP f is heterozonal to 
 
44 8 Crystallographic calculation. 
 
 A, B, and C, the arc PP* may be found by spherical trigonometry, 
 after the poles P and P f have been severally determined in 
 position relatively to C and CA as in the last article, by the 
 solution of the right-angled triangles PC A and P'CA. 
 
 431. A measurement between faces belonging to the prism- 
 zone, combined with one or more in a dome-zone, suffices for 
 determining the parameters of an ortho-rhombic crystal. Or it 
 may be that the angles between the faces of an octahedrid 
 form offer the only or the best images on the goniometer, and 
 an arc can generally be measured between the poles of two ad- 
 jacent faces of such a form ; as, for instance, between h k I and 
 h k /, where 
 
 \(hkl t hkl) = PH = go-AP. 
 When a second arc, such as 
 
 (hkl, hkl) = 2PL, or (hkl, h~kl] = 2 PK, 
 can also be obtained, this second arc conjoined with the first gives 
 the position of P; so that the parameters of the crystal can be 
 determined. 
 
 The Rectangular-axed Systems. 
 
 II. The Tetragonal System. 
 
 432. In a tetragonal crystal the morphological axis is an axis 
 of symmetry for all the physical characters of the crystal ; whereas 
 the axes of ortho-symmetry in which the trito-systematic zone- 
 plane intersects the proto- and the deutero-systematic planes are 
 only to be distinguished from the other possible zone-axes lying 
 in that zone-plane by the fact that they are the normals of actual, 
 not of abortive, planes of symmetry for the Crystallographic forms. 
 The symmetry of the crystal thus leads to the morphological axis 
 being taken for one, namely, for the JT-axis,- one of the other 
 pairs of perpendicular axes of symmetry being taken for those of 
 X and Y. 
 
 433. The di-prism. The fundamental equation (A) takes for a 
 tetragonal crystal the form 
 
 cos PA =cosP^ = CosPC; (1) 
 
Tetragonal system. 
 
 449 
 
 which for the pole L of a form {h k o} belonging to the zone [ooi] 
 becomes 
 
 AT k 
 AL -Jl 
 
 a . _ a 
 - cosAL = T 
 n K 
 
 or 
 
 434. The proto-prism {100}, and deulero-prism {no}, are 
 particular cases of the form {hko} in which the poles of the 
 form lie on the proto- and deutero- 
 
 systematic zone-circles respect- C D IT* A 
 
 ively. In the one case tan A L = o ; 
 in the other tan^4Z = i, and 
 
 435. The di-octahedron {hkl}. 
 If P (Fig. 365) be the pole of a 
 face of a general scalenohedral 
 form {h k 1} , H^ and H^ the poles 
 in which [AP], \JBC] and [BP], 
 
 [AC] respectively intersect, and Fig. 365. 
 
 L be the pole h k o in which the 
 
 zone-circles [ooi, h kl\ and [ooi] intersect, then, following similar 
 reasoning to that in the last section, or, directly from equation 
 (c\ (Art. 422) ; since in a tetragonal crystal a d, we have 
 
 cotAZ, = cotACP = -v> 
 
 tan PL = cot CP = -^- cos ACP. 
 ch 
 
 Similarly, tan PH^ = cot AP = j cos BAP, 
 
 } 
 
 (3) 
 
 tan PH 2 = cot BP = ~ cos CBP. 
 
 Equations by means of which, if the parametral ratio be given 
 and the symbol of a pole, the position of the pole can be deter- 
 mined, or its position being given its symbol can be computed. 
 
 436. The proto-octahedron and deutero-octahedron. H being a 
 pole of the proto-octahedron {h o /}, and A the nearest pole to it of 
 
45 Crystallographic calculation. 
 
 the form {100} on the same proto-systematic zone-circle, since 
 cosACP = i, we have 
 
 ............. (4) 
 
 and if D be the pole of the form (101) with the same signs to its 
 
 indices, a 
 
 cot DC - = tan AD, \ 
 
 / ............ (6) 
 
 and tan AH = -tan A D. ) 
 
 Evidently then DC may be taken as the arc-element of a 
 tetragonal crystal, since its tangent gives the value of the para- 
 metral ratio. 
 
 If K be a pole of the deutero-octahedron hhL and F the nearest 
 pole of the form {no}, 
 
 cos KCA cos 45 = -=> 
 
 A/2 
 
 and cotC^= -?-= ~ = tan KF ; .......... (6) 
 
 V 2 cn 
 
 and for O the pole of the form { 1 1 1 } with the same signs to its 
 
 indices as K, 
 
 cot CO = tan OF = 
 
 
 whence tan AD = - = \/2 tan OF. 
 
 c 
 
 From the expressions (5), the elements of the crystal being 
 given, the position of a pole on a proto-systematic zone-circle 
 can be determined if its symbol be known ; or its symbol be 
 found if its position on the zone-circle be known : and the equation 
 (6) gives the same result in the case of a pole situate on a deutero- 
 systematic zone -circle. 
 
 437. If P and P' be any two poles the symbols of which are 
 hkl and h'k'l', equations (/), Article 423, take the forms 
 
 cos AP = sin PH, = h C , 
 y Jb 
 
 cos BP = sin PH t2 k 
 cos CP = sin PL I 
 
 VF 
 
 a 
 
Tetragonal system. 451 
 
 where F = ~ = c 2 (h 2 + k 2 ) + a* / 2 . 
 
 And, if F' = ~ = c* (h'* + '*) + a" l 
 
 we have similar expressions for P' ; namely, 
 cos AP f = sin 
 
 ' 7F' 
 
 &c. &c. &c. 
 
 Whence also, as in equation (h], Article 424, we obtain 
 
 cosPP' = 
 
 (7) 
 
 a result directly deducible from the equation (ix), Article 396, by 
 making a b and a, /3, y quadrants. 
 
 438. By assigning special values to the indices in the symbols 
 of P and P f the arc-distance of any two poles may be computed 
 if the parametral ratio of the crystal be given. 
 
 And for any two poles mno and m'n'o in the equatorial zone 
 [ooi], we have 
 
 / , , ^ mm + nn 
 cos (mno, m'n o) = -; (8) 
 
 V(m* + n*}(m'* + n'*} 
 
 an expression which is independent of the parametral element 
 of the crystal; so that the normal-angles between faces per- 
 pendicular to the trito-systematic plane must be the same for 
 faces with the same symbols in all crystals of the Tetragonal 
 system, and depend solely on the indices in these symbols ; and 
 further, if mno and n'm'o are poles of the same form, in which 
 the two symbols have their similar indices in the inverse order, 
 but differ in the sign of one of them, 
 
 COS(;;ZTZO, n'm'o) = o, 
 
 and the arc-distance is for every form - The zone is in fact a 
 zone of abortive symmetry. 
 
 Gg 2 
 
452 
 
 CrystallograpkU calculation. 
 
 439. If P be a pole hkl, Q a pole mno, and hko be L, CQP 
 
 is a quadrantal triangle ; 
 
 = smCPcosLQ 
 
 hm + kn 
 
 = sinCP 
 
 440. If H and H' be two poles of forms {hoi} and 
 'and K r two poles of forms {hhl} and {h'h'l}, 
 
 ; cosC7/= 
 
 ' 
 
 441. To find the element of a tetragonal crystal. 
 
 et 
 
 and P / be poles of which the known symbols are hkl 
 
 and h'Kt respectively, 
 and let their arc-distance 
 from each other be given 
 (Fig. 366). 
 
 Let the zone-circle 
 [ooi] intersect [CP~] in 
 Z, i.e. mhko; [CP'] in 
 L', or tik'o ; and P/" 
 in Q, or wwo: Fa pole 
 
 Fig. 366. 
 
 at a quadrant's distance 
 from Q on [ooi] is 
 (nmo), and R or (uvw) 
 the pole in which [PP'] and [CF] intersect is a quadrant's distance 
 from (). The arc QP can therefore be found by the expression 
 in Article 52. Also, 
 
 T j and tan 
 h, 
 
 = i from (3); whence LQ is known. 
 m 
 
 Hence PL can be found from the right-angled triangle PLQ-, 
 thus CP the complement of PL is known. 
 
Cubic system. 453 
 
 Hence the position of P relatively to C and an arc CA (or CB) 
 is known, and by equations (3), 
 
 - = y tan PC cos A CP- 
 a ti 
 
 The Rectangular-axed Systems. 
 III. The Cubic System. 
 
 442. The general expressions, obtained in Articles 422 and 423, 
 for the relations of the faces of a crystal referred to rectangular 
 axes, assume their simplest forms when applied to the Cubic 
 system, in which, the parameters being equal, all the elements 
 of the crystal are fixed. 
 
 Thus the fundamental equations for the direction-cosines of a 
 pole P with the symbol h k I become 
 
 cos AP _ cos BP _ cos CP 
 ~~h~ T~ ~~T 
 
 A, B, C being as before 100, 010,001; these poles now belong 
 to a single form, the cube {100}. The equations (e) in Article 
 422 take the forms 
 
 cot CP = 
 
 tan ZAP = tmCBP = -- 
 
 and corresponding expressions give the position of P in respect 
 to the other poles of the cube. So that the position of P being 
 thus given, its indices can be determined; or, the indices being 
 given, its position is known. 
 
 443. As in equations (_/) of the general expressions for crystals 
 referred to rectangular axes, we have 
 
 Vs 
 
 cos CP = sin PL \ 
 
454 Crystallographic calculation. 
 
 where = ff + tf + P and H, K. L are poles of the form {hko}, in 
 which the zone-circles [AP] and [100], [BP] and [oio], [CP] 
 and [ooi] respectively intersect. 
 
 If be a face of the octahedron {in}, and H be a face 
 of that form symmetrical with in regard to a proto-systematic 
 plane and lying therefore in an adjacent octant, while (/ is a 
 face symmetrical with in regard to a deutero-systematic plane, 
 and lying therefore in an octant attingent to that containing O -, 
 since the pole of O is equidistant from three poles of the cube, 
 
 we have r 
 
 cos A = sin OD , 
 V3 
 
 D being a face of one of the three equidistant adjacent faces of 
 the rhomb-dodecahedron. 
 
 Hence the normal-angles between adjacent faces of the octa- 
 hedron and cube, and octahedron and dodecahedron, are 
 AO = 54 44-14', OD = 35 15-86'. 
 
 And, as has been otherwise proved, article 125, the normal-angle 
 between the faces forming the edges of the octahedron is 
 
 <9&= 2 0Z>=7o 31-73', 
 while 0(/= 2AO = 1 09 2 8 -2 f = flfl'; 
 
 the latter being the normal -angle between the faces of the 
 octahedron which meet oppositely at a quoin, and being equally 
 the normal-angle of an edge of the tetrahedron <r { 1 1 1 }. 
 
 444. To calculate the arc between two poles in the Cubic system. 
 If P be a pole of the form {hkl}, and P f a pole of the form 
 {h'tf?}, whereS=A 2 + / 2 + / 2 , and y=y 2 + ' 2 + f 2 ; by substi- 
 tution in the equation (h), Article 424, we have 
 cos PP' cos AP cos AP'+ cos BP cos BP' + cos CP cos CP r 
 hh'+kk' + ll' 
 
 an equation by which the normal-angle between any two faces 
 in the Cubic system of which the symbols are given can be 
 computed; the indices hkl, h' k'l' being taken absolutely as regards 
 sign and independently of relative magnitude. 
 
 In the following articles the values for the normal-angles 
 
Cubic system. 455 
 
 between faces of the different forms of the Cubic system will be 
 obtained by application of this equation. 
 
 445. The proto-systematic zone-circles S contain the poles 
 of the cube, the rhomb-dodecahedron, the tetrakis-hexahedron, 
 and its hemi-systematic form the pentagon-dodecahedron. If H 1 
 and // 2 be two faces the poles of which lie on one of these zone- 
 circles, we have for their normal angle 
 
 hh' + kk' 
 
 cos 
 
 If HI and H^ be faces of the same form and their poles lie on 
 the same zone-circle S, they may be symmetrical to a second 
 plane S and their symbols will only differ in the signs of one 
 index. Then for adjacent faces, 
 
 and for faces that are not adjacent the angle is supplementary, 
 
 "*>* = % 
 
 If the poles of H^ and ff 2 are situate as before on a zone- 
 circle S but are symmetrical to a deutero-systematic plane their 
 symbols differ only in the transposition of their indices, and 
 
 or their symbols also differ in the signs of both their indices, and 
 
 If the poles of H^ H 2 lying on a zone-circle S are not symmetrical 
 in respect to any systematic plane, the indices of their symbols are 
 transposed and differ in one sign, and cos H^H^ = o ; the two faces 
 are perpendicular and belong in fact to a zone of abortive symmetry, 
 a condition shewn to he true of every zone of the Cubic system. 
 
 For adjacent faces of the tetrakis-hexahedron, cube, and rhomb- 
 dodecahedron, we have 
 
 
 
 A/ 
 
45 6 Crystallographic calculation. 
 
 Since cos 2 AH = J?-^, 
 
 fa 2 Y COS A T~f J? 
 
 we have -^ ^= 9 , > or = tan AH: 
 
 h 2 cos 2 AH h 
 
 whence the normal-angle AH being given the symbol of H is 
 known. 
 
 446. The deutero-systematic zone-circles 2 contain the poles of 
 the cube, icositetrahedron, octahedron, triakis-octahedron, and 
 rhomb-dodecahedron. 
 
 If R and R be faces the poles of which lie on a deutero-sys- 
 tematic zone-circle, and have for their symbols mmn and mm'n', 
 we have for their normal-angle 
 
 2 m m' + n n' 
 
 V 2 m- + 7* 2 A/ 2 m' 2 + n 2 
 
 If R and R' belong to the same form, the poles R and R' lie on 
 the same zone-circle 2. If R and R' are symmetrical to a proto- 
 systematic plane, 
 
 , 2 m 2 n 2 
 
 cos RR = 5 * 
 
 2 m 2 + n 1 
 
 Also we have cos CR = ==== = sin RD ; 
 
 V 2 m 2 + n 2 
 
 whence cos OR = 
 
 V%\/2 m 2 + n 2 
 
 m tan CR 
 
 Also - = - . 
 
 n </ 2 
 
 447. The cube, octahedron, and rhomb-dodecahedron. Of these 
 three fundamental figures of the Cubic system it has been already 
 shewn, with regard to the cube and octahedron, that their edges 
 are truncated by the faces of the rhomb-dodecahedron, and that 
 the quoins of the one form are truncated by the faces of the other ; 
 and further, that each face of the cube is perpendicular to a zone 
 in which lie four faces of the cube and four faces of the dodeca- 
 hedron: each face is also inclined at a normal-angle of 54 44-14' 
 on four faces of the octahedron, and of 125 15-86" on the remain- 
 ing four faces of that form. 
 
Cubic system. 457 
 
 Each face of the octahedron is perpendicular to a zone in 
 which six faces of the rhomb-dodecahedron lie, and is in- 
 clined at 35i5'86' on three faces of that figure symmetrically 
 grouped round it, and 14 4 4 4. 14' on the faces opposite to 
 these. 
 
 The normal-angle of an edge of the rhomb-dodecahedron is 60, 
 since cos (no, 101) = \; and that of the faces which meet op- 
 positely at a tetragonal quoin is 90. The edges of this form are 
 truncated by those of the icosi-tetrahedron {211}. .They are 
 bevilled by the faces of a hexakis-octahedron of the series (Art. 60) 
 in which the symbols are of the form {/* + A, //,, A}. 
 
 The following table gives the normal-angles between adjacent 
 faces in the zone [no, ioi]of certain of these forms : 
 
 {321}, 321, 312 = 21 47' 12"; 321, 23! = 38 12' 48"; 
 
 {43 1 }* 43 r > 4i3 = 32 12' 15"; 43 1 * 341" = 2747 / 45 // ; 
 
 {532}, 532, 523 = is "' 25"; 532, 352 = 464 9 '35"; 
 
 {54i}, 54i, 5i4 = 38 12' 48"; 54i, 451 = 2i47'i 2 "; 
 
 {743}, 743, 734 = 9 25' 48"; 743, 473 = 50 34' 12"; 
 
 {752}, 752, 725 = 2747 / 45' / ; 752, 572 = 32i2'i5' / . 
 
 448. Of the three fundamental forms of the Cubic system the 
 octahedron is the only one which can undergo a hemi-symmetrical 
 suppression in the number of its faces, the tetrahedron cr { 1 1 1 } 
 being its haplohedral semiform ; the normal-angle of each of the 
 six edges D of the tetrahedron is 109 2 8. 2 7'. 
 
 449. The tetrakis-hexahedron. The edges of the cube are be- 
 villed by the faces of the cube-pyramidion or tetrakis-hexahe'dron 
 {hko}. This figure has two kinds of edges, each face of the 
 form being an isosceles triangle, the edge H, which forms the 
 base of the triangle, being parallel to one of the crystallographic 
 axes, but lying, like each of the remaining edges D, in a deutero- 
 systematic plane. 
 
 The following angles for different varieties of the form follow 
 from the equations 
 
45 8 Crystallographic calculation. 
 
 For the form {210}, H= 36 52' 12" = D; 
 
 {530}, ff= 2 S 4 '2i", D = 4 2 4 o' 5"; 
 {320}, #=2237'i2", .0 = 46 n' 13"; 
 {520}, #=46 2 3 '5o", >= 3 o27' i"; 
 {310}, ^=53 7' 48", -#=25 50' 31"; 
 {410}, #=6i55'39", ^= i9 45' o"; 
 
 {720}, #= 5 8 6' 33", Z>=222 4 'IO". 
 
 The pentagon-dodecahedron. The only hemi-symmetrical form 
 of the tetrakis-hexahedron exhibiting a defalcation in the number 
 of the faces of that figure is the pentagon-dodecahedron 77 {kk o} 
 or pyritohedron (Article 183). The symmetrical pentagons which 
 form its faces are euthy-symmetrical to a proto- systematic plane 
 and have one edge O dissimilar to the other four : it lies in a 
 proto-systematic plane perpendicular to the former plane. The 
 remaining edges lie in no systematic plane. 
 
 45O. The triakis-octahedron. The faces of the triakis-octahedron 
 or octahedrid pyramidion {hhk} bevil the edges of the octahedron: 
 the edges O, which are parallel to those of the octahedron, lie 
 each in a proto-systematic plane S; the edges D, which form the 
 isosceles sides of the faces of the form, lie in the 2-planes : 
 Article 174. 
 
 The equations 
 
 _ 2h*-& n h*+2hk 
 
 COS O = j- rs 5 COS D = r^ 
 
 2h z + k z 2 h* + & 
 
 give the following normal-angles for the two kinds of edges in 
 some of the more usual varieties of the form : 
 
 {332}, 0-50 28' 44", 2= 17 20-29^; 
 
 {221}, 6>= 3 8 56 / 3 3 // , Z>=27i 5 / 58 // ; 
 
 {331}, 0=26 3 i' 3 i", ^= 37 5i / 49 // ; 
 
 {441}, 0=20 3' o", JD = 43 30". 
 
 The hemi-triakisoctahedron or twelve-deltohedron a {hhk}, Article 
 179, is the only form hemi-symmetrically derived from the triakis- 
 octahedron by the extinction of half its faces. It has two sets of 
 edges, one identical with the edges D of the triakis-octahedron. 
 
Cubic system. 459 
 
 and the other denoted by A, the normal-angle of which is deter- 
 mined by the equation 
 
 The following are the normal-angles between the faces forming 
 the edges A for different values of h and k : 
 
 ^{332}, A = 97so'i6"; 
 
 a {221}, A = 90; 
 
 er{ 33 i}, A = 8o54'55"; 
 
 o-{ 44 i}, A = 75 58' 13". 
 
 In consequence of the edges A of the form cr { 22 1 } being right- 
 angles, it will be seen that the faces of that form are those of four 
 cube-quoins symmetrically grouped in respect to the axes of sym- 
 metry of the system. Their poles are at a normal-distance of 
 i548 / from those of the adjacent faces of the octahedron, at 
 70 31' from the more remote of the three nearest poles of the 
 cube ; furthermore, each pole of the dodecahedron is 45 from four 
 poles, and each pole of the form {411 } is 45 from two poles of 
 the form {221}. 
 
 451. The icositetrahedron. The faces of the icositetrahedron \hkk\ 
 replace symmetrically the quoins of the octahedron and of the 
 cube (Art. 173). 
 
 Of the two kinds of edges and H which form the sides of 
 the deltoid faces of a form {hkk}, the normal- angle of the faces 
 meeting in an edge is given by the equation 
 
 the two faces being symmetrical to a plane S, that of a pair of 
 faces symmetrical to a plane 2 and meeting in an edge H is 
 obtained from the expression 
 
 the values of these angles for some of the more common varieties 
 of the icositetrahedron are, 
 
 {322}, 0= 5 82 / 3 // , 
 {211}, = 4 8 n' 23 
 
 = 35 5' 48", 
 
460 Crystallographic calculation. 
 
 {411}, 6>=27i 5 '58", ^=60; 
 {511}, (9 =22 n' 30", ^=6 5 57'2 9 "; 
 {611}, (9= i84o'i8", H=6 9 59' 41". 
 
 The only hemi-symmetrical form of the icositetrahedron pre- 
 senting a defalcation in the number of faces belonging to that 
 figure is the hemi-icositetrahedron or tetrahedrid-pyramidion cr {h k k} 
 (Art. 180). Its faces are isosceles triangles, and the edges H 
 forming the similar sides of each face are identical with the edges 
 H of the holo-symmetrical figure. 
 
 The normal-angle of the remaining edge, D, of each face may 
 be computed by the formula 
 
 Thus for the forms, <r {322}, D = 86 3 7' 40", 
 cr{ 2 ii}, Z>=7o 3 i'44", 
 
 o-{ 3 n}, Z>=5o28' 4 4", 
 
 o-{ 4 n}, >= 3 8 56' 33", 
 
 <r{ 5 ii}, Z?=3i35'n" s 
 
 o-{6n}, Z>=26 3 i'3i". 
 
 452. The forty-eight scalenohedron or hexakis-octahedron, {hkl} 
 (Art. 175). Each face of this form is a triangle, scalene in the 
 sense that its three edges are crystallographically dissimilar, there 
 being twenty-four edges of each different kind. If represent 
 any edge formed by two adjacent faces of which the symbols 
 differ in a sign, D an edge in which the highest index retains the 
 same position in the symbols for the two adjacent faces, and H an 
 edge in which the lowest index remains unchanged in its position 
 in the two symbols ; the normal-angles of the three kinds of edges 
 are given by the equations 
 
 = 
 
 and the following are the values for the edge's of some of the more 
 frequently occurring of these forms : 
 
 {321}; =106 36' 6", Z>= 2i 4 7 / i2", J7= 2 i47'i2"; 
 {421}; 0= 25 12' 31", Z>=i745'io", ^ r =3557 / *" \ 
 {531}; 0= 19 27' 47", .0=27 39' 38", H =2f 39' 38"; 
 (543); 0= 50 1 2' 29", D= 11 28' 42", H= 11 28' 42". 
 
Cubic system. 461 
 
 The distances of a pole of the hexakis-octahedron from the 
 nearest, the next, and the most remote of the three least distant 
 poles of the cube are given severally by the equations 
 
 h k I 
 
 Vs Vs Vs 
 
 453. Special varieties of the forty-eight scalenohedron have 
 been already considered, Article 175. In one series of these the 
 poles lie in zones with those of the rhomb-dodecahedron, the 
 indices being determined by the equation h = k + / : the twenty- 
 four edges D of such a form are parallel in groups of six to 
 the four trigonal axes O of the system. In the other series 
 the poles lie on zone-circles bisecting the angles between the 
 deutero-systematic zone-circles 2, and the forms correspond to 
 those which in a trigonal system have been designated by the 
 symbol {min}, where 21 = m-\-n\ and in a form of this kind 
 normal-angles of the edges H and D have the same values 
 though the edges are not of equal length nor otherwise 
 similar. 
 
 Forms of the first series would be {321}, {532}, {431}, &c. , 
 and of the second {321}, {53 1 }, {43 2 }> &c- 
 
 454. It has been seen that, from the nature of the symmetry 
 of the Cubic system, it is only the general scalenohedron that is 
 capable of undergoing a hemi-symmetrical partition of its faces in 
 accordance with both of the laws designated by the symbols a- {h k /} 
 and Ti{hkl}. The former of these symbols is that of the hexakis- 
 tetrahedron, the latter is that of the trapezoid dodecahedron or 
 diploheclron. 
 
 The hexakis-tetrahedron v{hkl}, Article 181, has three sets 
 of edges, the normal-angles of which are determined by the 
 equations 
 
 tf-2kl 
 
 _ 
 
 cos D = ^ > cos H = ^ > cos A = ^ 
 
 O O O 
 
 where the angles D and H are those belonging to the respective 
 forms of the hexakis-octahedron. 
 
 The following are the normal-angles for the edges A of 
 various hexakis-tetrahedra, A being the edge in which the signs 
 
462 Crystallographic calcidation. 
 
 of two of the indices of the faces forming the edge are dif- 
 ferent : 
 
 o-{ 3 2i}; A= 158 12' 47"; 
 
 .{421}; A- 55 9' o"; 
 
 -{53i}; A- 57 7 'i8"; 
 
 -{543}; A= 88 51' 14". 
 
 7/k? diplohedron, TT {hkl} : Article 184. The edges which form 
 the four sides of the trapezoidal faces of this figure are of three 
 kinds, representing three different normal-angles. The edges G 
 which meet in a trigonal quoin do not lie in the 2-planes, and the 
 normal-angle of the faces meeting in each such edge G is given 
 
 by the equation hk + kl + lh 
 cosG = y- 
 
 The two other edges and li of each face meet in a tetragonal 
 quoin, and lie in proto-systematic planes perpendicular to each 
 other. They are formed by faces the symbols of which in each 
 case differ in one of their signs ; and their normal-angles may be 
 computed from the equations 
 
 AH* 2 -/ 2 tf-*H/ 2 
 cos 12 = ^ 5 cos = = ; 
 
 o o 
 
 where >Q is the edge representing the longer and that represent- 
 ing the shorter side of the trapezoid. 
 
 The following are the normal-angles for G, O, and for various 
 diplohedra : 
 
 77(321}; G = 3i O'IQ", 11 =106 36' 6", 0=6 4 37'23"; 
 77(421}; = 48 n' 2 3", a=25i2 / 3 i' / , = 5i45'i2"; 
 77(531}; G = 4 855' 4", &= 19 27' 47", = 60 56' 27"; 
 "{543}; ,G= i 9 56 / 54 // , G = 5oi2 / 2 9 // , 0=68 53' 59". 
 
 The pentagonal icositetrahedon^ a {hkl}: Article 177. Of the 
 forms in the Cubic system the general scalenohedron is the only 
 one which can undergo a hemi- symmetrical suppression of its faces 
 in accordance with the law denoted by the symbol a ; the resulting 
 semiform being the pentagonal icositetrahedron a [hkl\. 
 
 The semiform a {hkl} has three sets of edges G, V, W (see 
 
Hexagonal system. 463 
 
 Figs. 83 and 84, Art. 177), the normal-angles of which are deter- 
 mined by the equations 
 
 hk + kl+lh tf 2hk-P 
 
 Thus for 0(785}; 
 
 G=i8i 9 '39", F= 62 22-10", PF= 5 o55'4"; 
 for a{ 9 86); 
 
 G= 15 59' 12", F= 63 24' 57", FT =53 2 2V'. 
 Four edges F meet in each tetragonal quoin ; three edges G in 
 each trigonal quoin ; and each edge W would be truncated by a 
 face of the form {no}. 
 
 The Hexagonal System. 
 
 455. As referred to three axes lying in the proto-systematic 
 planes S symmetrically with regard to the zone-axis of these 
 
 Fig. 367. 
 
 planes, a crystal of the Hexagonal system has for its parametral 
 plane a face of the pinakoid form {in} which is parallel to the 
 equatorial systematic plane C : this last plane being also the plane 
 of the zone-circle [in]- 
 
 The only variable element of the system is the axial angle f. 
 
 If X, F, Z (Fig. 367) be the axial points, A lt A z , A 3 the poles of 
 
464 Crystallographic calculation. 
 
 the axial planes YZ, ZX, and XY severally ; then A 1 is 100, A 2 is 
 oio, A 3 is ooi, and the arcs A 1 Y, A^Z, A%X, A^Z, A 3 Y r , A 3 X 
 are quadrants ; and the triangles A^A 2 A 3 and XYZ are polar, and 
 by the symmetry of the construction A 1 and X, A% and Y, A 3 
 and Z lie on the same side of (in), on quadrants of the great 
 circles S which meet in that pole. 
 
 Further, OX = OY = OZ, 
 
 and OA l = OA^ = OA 3 , 
 
 " 1 " 2 ~~~ 23 ^~ 3 1 ' 
 
 XOY - YOZ = ZOX = 120. 
 
 Since the axes X and Y are symmetrically situate in respect to 
 a great circle S S) the arc XY is bisected, say in V, by that great 
 circle ; and in the triangle XO V, 
 
 sin VX = sin OX sin VOX, 
 or sin J f = sin OX sin 60, 
 
 sin OX = sin 6>J^ = sin OZ = -^~ sin \ f . (i) 
 
 V3 
 
 456. Let /* be the pole hkl of a discalenohedron (Fig. 367). 
 Then the pole symmetrical with it in regard to S 1 is hlk, and the 
 edge of the faces hkl and hlk would, Article 133, be truncated by a 
 face R with the_symbol (2/1, k + l, k + l), which is of the form h'k'tf. 
 The faces hkl t efg which are symmetrical to the equatorial plane C 
 and lie in the zone [in, hkl\ are truncated by a face N for which 
 the symbol is obtained by the addition of the symbols (%h, 3 k, 3/) 
 and efg', that is to say, the symbol is 
 
 (2h kl t 2k lh t 2lhk), 
 
 or (pqr), where p + g + r = o, and the face thus belongs to the 
 equatorial zone of the crystal. 
 
 457. Let R be the pole hkk of a rhombohedron (Fig. 367). 
 Since the parameters are equal, 
 
 cos RX cosRF cosXZ 
 T = 7 = and cos Rx cos KZ, (11) 
 
 and cos RX = cos ( OR - OX) = cos OR cos OX + sin OR sin OX, 
 also cos RY = cos OR cos OY+ sin OR sin OY cos 120, 
 = cos OR cos OX- \ sin Oft sin OX] 
 
Hexagonal system. 
 
 465 
 
 whence by substitution in (ii), 
 
 (- + jfy sin OR sin OX = (h-k) cos OR cos OX, 
 
 and tan OR = 2 7- cot OX (iii) 
 
 h + *k 
 
 Hence, for a pole A l of the form {100} on the hemisphere which 
 is positive relatively to the zone [m], 
 
 tanO4 = 2 cot OX; ................ (iv) 
 
 and if R is a pole of a form \hkk\ lying also on the positive side 
 of [in] and on the same great circle with A lf 
 
 tan OR = 
 
 tan OA 
 
 OR and OA having the same or opposite signs according as R 
 and A are on the same or opposite sides of 0\ i.e. according as 
 the rhombohedron {hkk\ is direct or inverse. 
 
 And since, from equation (i), sin OX is known when f is given, 
 it is seen from equation (iv) that tan OA determines f : and OA 
 may therefore serve as the arc-element of a crystal in this system. 
 
 458. Every pole of a form {hkk} is at a quadrant's distance 
 from two opposite poles of the form {oil} (Fig. 183) ; so that khk 
 and kkh being poles of the form {hkk} on the same side of [m], 
 the great circles [khk, kkh] and [i 1 1] intersect in the poles oil and 
 oil, and from either of 
 
 these poles the pole hkk 
 is distant a quadrant; the 
 arc-distance of khk and 
 kkh from them is either 
 9o-f-^A or 90 J A, 
 where A is the arc joining 
 any two of the poles hkk, 
 khk, kkh. 
 
 459. R being one of 
 
 Fig. 368. 
 
 the above poles of the form {hkk} and T a pole of the form 
 truncating by its faces the edges of {hkk}, the symbol of the 
 truncating form is \2k, h + k, h-\-k\, and we have evidently, equa- 
 ting values of tan OA given by (v), 
 
 tan OR =-2 tan 0T 7 . 
 nh 
 
466 Crystallographic calculation. 
 
 The zone-circles [in] and [in, 100] or [SJ intersect in M 
 or 2 IT, a pole of the proto-hexagonal prism, and [2J or [2!!], 
 the zone-circle perpendicular to [S'J, intersects [in] in oil, a pole 
 of the deutero-hexagonal prism. 
 
 For NI a pole (pqr) of the dihexagonal prism lying on the 
 zone-circle [in] (Fig. 368), 
 
 cos NX _ cos NY 
 
 P q 
 
 From the triangles XON, YON, ON being a quadrant, 
 cos NX = sin OX cos XON, 
 cos NY = sin 01" cos .TOW = sin O^T cos (120 - 
 
 and since Af)N is MN we have - = \ (\/3 tan MN i) ; 
 
 but since p + q f r = o, 
 
 I q T 
 
 Vz P 
 
 Now the zone-circle [in] is an isogonal zone with abortive 
 symmetry (Article 99), and every face in it has for its pole the 
 pole of a possible zone, the indices in the symbols for the zone 
 and for the face being identical : hence every face in the zone 
 [in] is perpendicular to another possible face of the zone. 
 If For/'/r' be the face at a quadrant's distance from N t 
 
 tan MV = tan (MN+ 90) = - 
 
 I // r 7 .- /> 
 
 and 
 
 A/3 P r ~V 
 
 P' y~ r i / 2 r 
 whence -7 = and --7 = , (vii) 
 
 If now ^V lie in the zone [in, hkl\, its symbol (/^r) is 
 
 (2h k /, 2k / h, 2/ h >), and = - ~ -\ 
 
 p 2/1 K / 
 
 so that, P being h k /, 
 tan A, OP = 
 
 2h-k-l 
 
 If 7? be a pole h' k' k' of the dirhombohedron the faces of which 
 truncate the alternate edges hkl, hlk, &c. of the discalenohedron 
 
Hexagonal system. 467 
 
 {hkl}, the angle at R in the triangle FOR is a right angle, and 
 tan OR = tan OP cos ROP 
 
 and substituting values of hkl for h' K k' (Article 466), 
 
 tan OX tan OP cos MOP = * k k ! , 
 
 i tan OA j 2hkl 
 j 
 
 which may be written 
 
 ~ . h + k + l cos 
 
 tan OA, 
 
 
 (ix) 
 
 460. The position of a pole P or hkl\s determined by the arc 
 OP and the angle MOP, i.e. the arc MN, where ^V is 
 
 (2h-k-l y 2k-l-h, 2l-h-k\ 
 
 And if the position of P is known, its symbol can be determined 
 from equations (viii) and (ix). For since the terms of the ex- 
 pression 2hkl may be represented by any numbers, whole or 
 fractional, in the ratios of 2^, k, and /, we may take 
 
 2h-k-l= i; ................ (x) 
 
 ten A, OP 
 then from (vin), kl - ~ ; 
 
 V3 
 
 tan OA, 
 and from (,x), 
 
 , tan 
 
 _ 
 while 3 (/&/) = -/s tan ^4 X OP ; 
 
 two equations from which and from equation (x) the symbol of 
 P is readily obtained. 
 
 If the symbols were those of two poles of the discalenohedron 
 belonging to the semiform {efg\ correlative with {hkl}, for 
 instance, the poles gfe and g ef symmetrical on S lf expressions 
 identical in form lead to similar results. 
 
 H h 2 
 
468 Crystallographic calcination. 
 
 461. For a pole K lying on a deutero-systematic zone-circle 2, 
 and having a symbol m in; since A^OK = - and, by Article 121, 
 
 - = z\ we have, for OK where the symbol is given, 
 
 and for the ratios of the indices of the symbols, where tan OK 
 is known, 
 
 m _ tan OA + A/3 tan OK i _ tan OA 
 
 n ~~ tan OA - A/3 tan6>^ ' ~~ tan O4- -/Stan OAT* 
 If tan 0= A/3 tan OK, we have 
 m sin ( OA + 0) 
 
 72 ~sn 
 a form adapted to logarithmic computation. 
 
 462. To determine the arc joining two poles P and P' not 
 lyino on a systematic zone-circle, where the symbols hkl and h'k'l' 
 of P and P', respectively, are given ; the angles A OP and A OP', 
 and therefore POP', are to be found from equation (viii), and the 
 arcs OP and OP' being found from equation (ix), the third side 
 of the triangle POP', that is to say, the arc PP'. is directly found 
 by spherical trigonometry. 
 
 Or we may proceed as follows : 
 
 Let [u v w] = [h k /, h'k'l'\ and the zone-circle PP' or [u v w] 
 meet the zone-circle [in] in a pole Q or efg ; then 
 
 e = v w, f=. w u, g = u v; 
 and it may be shown that if [OP] meet [i 1 1] in TV 7 , 
 
 ^.^ 
 
 Similarly, tan (W = 
 
 Then, PP'is known, for 
 cos <2P = cos QN sin OP, and cos QP' = cos QN' sin 
 
Hexagonal system. 469 
 
 463. To find the arc -element of a hexagonal crystal. 
 
 Where the pole of any face of the crystal has been given in 
 position with respect to the pole (m) and the zone-circle [in, i oo], 
 or [0-4J, or the zone-circle passing through and the nearest 
 pole of the form {100}, the equations (ix) give the arc-element 
 directly ; provided that the symbol of the face is given, and that the 
 face neither belongs to the zone [in] nor is the face (in) itself. 
 
 Where however we have only symbols and one measured arc 
 of the crystal for data, the problem takes the form of having to 
 
 Fig. 369. Fig. 370. 
 
 
 
 determine the arc-element from the measured arc-distance between 
 two poles whereof the symbols only are given or can be deter- 
 mined. Here then we have the following cases : first, P' one of 
 the poles may lie on the zone-circle [in], P the other lying 
 either in or out of the zone containing and P'. Or secondly, 
 P f may lie on the zone-circle OP but not on [in]. Thirdly, 
 both P and P' may be discalenohedral poles and their zone lie 
 obliquely to the zone [in]. 
 
 I. (a) Let P be the pole hkl, P' the pole/^r lying on the zone- 
 circle [in] (Fig. 369). In this case (pqr) is 
 
 ( 2 h-k-l, 2k-l-h, 2l-h-k). 
 
 From equation (viii) AflP can be determined, since 
 
 k-l 
 
 Also PP' being given, OP is known, for tan OP = cot PP' ; 
 hence AO the arc-element can be found by equation (ix). 
 
470 Crystallographic calculation. 
 
 I. (f) In the case where P' is not in the zone OP, let N 
 be the pole in which [OP] and [in] intersect (Fig. 370). Thus 
 NMoiAOP is known, from (viii) ; so is P'M and therefore P'N; 
 hence PN in the right-angled triangle PNP' can be computed. 
 Thus OP also is known and P given in position ; so that A 
 can be calculated from equation (ix). 
 
 If the symbol of P' had not been given but only that of [uvw], 
 
 a zone traversing PP 7 , then the symbol of P' is |i ^ ^1 
 
 II. If neither pole is on the zone-circle [in] but both are in 
 a zone with (in), the arc PP' and the symbol of P being given ; 
 then, if Nbe the pole in which the zone-circles [OP] and [in] 
 
 Fig. 371- Fig. 372. 
 
 intersect, the symbol of N can be found, and it determines, as 
 before, the angle A OP (Fig. 371). The four poles N, P', P, O 
 lie in a zone, and of these the symbols of O or (in) and P 
 or (hkl) are given. Also N lies in the zone [m]; and let 
 the symbol [uvw] of a zone traversing P' be also given. Then, 
 by Article 52, 
 
 sin (2 PNPP') = (2 A- 1) sin PP', 
 
 where A = 
 
 an equation by which NP, and therefore tan OP, can be com- 
 puted. So that P is given in position and the element can be 
 found. 
 
 III. If P and P' (Fig. 372) are both independent poles hetero- 
 zonalto 0, let the zone-circle [111] intersect [PP'] in Q and [OP] 
 
Hexagonal system. 471 
 
 in N. Then the symbols of Q and N are known. Let W be the 
 symbol of a pole at a quadrant's distance from Q on the zone 
 [in], of which therefore the symbol may be found by equations 
 (vii), and let R be the pole in which [OW] and [QP] intersect. 
 Then, of the four poles Q, P, P / , R let the known symbols of 
 P and R be hkl and efg\ the symbols also of the zone [in] 
 traversing Q, and of a zone [uvw] traversing P', which may be 
 [OP'], are known, as are the arc QR which is a quadrant, and 
 the arc PP' which has been found by measurement. 
 The arc QP can thus be found by the equation 
 
 = (2A-i)sin/V, 
 
 where A= -- z -- 
 
 e+f+g 
 
 QM and NM, and therefore also QN, can be computed by equa- 
 tion (viii); and thus PN in the right-angled triangle PNQ can 
 be found. 
 
 Hence OP and A OP are determined, and the arc-element and 
 the axial angle of the crystal can be computed by equation 
 (ix). 
 
 464. The least unsymmetrical mode of referring a hexagonal 
 crystal to axes of which one is at right angles to the rest (Art. 118) 
 is that employed by Bravais, and adopted in recent times in the 
 Zeitschrift fur Krystallographie. 
 
 The axial system adopted in this method assumes for the four 
 axial planes the systematic planes 2 and C, the axial system con- 
 sisting of three lateral axes which are the zone-lines [2 X C] [2 2 C] 
 and [2 S C] and of a vertical axis which will be the zone-line 
 
 P.S,]. 
 
 The JST-axis and the J^-axis are the first and third of these, the 
 
 fourth or zf-axis being the morphological axis perpendicular to them. 
 while the redundant lateral axis may be designated by the letter U\ 
 the general symbol for a form being {h k i /}. 
 
 The directions of the three lateral axes are so taken that the 
 poles designated in this treatise as oil, ilo and Toi lie on the 
 positive side of the origin, upon the three axes respectively. 
 
472 
 
 Crystallographic calculation. 
 
 The constant relation of the index *, corresponding to the re- 
 dundant axis U t to the indices on the axes X and Y may be easily 
 established. If in Fig. 373 the face flKL intersect the axis U in 
 a point / we have for the intercepts of this plane 
 
 a a a c 
 
 v Y 7 j r 
 
 Since area HOI+ area KOI = area HOK, we have 
 
 OH. 6>/sin6o+ OK . OIsm 60 = OH . OKsm 120. 
 Hence VH . 01+ OK. 01= OH. OK; or, if we regard lengths 
 as negative when measured on the negative part of an axis, 
 
 OH. 01+ OK . 01+ OH. OK = o, 
 i i i 
 
 and 
 
 but 
 
 OH= 
 
 n 
 
 k 
 
 In the symbol (h k 2 /) of the face HKIL, therefore, the third 
 
 index is equal to the alge- 
 braical sum of the first two 
 with its sign changed. 
 
 For the derivation of sym- 
 bols for zones from those of 
 faces, or the inverse process, 
 the usual operations are re- 
 sorted to; the determinant is 
 
 Fig. 373- 
 
 taken of the symbols of any 
 
 two faces but with one of the first three indices, and of course the 
 same one throughout, omitted from the face-symbol. 
 
 The symbol of a face obtained from those of two zones has then 
 to be completed by the introduction of the fourth index in accord 
 with the relation h + k + i = o. 
 
 Fig. 374 represents a projection of some of the forms of a hexa- 
 gonal crystal referred to such an axial system, the four-indexed 
 symbols representing the poles which in this treatise have the three- 
 indexed symbols conjoined with them in the figure. 
 
Hexagonal system. 473 
 
 Had the S-planes, instead of the deutero-systematic planes, been 
 selected for the axial planes, the symmetry of the distribution of 
 
 - 374- 
 
 the scalenohedral forms, and of the indices in their symbols, would 
 have presented greater symmetry and more geometrical elegance. 
 
CHAPTER IX. 
 
 THE REPRESENTATION OF CRYSTALS. 
 
 465. The representation of a crystal by a model 'in the 
 round ' may best be effected by apparatus specially constructed 
 for cutting sections of the wood, or other material employed, in 
 directions inclined to each other at angles identical with those 
 of the edges of the crystal ; or they may be cut by the help of 
 determinations deduced from the projection of the crystal on a 
 plane surface, whereby the angles are given at which the edges 
 meet in each quoin of the crystal. 
 
 But for use in written description recourse is had to representing 
 the crystal by a drawing of its edges, which are projected on a 
 plane, as on a screen, by lines emanating from the eye and traversing 
 each point to be projected. Where the eye is near the crystal 
 the lines are more or less divergent, and the figure on the screen 
 obeys the ordinary rules of perspective. In proportion as the eye is 
 removed to a distance, these rays approximate to parallelism ; so 
 that, by conceiving the eye to be placed at an infinite distance, 
 a figure of the crystal may be projected by means of parallel rays. 
 
 Perspective figures of the former kind involve complex processes 
 in the drawing from which those of the latter kind are free, while 
 also the projection by the aid of parallel rays offers some important 
 advantages. 
 
 466. Parallel projection. A projection of this kind offers the 
 following advantages : 
 
 I. Parallel lines on the crystal, the edges for instance belonging 
 to a zone, are represented as parallel lines in the projection. 
 
 The converse however is not true, since lines parallel in the 
 projection do not necessarily represent lines parallel on the crystal ; 
 for, if we consider a plane passing through any two rays, all lines 
 lying in that plane will be seen to be represented on one and the 
 same line in the projection, whatever may be the angles at which 
 they are inclined to the two rays and to each other. 
 
Projection on a systematic plane. 475 
 
 II. A second important property of the parallel projection 
 consists in this that a straight line on the crystal, if divided in any 
 ratio, is represented in the projection by a line divided in the same 
 ratio. Thus AB, BC, divisions of a line AC inclined at any angle 
 to the plane of projection, are represented by ab, be on ac, the 
 projection of AC; and Aa, Bb, Cc being parallel to each other, 
 we have ab AB 
 
 It follows from this that, if the axes and parameters of a crystal 
 are given in projection, intercepts on the actual axes will be repre- 
 sented on each projected axis by intercepts proportional to the 
 original intercepts on that axis ; and, from the indices of a plane, 
 its projected intercepts on the axes can be taken. 
 
 It will also be seen that equal parallel lines on the crystal will be 
 projected as equal parallel lines ; and, further, a line parallel to a 
 visual ray will be represented by a point, and a plane containing 
 a visual ray will be projected as a line. 
 
 Furthermore, all lines lying in planes parallel to the plane of 
 projection if equal will be projected as equal lines, and if unequal 
 will preserve their ratios in the projection. 
 
 467. Orthogonal projection : the plane of projection a systematic 
 plane. Systematic projection. For a simple representation of a 
 crystal, it is best to take the plane of projection perpendicular to the 
 direction of the visual rays, i.e. to employ an orthogonal projection. 
 The crystal may now be placed in any position in the path of the 
 visual rays with a view to its being orthogonally projected ; and 
 the most suitable position will of course depend on the purpose of 
 the projection. Sometimes this purpose is sufficiently fulfilled by 
 making a systematic plane the plane of projection ; the faces of 
 the zone perpendicular to this plane becoming lines which bound 
 the figure, while the edges of that zone are represented by the 
 points in which these lines meet. An orthogonal projection of 
 this kind may be termed a systematic projection. 
 
 Such a projection presents great facilities to the draughtsman, 
 and is employed, except for the Cubic system, almost exclusively in 
 Professor Miller's classical treatise on Mineralogy. It constitutes 
 in fact a sort of graphic counterpart to the representation of a 
 
476 
 
 Representation of crystals. 
 
 crystal by the stereographic projection of its poles. If the stereo- 
 graphic and the systematic projections of a crystal be constructed 
 with the same systematic plane for the plane of projection in each, 
 it will be seen that for the faces of the zone perpendicular to that 
 plane the poles will be distributed on the circle of projection in the 
 one case, while they are represented by lines in the other case : and 
 these lines will evidently be parallel to tangents drawn through the 
 poles on the circle of projection, if the two projections are similar 
 in orientation. The direction in which the edge of any two faces 
 of the crystal would be projected can be determined by tracing the 
 zone-circle containing the poles of the two faces and drawing a 
 tangent to the projection-circle through the pole in which it and 
 the projection-circle intersect. 
 
 Fig- 375 () 
 
 I 
 
 Fig. 375 
 
 All the lines that are to represent the edges of the front half of 
 the crystal can thus be determined so far as their directions are 
 concerned. Their actual position on the projection will depend on 
 the relative magnitudes to be given to the several forms, and for 
 this a small free-hand drawing of the crystal or a view of the 
 crystal itself is desirable. Where faces of the same form are symme- 
 trically repeated, care has to be taken that the area of corresponding 
 faces is maintained constant by keeping corresponding symmetrical 
 lines on the crystal everywhere of equal length in the projection. 
 In practice it will be found that the parallelism of the edges of the 
 faces belonging to each zone rapidly indicates the contour of the 
 
Orthogonal projection. 477 
 
 faces enclosed by the various edges ; and in fact, when the stereo- 
 graphic projection of the crystal is once correctly figured, the 
 systematic projection is produced by a simple process of evolution 
 from it. In this way the systematic projection of a crystal of 
 bournonite given in Fig. 375 (b} can be immediately obtained from 
 the stereographic projection of the poles given in Fig. 375 (a). 
 
 468. Orthogonal projection : the plane of projection not a syste- 
 matic plane. In order, however, to obtain a more complete view of a 
 crystal than is given by the systematic projection, it is necessary to 
 be able to represent the various forms more as they would be seen 
 in a perspective drawing, as well in order to give an aspect of 
 solidity to the figure as to indicate the relative importance of the 
 different forms. 
 
 The systematic projection, for instance, indicates only by its 
 boundary-lines, and therefore in fact obliterates as faces, the planes 
 in the zone of which the edges are parallel to the line of sight;' 
 and these frequently include important pinakoid-, dome-, or prism- 
 faces, to which the crystal may owe its most characteristic aspect. 
 
 The drawings are therefore generally made in such a way as 
 that the plane of projection is not in parallelism with any systematic 
 plane, nor, intentionally, with any face. The crystal is in fact 
 placed in such a position that while its orientation gives to the axes 
 nearly the positions in space usually assigned them in geometry, 
 certain rotations through small angles, forward and from right to 
 left respectively, bring into view, and as it were open out, faces that 
 in the systematic projection are represented by lines only; while 
 the idea of solidity is further imparted to the figure by the difference 
 in dimensions that faces of the same form assume when thus 
 moved into positions in which they are differently inclined to the 
 line of sight. 
 
 469. Cubic system. Let Ox, Oy, Oz be three rectangular lines 
 of reference fixed in space, Ox and Oy being horizontal and Oz 
 vertical ; let xOz be the plane of orthogonal projection, and the 
 visual rays have directions parallel tojyO. Taking as the simplest 
 case a cubic crystal, we may suppose the initial position of the 
 crystal to be such that the crystal-axes OX, OY, OZ are coincident 
 with the fixed lines Ox, Oy, Oz, respectively : in the corresponding 
 
Representation of crystals. 
 
 projection, the projections of OX, OZ will coincide with the lines 
 Ox, Oz respectively, and the projection of OY will be the point O 
 (Fig. 376). 
 
 Next suppose the crystal to be rotated, in the direction of the 
 hands of a watch, through a small angle 8 round the fixed line Oz. 
 
 B 
 
 $ y Y A. x 
 
 Fig. 376. 
 
 The .F-axis will now be represented by a line coincident in direc- 
 tion with the line Ox ; and A', the projection of any point A on 
 the axis OX, will move along Ox towards the origin while the 
 rotation is proceeding (Fig. 377). Since the parameters are equal 
 
 S' 
 
 A r 
 
 y 
 
 o 
 
 Fig. 377- 
 
 we may take in the initiatory position OA = OS = OC = a for 
 the parameters on the axes X, Y, and Z. During the rotation of 
 the crystal, A describes a circle : the visual rays are in the plane of 
 this circle and are perpendicular to the initial position of OA. 
 If we describe a circle (see Fig. 378) through A and B with its 
 centre at the origin, we may determine by a simple construction 
 the values of OA ' and OB', representing the projected lengths 
 a on the -X"-axis and 7-axis, after the rotation of the axes through 
 the angle 5. 
 
Projection of rectangular axes. 
 
 479 
 
 Making A OD = b, we may suppose A to have moved on the 
 circle to D ; when seen by visual rays in the plane of the circle 
 
 and perpendicular to the initial position of OA, A will appear as at 
 A', where DA' is perpendicular to OA. 
 
 Similarly, after the rotation, B will have moved to E and be 
 seen as at B' : and OA' = a cos b, OB' a sin 6. 
 
 The projections OA', OB', OC' of the lines OA, OB, OC on the 
 plane xOz, after this single rotation of the crystal through an angle 
 8 about the line Oz, are given in Fig. 377. 
 
 B' 
 
 Fig- 379- 
 
 Let now the system further revolve round the line Ox through 
 a small angle e, the point C or C' moving forward, and its projec- 
 tion C" downward along the axis Z. Then the projections of the 
 axes X and J^on the plane xOz will become disengaged from their 
 former coincidence with the line Ox, and will take new directions 
 
480 
 
 Representation of crystals. 
 
 in OX" and OY" : the points A' and B' will appear in the pro- 
 jection to move along lines perpendicular to Ox to the new 
 positions A" and B", while C moves to C" (Fig. 379). 
 
 The axial system has now taken a new aspect. The axes X 
 and J^come to be represented by lines from the origin through A" 
 and B") and the parameters are measured on these axes and on Oz 
 by the ratios OA" : OB"-. OC" ; 
 
 which, it will be seen by the constructions in Figs. 378 and 380, 
 may be determined by the expressions 
 
 B'asinS 
 
 C 
 
 C" 
 
 ctcoss 
 
 /r 
 
 Fig. 380. 
 
 OA' = acosb ) 0.5' = 0sin5 1 
 B'B" = a cos 8 sin e) 
 
 = a sm sin 
 
 = a cos 
 
 //2 = a* (i -sin 2 8 cos 2 e), 
 
 and 
 
 For the angles between OX", OY", and OZ", the projections of 
 the axes OX, OF, and OZ, we have (Fig. 379) 
 
 = cotan 
 
 cot Z" OY" = cotan 
 
 A' A" 
 
 = yjr : = -tan 8 sine: 
 
 = -\.mxOY" 
 
 B'B" 
 
 = nD , = cotan 8 sin ; 
 
 whence also X"OY" can be computed. 
 
Projection of rectangular axes. 
 
 481 
 
 470. The values of 6 and e th*at give to a crystal the most 
 advantageous position for exhibiting its features have been differ- 
 ently taken by different crystallographers. 
 
 Haidinger, who has been very generally followed, took tan 5 = \ 
 and sin 6 = J, corresponding to 6 = 18 26' and e= 7 n', nearly; 
 the projected parametral ratios being 
 
 a:b:c = 0-957:0-3404: i, 
 and the axial angles as seen in projection 
 
 9 2 Q 2 3 ' and m?= iio 33 $'. 
 
 Fig. 381. 
 
 471. Cry sf allographs. When the projection of the axes has been 
 determined for the Cubic system, that for any other system is 
 readily deduced from it. And it is very convenient to have an 
 instrument in the form of Fig. 381, made of brass or other metal of 
 about -|- in. thick, cut to the angles of the projected axes, and 
 graduated along each of its three limbs with divisions subdivided 
 to tenths and in the ratios of the projected parameters of a cubic 
 crystal. Such an instrument or crystallograph serves for laying 
 down on a piece of card, or a sheet of smooth paper strained on 
 
 i i 
 
482 
 
 Representation of crystals. 
 
 a frame, the projection of a cubic axial system, from which that of 
 the axes and parameters for any other crystallographic system can 
 be deduced in the following manner. 
 
 (a) Orthosymmetric systems. The Tetragonal and Orthorhombic 
 systems are easily dealt with. In the latter, for instance, the 
 orientation of the axes as projected by the crystallograph remains 
 unchanged, but we have to find parameters on each of the three 
 axes proportional to those of the crystal, but interpreted in accord 
 with the parametral units of length on each limb of the crystallo- 
 graph. In the Tetragonal system, only the parametral length on 
 the ^-axis has to be so adjusted. 
 
 (b) Hexagonal system. For the purpose of transforming a 
 projection for a cubic axial system into one for a hexagonal 
 crystal, let the axes of the cubic system be XFZ, those of the 
 hexagonal crystal X'Y'Z'. 
 
 Fig. 382. 
 
 If now a projected cubic system of axes be drawn, and the axis 
 Z be taken as the morphological axis of the hexagonal crystal, the 
 axes X and Y being taken for the symmetry-axes [S s C] and \_S 9 C] 
 or [ilo] and [112] respectively, the axis Z^of the hexagonal 
 crystal will lie in the YZ plane between Z and Y, and the axes X' 
 and Y f will lie in the systematic planes S l and S 2 which are inclined 
 at 30 to the plane 2 3 and at 60 to S 3 (Fig. 382). 
 
 Let ABC be any three points on X'Y'Z' equidistant from the 
 origin of the system. Let a perpendicular from A meet the plane 
 XOY"m M, and draw through J/a line parallel to the axis J^and 
 cutting the axis X in N. Then ON, NM, MA are the rect- 
 
Projection of the axes of a hexagonal crystal. 483 
 
 angular coordinates of A, as referred to the rectangular axes OX, 
 OF, OZ-, and 
 
 OM= OA cosAOM = OA sinAOZ, 
 ON = OMcos NOM = OA cos 30 sin A OZ, 
 NM = OMsm NOM = OA sin 30 sin A OZ, 
 MA = OA s'mAOM = OA cosAOZ : 
 
 whence, dividing each by OA sin A OZ, the coordinates of A may 
 be taken as 
 
 on axis X; cos 30 or 0.866 ; 
 
 on axis .F; sin 30 or 0.5; 
 
 on zxisZ; cotan .4 OZ: 
 
 where cotan AOZ \ tan (ioo-iii), an angle which measures 
 the characterising element of the particular crystal (Art. 457). 
 
 Similarly, after dividing by OA sin A OZ as before, the co- 
 ordinates of B are found to be 
 
 on axis OX ; 0-866 ; 
 
 on axis OF; 0-5; 
 
 on axis OZ ; cotan A OZ : 
 and the coordinates of C are 
 
 on axis OX ; o ; 
 
 on axis OF; i ; 
 
 on axis OZ ; cotan A OZ. 
 
 The positions of the projected axes of the hexagonal crystal can 
 now be determined by means of the crystallograph ; lengths cor- 
 responding to the above coordinates are measured along the 
 projected cubic axes, the unit being in each case determined by 
 the limb along which the measurement is made (Art. 466). 
 
 (c) Oblique systems, (i) For the graphic representation of a 
 crystal in the Monosymmetric system by aid of the crystallograph, 
 the directions of the J^-axis and of the Z-axis are identical in the 
 projections of the crystal and of the cubic axial system, and that 
 of the Z-axis is retained in a vertical position. 
 
 The elements of the monosymmetric crystal being a\b\c, TJ, it 
 is necessary, as in the last Article, to find the rectangular coordi- 
 nates of A, B and C relative to the cubic axes OX, OF, OZ, where 
 
 i i 2 
 
4 8 4 
 
 Representation of crystals. 
 
 OA=a, OB = b, OC = c and AOC = rj. If AH be drawn 
 parallel to OZ meeting OX in H (Fig. 383), it will be seen that 
 the coordinates of A in space are 
 
 on axis A, OH = OA cos (rj - 90) ; 
 on axis Y, o ; 
 
 on axis Z, HA = OA sin (77 - 90). 
 The coordinates of B are 
 
 on axis X, o ; on axis Y, b ; on axis Z, o. 
 The coordinates of C are 
 
 on axis A", o ; on axis Y, o ; on axis Z, c. 
 By measuring along the limbs of the crystallograph lengths 
 representing these coordinates, the projected axes of the mono- 
 symmetric crystal are de- 
 termined as in the last 
 Article. 
 
 (2) The projection of 
 the axes and parameters 
 of an anorthic crystal can 
 best be obtained by finding 
 the values of the coordi- 
 nates for a point at para- 
 
 l 33 ' 
 
 metral distance from the origin for each axis. 
 
 The Z-axes for the cubic axial system and the anorthic crystal 
 may be taken as vertical, and the A-axis of the crystal may be 
 taken to lie in the plane ZX of the cubic system. Let the cubic 
 axes as projected be X' Y r Z '. 
 
 The elements of the anorthic crystal being a : b : c, f, 17, let 
 OH, OK, OL be the parametral lengths of its axes. The axial 
 angles of the crystal are f, 77, f, and the angle of 6 between the planes 
 HOL, KOL = i8o-A, A and B being the poles of the faces 
 100 and oio ; we have also, for the determination of the value of 
 6, the expression 
 
 where S= 
 
 cos S cos (S 
 
Projection of the axes of an anorthic crystal. 485 
 
 OH lies in the plane XOZ: drawing HS parallel to ZO and 
 meeting the axis OX in S, we have (Fig. 384) 
 i. for the point H the coordinates in space 
 
 on the ^XT-axis OS = OH cos SOH = a sin rj, 
 jT-axis o, 
 
 Z-axis SH = OH sin SOH = a cos r? ; 
 
 ii. for the point K, draw the line KR perpendicular to the plane 
 XOY and meeting it in R\ draw RT perpendicular to OX and 
 meeting it in T, then 
 
 OR = 6>^cos ROK = OK sin KOL = b sin 
 
 Fig. 384. 
 
 Further, 7Y9 and 7?0 being both perpendicular to OL, the angle 
 TOR is the angle between the planes TOL and 7?0Z, that is, the 
 supplement of the angle between the planes HOL and KOL. 
 Hence TOR = AB. 
 
 We thus have for the coordinates of K, 
 
 on axis X, OT= OR cosKOT= -sin f cos ^^, 
 r, RT= OR$mROT = + sin f sin ,4 , 
 Z, RK = OK cos KOL = b cos ; 
 
 and iii. for the coordinates of the point L we have, on the axes of X 
 and Y, o, on the axis of Z, c. Measuring these lengths along 
 the limbs of the crystallograph, the projections of OH., OK, OL are 
 immediately determined. 
 
 472. Projections of twin-crystals. In drawing a twin-crystal three 
 preliminary steps have to be taken. We have to determine by the 
 
486 
 
 Representation of crystals. 
 
 Fig. 385. 
 
 above methods the axial system for one of the individuals as seen in 
 projection ; next the twin-plane has to be represented ; and, finally, 
 the axial system for the second individual has to be determined. 
 
 We may first suppose the two crystals in apposition, and 
 the twin-face to be the face of combination. We now need to 
 find the position of the twin-axis relatively to the axial system of 
 
 crystal No. I, or, which is the same 
 thing, to find the point in which this 
 axis meets the twin-face to which it is 
 the normal. 
 
 Let the twin-plane parallel to the 
 face hkl intersect the actual axes in 
 H l K^ Zj ; and let the twin-axis, the 
 normal to the plane H^K 1 L^ J meet 
 the plane itself in the point D l (Fig. 
 385). By the aid of the crystallo- 
 graph it is possible, as already explained, to determine OH, OK, 
 OL, the projections of O l ff 1 , O^K^ O l L l : it is now requhed to 
 find OD, the projection of the normal O i D l . 
 
 Join D l to the angles of the triangle ff l J l L l) and produce 
 the lines to meet the sides of the triangle in M 1 N^ R l , respectively. 
 By calculation or construction, determine the ratios 
 
 K.M.-.M^L^ L^N.-.N^H^ H.R,: R,K,. 
 These ratios being unaltered by parallel projection (Art. 466), the 
 projected points M, JV, R can now be found by dividing the projected 
 lines KL> LH, UK in the same proportion. The intersection of 
 the lines HM, KN, LR is the projection D of the extremity of the 
 twin-axis. 
 
 473. When the axes are rectangular the construction of the 
 above ratios is simple, for the lines O 1 M v O l N v O 1 R l are then 
 perpendicular to the lines K^ Z 15 Zj H v H, K^ respectively, as may 
 be thus proved : K l Z x being the line of intersection of the planes 
 ff l K^ Z x and O l K^ L^ is perpendicular to the lines O l D l} O H^ , 
 normal to those planes, and is thus at right angles to the plane 
 HI O l Z> x and to every line in it ; hence K Z x is perpendicular to 
 6\ M l : similarly Z a H^ is perpendicular to O l N^ , and H^ K l to 
 O.R,. 
 
Projection of a twin-crystal. 
 
 487 
 
 To construct the ratios we may proceed as follows : con- 
 struct three triangles k^ o l / x , ^ Q I h^, h^ o^ k^ equal to the triangles 
 K^O.L^, L^O^H^ H^O^K^ as illustrated in Figs. 386 a, b, c\ 
 
 draw the perpendiculars o l 
 
 throuh 
 
 in the 
 
 respective triangles draw lines ^ /, / a h, h^ k having lengths identical 
 with those of the projected lines KL, LH, HK; join / a /, h^h, k^ k 
 
 Fig. 386 (a). 
 
 Fig. 386 (J). 
 
 and draw their parallels m^ m, n^ n, r l r. The positions of the 
 points 71/5 N, R on the sides of the triangle HKL are then deter- 
 mined by setting off lengths KM, LN, HR equal to the lengths 
 X' x m, /j n, h^ r. 
 
 474. In order to represent the axial system of crystal No. II, we 
 have recourse to the following 
 construction : 
 
 Project the axial system for 
 crystal No. I (Fig. 387). Deter- 
 mine on the axes the parametral 
 lengths OA, OB, OC for the face 
 of the form { 1 1 1 } belonging to 
 the octant, and also the inter- 
 cepts OH, OK, OL of the twin- 
 plane hkl. 
 
 Find the normal OD of the 
 plane HKL, meeting HKL in D, 
 and continue OD to (/ making 
 
 ig. 387- 
 
 DO' = OD. Then (/ will represent the origin of the axial system 
 of crystal No. II ; and if O'H, O'K, &L be drawn they will repre- 
 sent in projection the directions of the axes OX' , OF, OZ f of the 
 
488 Representation of crystals. 
 
 axial system No. II, and they will further represent on those axes 
 the parameters of the plane HKL. 
 
 If now through A,JB and C, the parametral points of a plane of 
 the form {in} in crystal No. I, parallels to O O f be drawn cutting 
 O'H, UK, (/L, in A', B\ C', these last points will be the points in 
 which the corresponding plane of the form {in} of crystal No. II 
 will meet its axes, O'A', O'B', O'C', being the parametral lengths as 
 represented in projection. The actual face of the form {in} 
 represented by the plane A' B f C' and the axial octant within which 
 it lies will have opposite signs to those for the face ABC of crystal 
 No. I, since the two crystals are conceived .in parallel orientation 
 previous to the assumed rotation of one of them round 0', so that 
 the plane A' B' C' would before and after the rotation lie in an 
 octant opposite to that on crystal No. II which contains the face 
 with the same symbol and signs as (hkt) on crystal No. I. 
 
 475. The various kinds of twins will require the axial systems 
 for the two individuals to be transferred relatively to each other 
 into new positions, though their orientation relatively to the direc- 
 tion of the twin-plane and twin-axis is not altered. 
 
 Thus for an interpenetrant twin the origin for both crystals 
 may be at the same point, or for the one crystal the points and 
 0' may be slightly removed from each other along the direction 
 of the line OD. For the condition of juxtaposition, the parallel 
 shifting of the axial system for crystal No. II will be such that 
 C/ D f is parallel to OD. In every case the new axes for crystal 
 No. II are parallel to the directions just found for the axes of 
 that crystal. 
 
 It may happen that we wish to project the two crystals on a 
 plane perpendicular to the twin-face, this face being then repre- 
 sented in the figure by a line through D perpendicular to OO'. 
 The line thus representing the twin-plane is usually, in such a 
 projection, taken as a vertical line. 
 
 476. General rules. For the projection of a given crystal 
 we have to first ascertain the requisite data ; which are, the sym- 
 metry and the system of the crystal, its parameters, the forms it 
 presents, and approximately the relative magnitudes with which 
 the faces of these forms are developed on the crystal. 
 
General rides. 489 
 
 It is in fact very desirable, and indeed, if the crystal presents 
 several forms, necessary to have a free-hand drawing of it, con- 
 structed with sufficient care and accuracy to show the relative im- 
 portance of the faces, and the consequent developement of the 
 edges in which the various minor forms meet with each other and 
 with the more important forms. 
 
 It will be necessary however in all cases to commence by laying 
 down the axial system in projection by aid of the crystallograph. 
 The axes should be drawn in considerable extension, the para- 
 meters and the intercepts for important faces being marked off at 
 some considerable distance from the origin. If now the edges of 
 two or three of the most prominent forms be drawn on this larger 
 scale and the intersections of the faces of these forms with one 
 another be determined, we have lines of appreciable length, to 
 which the shorter lines nearer to the origin that are actually to 
 represent these edges in the drawing can be made parallel with 
 great accuracy. The edges formed by faces of less conspicuous 
 forms, which are generally those with more complex symbols, and 
 on this account often more numerous from the symmetrical repeti- 
 tion of the faces, can also be now drawn in over the lines repre- 
 senting the edges of the first drawn forms. 
 
 477. In order however to make any advance we have to be 
 able to determine the direction of the edge for any two faces. 
 
 For this we may find the lines in which the two planes severally 
 would intersect the axial planes. Let the two planes (hkl) and (pqr) 
 intersect the axes in HKL and PQR (Fig. 388). HK and PQ 
 lie in the plane OHK, and if produced will meet in a point U 
 unless both are parallel to an axis. So also RQ and KL will meet 
 in a point S, and HL and PR in a point T; and the points S, T, U 
 are all points common to the planes hkl and pqr t and will lie in 
 a straight line which is the edge of intersection of those two planes. 
 
 Another method, and in general the most satisfactory one, is 
 really a particular case of the preceding. Since the direction of a 
 plane depends, not on the absolute, but on the relative values of the 
 intercepts, and the three intercepts of any plane on the axes may be 
 multiplied by any arbitrary number, the intercepts of the two planes 
 of which the edge is required should be so multiplied that the two 
 
490 
 
 Representation of crystals. 
 
 intercepts on one of the axes say the ^T-axis are identical for 
 both planes. The points corresponding to T, [7, P and H then 
 coincide, say at H f (Fig. 389), and a line joining this point to the 
 
 Fig. 388. Fig. 389. 
 
 point of intersection of lines QfR' and K'L', corresponding 
 to QR and KL, will have the direction required. Thus for 
 the planes (362) (313) the intercepts of the first plane on the 
 
 axes will be in the proportion -:-:-or2#::3r; those 
 of the second plane will have the ratios - : - : , or a : 3 b : c. If 
 
 O J 
 
 the second set be doubled, the intercepts of the second plane will 
 be 2 a, 6<5, 2c. 
 
 Hence, taking OH', OK', OL', OQ', OR', as 2 a, b, 3^, 66, 2c 
 respectively, draw Q'R' and K'L' ; the line H' S' , joining S' their 
 point of intersection to H' t has the direction required (Fig. 389). 
 And of course the direction of the edge may be likewise found 
 by determining the coordinates of a 
 point in the edge considered as an 
 origin-edge. Thus [uvw] being the 
 symbol of the edge, take off (Fig. 390), 
 
 OU on axis X u<z, 
 UV parallel to axis F=vb, 
 VW parallel to axis Z = w<r. 
 
 o 
 V 
 
 Fig. 390. 
 
 Then OWis the direction of the zone-axis in question (Art. 20). 
 
 478. Having thus the means of determining the direction of 
 any edge to be represented, we have to introduce the edges parallel 
 
Mer asymmetrical crystals. 491 
 
 to such directions in their proper positions and with appropriate 
 lengths. These depend on the relative importance to be given in 
 the drawing to the different forms presented by the crystal. It is 
 advisable in general to arrange that the crystal-figure shall be at a 
 part of the paper not too near the origin ; otherwise, as the drawing 
 progresses, unnecessary confusion results through the frequent 
 intersection of the growing figure by lines only drawn as aids to its 
 construction. As in most cases the crystal is to be represented ' in 
 equipoise/ care has to be taken that all the faces of each form are 
 projected in equivalent magnitudes. For this purpose it is best to 
 draw conspicuously the directions of the axes of symmetry in their 
 due projection. 
 
 The edges which are not lines of symmetry but belong to 
 conspicuous forms are then put in, and care is taken in doing so 
 that the lengths of the lines representing corresponding edges are 
 adjusted in accordance with the symmetry of the crystal as seen in 
 projection. 
 
 The parallelism of all edges formed by the faces of a zone with 
 each other simplifies very much the introduction of the lines repre- 
 senting such edges. It is generally advisable at first to draw the 
 crystal with the omission of the smaller faces, and to introduce 
 these subsequently ; and to do this notwithstanding the necessity 
 it involves of afterwards erasing lines already put in, when the 
 faces of the less conspicuous forms are introduced. But the time 
 lost is more than saved by the greater facility with which the proper 
 positions and distances from the origin of the various faces and 
 edges are maintained, while the projection of the crystal is growing 
 under the hand. 
 
 479. Drawing of mer asymmetrical crystals. In drawing a hemi- 
 or tetarto-symmetrical crystal it is generally advisable to draw the 
 crystal holosymmetrically and in equipoise, and then to deduce 
 from this the hemisymmetrical projection by the omission of some 
 and the extension of others of the edges of the crystal in accord- 
 ance with the symmetry. Where the merosymmetrical forms 
 occur only as replacements by small faces of certain of the quoins 
 and edges of the crystal, these are best introduced after the con- 
 struction of the holosymmetrical figure. Often however we have 
 
49 2 Representation of crystals. 
 
 to note that some hemisymmetrical forms concur which are 
 correlative and unequally developed on the crystal, and these 
 will obviously have to be treated as separate forms, each with its 
 distinct developement, and to be deduced from distinct imaginary 
 holosymmetrical forms. 
 
 480. Methods of verification. For the verification of the exacti- 
 tude with which the edges, and their intersections in quoins, have 
 been projected, various methods may be had recourse to. Exact 
 measurements of the lengths of corresponding edges should give 
 those lengths precisely equal. 
 
 Further, inasmuch as where a crystal is centro-symmetrical any 
 plane that divides it in half divides it, in the general case, into two 
 antistrophic or enantiomorphous halves, it is evident that if half a 
 holosymmetrical crystal be drawn accurately the other half can be 
 added to it by the simple method of taking on tracing paper a 
 tracing of the half-projection, rotating it in the plane of the drawing 
 through 1 80, and taking the quoins through by a pricking point 
 on to the paper carrying the original half-projection. The lines 
 joining these points should exactly correspond to the lines of 
 the half-figure already drawn. This method, of little general use 
 for the projection of the half of the front part of the crystal, is 
 of great use in putting in the dotted lines that usually represent the 
 edges of the other side : by thus repeating in reverse the lines and 
 points of the projection on the front half, it is at once seen if they 
 do not meet exactly, as they should do, when the two antistrophic 
 semiprojections are thus brought into unison. 
 
 In the case of a twin-crystal, the equal developement of the 
 two individuals is best secured by testing the position of corre- 
 sponding points on the two crystals by drawing through them, 
 or one of them, a parallel to the twin-axis, determining on it 
 the point of intersection with the twin-plane, and noting that the 
 corresponding quoins, or other corresponding points on the two 
 edges to be compared, are exactly equidistant from the said point. 
 
 481. Gnomonic projection. In the stereographic projection the 
 point of sight was taken at the surface of a sphere, and was the 
 pole of a great circle in the plane of projection. In the gnomonic 
 projection the centre of the sphere is the point of sight, and a 
 
Gnomonic projection. 493 
 
 tangent plane to the sphere is the plane of projection touching the 
 sphere at a point termed the centre of projection. The planes of 
 all great circles pass through the point of sight, and can only 
 intersect the projection-plane in 
 straight lines which will be of infinite 
 length. 
 
 In Fig. 391 and subsequent Figures 
 let be the point of sight, P the 
 centre of projection, and OP = r. 
 OU, OF and <9PFare three normals 
 to faces of the system, of which the 
 mutual inclinations are 
 
 [7V=0, VW=<$>, WU=^. 
 
 Then OP is the normal to the projection-plane, which may be 
 any plane arbitrarily chosen but must intersect 017, 0V, OW giving 
 intercepts Ou, Ov, Ow : OP is inclined on those normals at known 
 angles, viz. 
 
 (3, POW=y. 
 
 Then Ou = --, Ov = 
 
 cos a cos/3 cosy 
 
 vw = r cos a Vcos 2 /} + cos 2 y 2 cos /3 cos y cos \j/, 
 wu = r cos /3 >v/cos 2 y + cos' 2 a 2 cos y cos a cos $, 
 uv = rcos y \/cos 2 a + cos' 2 /3 2 cos a cos /3 cos 6, 
 
 expressions by the aid of which the points u v w and P can be 
 projected on the surface of the drawing which is the plane of pro- 
 jection. The plane selected for the purpose, and which is deter- 
 mined in relation to the crystal by the angles a, /3, y, is arbitrary, 
 but usually so chosen as in each different system to exhibit the 
 zones belonging to at least one systematic triangle ; the normals for 
 reference being also selected so as to enable these zones to be laid 
 down within convenient limits. 
 
 If a sphere of radius r be supposed described round as centre 
 and a great circle pass through two poles or other points A' and B f 
 on the sphere, these points will lie on a straight line upon the plane 
 of projection, which will be the projection of the great circle ; and, 
 
494 Representation of crystals. 
 
 if the points be poles, will be the projection of their zone-plane and 
 zone-circle. 
 
 482. To measure the angle between two lines of which the points of 
 projection .are given. 
 
 P being the centre of projection, let A B be the line on which 
 
 7) JB A the two P oints A/ and B ' are projected 
 
 in A and B (Fig. 392). 
 
 Then lies on a line vertically over 
 P, and AOB is the actual angle at 
 between the two normals OA / OB' . In 
 order by a graphic method to measure 
 this angle it will be requisite to lay down 
 a triangle congruent with A OB on the 
 lg ' 3 9 2 ' plane of the drawing. 
 
 Through P draw a perpendicular to AB, produced if necessary, 
 and meeting AB in D; also a line Pa> r, perpendicular to PD : 
 join Z>co, and on DP or its continuation take Ii such that 
 
 The triangle DOP will evidently coincide with the triangle 
 if it be turned on its edge DP till falls on oo, and 
 
 Similarly the triangle DO A becomes coincident with D IA after 
 turning round AD, and the angle DIA is the same angle as D OA 
 which is measured in the projection by the line AD. Hence the angle 
 between any lines represented by points on the projected line 
 AB is the angle between two lines joining these points with a point 
 H found by the above construction. 
 
 483. Two zone-planes being projected in CA , CB, intersecting in C, 
 to find the angle of their inclination ; or. conversely , given one zone- 
 plane in projection, to project a second zone-plane tautohedral with it at 
 a given angle in a pole C given in projection. 
 
 If OC be supposed drawn, and also PC, and a perpendicular to 
 PC and therefore to the plane OPC, be drawn through P inter- 
 secting the two zone-projections in A and B, and continued to o> 
 at a distance = r from P, it will be seen that the triangle Pa>C 
 will be congruent with the triangle POC if the latter be turned 
 round PC till it falls on the projection-plane. 
 
Gnomonic projection. 495 
 
 If now K, outside the plane of the paper, be the point in which 
 a plane AKB would be intersected by OC perpendicularly, the 
 angle AKB would be the angle of the edge OC formed by the two 
 zone-planes OA C and OB C. 
 
 Draw PD a perpendicular to Co) (Fig. 393) ; the line PK being 
 perpendicular to OC would fall into congruence with PD when the 
 triangle POC falls on the projection- ^ 
 
 plane, after rotation round PC. 
 
 In PC take P& = PD, and draw 
 
 Then the triangle AKB when turned 
 round AB till it falls on the projection- 
 plane is congruent with AQJB, and 
 
 
 the angle ALB is the actual angle Fi g- 393- 
 
 between the zone-planes CA, CB. Hence the simple graphic con- 
 struction by which PD, and thus Pl, is determined enables us to 
 find the angle between any two zones intersecting in C, or to 
 project a zone intersecting a zone given in projection at any 
 required angle. 
 
 484. In the application of these artifices of construction to the 
 projection of a crystal, that is to say to the projections of its zone- 
 planes, it is clear that the area included in the projection must be so 
 limited as not to embrace zones which by their angular distance 
 from the centre of projection would be represented by lines too far 
 removed from that centre. A distance representing 90 of angular 
 measurement from P is infinite in length, and therefore an area 
 the greatest diameter of which would represent an angle not larger 
 than about 45 on each side of P, or 90, is as large as can be 
 conveniently employed. 
 
 This area of projection needs further to be so chosen as to 
 exhibit as far as possible the essential features of the crystal in 
 regard to symmetry : and for this a systematic triangle or a group 
 of these naturally presents itself as offering the conditions required. 
 
 In the Ortho-symmetrical systems an octant bounded by three 
 systematic planes, and in the Hexagonal system two adjacent sys- 
 tematic triangles included by two planes -S 1 and the plane C, are the 
 usually projected portions of a crystal. 
 
496 
 
 Representation of crystals. 
 
 The normal OW would in the Ortho-symmetrical systems be the 
 Z-axis [ooi], and in the Hexagonal system the zone-axis [in] ; 
 U and V would be normals lying in the zone-plane [ooi] or [in], 
 in the two cases. 
 
 The intersections of two of the 2- or two of the ^-systematic 
 planes with this zone-plane are conveniently taken for the two 
 normals in the Cubic, Tetragonal and Hexagonal systems ; and, in 
 the Orthorhombic system, the axes X and Y, or two normals sym- 
 metrically situate in regard to X or Y. The projection-plane is in 
 all such cases taken more or less symmetrically with regard to the 
 normals OU, 0V, OW, and therefore in some cases is not a crystal- 
 face. 
 
 W 
 
 V 
 Fig. 394- Fig. 395. 
 
 Thus in all the systems in which three perpendicular normals 
 are allowed by the symmetry we may take < = ^ = , and the 
 
 plane uvw as equally inclined on the normals OU and 0V. The 
 centre of projection P will lie on WL, L being the middle point 
 of UV. 
 
 then cos a = cos/3 = sin y cos 1 = sin y cos e ; 
 
 and in Fig. 394, 
 
 WL = 
 
 sin y cos 7 
 
 OL = 
 
 smy 
 
 tan UWL = : 7 = tan e cos y. 
 
 tan 6 
 siny 
 
 The triangle UVW can be constructed by the aid of this last 
 
 ratio. 
 
 PL r sinycosy_ 2 
 
 WL ~~ tan y r 
 
 Also, 
 
Gnomonic projection. 497 
 
 so that P can be found on WL, 
 
 _PW_PU PU 
 "" 5 r PW 
 
 In the Cubic, Tetragonal and Orthorhombic systems the angle 
 c is taken as 45. 
 
 Generally y is taken = 45 and cosy = ? 
 
 V 2 
 
 /. cos a = J and a = 60. 
 
 Then, 7Z= - ^ PZ = i PFZ, P*7 = 
 
 V 2 
 
 If however we take a = (3 = y and 6 = 45, 
 then, Fig. 395, P7=/>F=PPF; 
 
 cos a = sin a cos 45, tan a = A/2, 
 
 a =54 44i'5 
 
 V3 
 In the Hexagonal system e = 30. 
 
 Here cos y is taken = -= 
 
 ^3 
 
 Then cos a = and a = 45. 
 
 v 2 
 
 UL = IWL=PL; PU= ^2 PW. 
 
 485. One of the advantages of the gnomonic projection lies in 
 its application to the drawing of crystals in orthogonal projection. 
 The lines in a gnomonic projection are the projections of zone-planes, 
 and therefore the edges of the faces belonging to any zone are per- 
 pendicular to the line in which the zone-plane is projected ; so that 
 the projections of all the edges of a crystal can be drawn in the 
 directions they would have were the crystal projected orthogonally 
 on the plane taken for the gnomonic projection-plane. 
 
 486. To obtain a gnomonic projection corresponding to the 
 axes used in Art. 470, we proceed as follows. Let the rect- 
 angular axes fixed in space and passing through the centre 
 of the sphere be OX, OZ, respectively horizontal and vertical, 
 parallel to the plane of the paper, and OF perpendicular to the 
 
 Kk 
 
498 
 
 Representation of crystals. 
 
 plane of the paper (Fig. 397). Let the paper remain fixed, and the 
 cubic axes OA, OB, OC, which initially coincide with OX, OY, OZ, 
 move under it by first a rotation 6 about OZ and then a rotation e 
 about OX. The first rotation brings the pole (oio) to v where 
 PB 1 = rtanb along the horizontal line; the pole (ooi) remains 
 at an infinite distance along the vertical line PZ, and (100) travels 
 from an infinite distance towards P along the horizontal line. The 
 second rotation brings (oio) to B, found by taking PQ = rtane, 
 and $#=rtan8, as may be seen from the construction in 
 
 Fig. 396, where OQ = - -, QOB = POB l = 8, 
 
 rtand 
 
 cose 
 
 QB = OQ tan 6 = 
 
 cose 
 
 \ 
 
 Fig. 396. 
 
 Fig. 397- 
 
 The line BQ is then the zone-line [oio, 100]; BD the zone- 
 line [oio, oo i ] may be drawn as a line inclined to BQ at an angle 
 representing 90 by the construction given in Art. 483. The poles 
 (100) and (oio) may then be set off along these lines at distances 
 BA, BC, representing 90 by the construction given in Art. 482. 
 
 The above projection may be more conveniently drawn directly 
 from the crystallograph by drawing the lines CA, CB perpendicular 
 to the B and A arms of the latter respectively and setting off a dis- 
 tance CP = r cote along a line parallel to the C arm (Fig. 397). 
 Then through Q t where PQ = r tan e, draw BA perpendicular to 
 CQ\ ABC are then the extremities of the rectangular axes, or the 
 poles (100) (oio) (ooi) in the Cubic, Tetragonal, or Orthorhombic 
 
Drawing implements. 499 
 
 systems. The complete projection for a crystal belonging to either 
 of these systems is constructed by laying down the poles of (no) 
 (on) (101) upon the sides of the axial triangle at distances 
 corresponding to their inclinations to ABC. 
 
 The rhombohedral or hexagonal projection may be derived from 
 the triangle by taking ABC as the poles (iol) (ill) (m) respect- 
 ively, and laying down other poles upon these zones by their 
 angular distances from ABC. The projection of a monosymmetric 
 crystal is obtained by taking C (ooi) at a distance from B repre- 
 senting the angle (100:001) along the line BC, and then laying 
 down other poles in the axial zones as before. 
 
 The projection of an anorthic crystal is obtained by taking B^ 
 (oio) at a distance from A, representing the angle (100:010) along 
 the line AB and drawing the zone-lines B^ C l and AC^ such that 
 their inclinations to AB represent the angles 77 and f respectively ; 
 and finally laying down other poles in the axial zones as before. 
 
 487. Drawing implements. It remains to notice the implements 
 and materials requisite for carrying on the processes described for 
 the projection of a crystal. 
 
 Good smooth drawing paper or card is the best material on 
 which to carry out the drawing. If paper is used it should be 
 mounted on a drawing board, and be fastened thereto at the 
 corners by a little gum. It should be of sufficient size for carrying 
 out the projection of the axial system of the crystal with axial 
 lengths equivalent to from 5 to 10 inches according to the 
 complexity of the crystal to be drawn. 
 
 It is generally well, where many faces have to be represented, to 
 draw the crystal-projection at some little distance away from the 
 axial projection obtained by aid of the crystallograph, as already 
 pointed out, and even sometimes to project more than one such 
 subsidiary axial system, all of course in absolutely parallel orienta- 
 tion with one another and with the corresponding lines in the 
 crystal. 
 
 This, and indeed the ruling of all ordinary straight lines on the 
 drawing, is best effected by help of two similar flat pieces of 
 vulcanite or thin steel, each in the form of a triangle with angles 
 of 90, 60 and 30, and having a hypotenuse 7 or 8 inches long ; 
 
 K k 2 
 
500 Representation of crystals. 
 
 for many figures a length of 2 or 3 inches is however sufficient. 
 The sides of the triangles should be as straight as they can be 
 made: the two implements when used together, the hypotenuse 
 of one sliding in contact with that of the other, form an excellent 
 parallel ruler. 
 
 Pencils of various degrees of hardness and blackness are needed, 
 and they must be used with a very fine point. In use, fainter lines 
 are employed for the purely structural lines, that form as it were the 
 scaffolding of the projection, and these can be rubbed out when 
 the final lines representing the complete drawing have been 
 introduced. A hard pencil, as dark as is consistent with hardness 
 and the retention of a fine point, is needed for the last process, 
 unless a ruling pen with Indian ink is substituted for the pencil. 
 The pen or pencil should lean away from the face of the ruler so 
 that the point may be in close contact with the edge formed by the 
 ruler and the paper. The dotted or broken continuity of line that 
 is employed to indicate the edges of the back-half of the crystal 
 needs an even and careful use of the pen or pencil point to give 
 it uniformity, and the delicacy requisite to prevent its obtrusion on 
 the eye. 
 
 The simplest way of introducing this representation of the 
 back-half of the crystal is that of the use of tracing paper already 
 described. The projected front-half is traced through and on to 
 the tracing paper, and this traced figure is then turned round in 
 its plane through 180 till corresponding points on it and on the 
 original projection are brought to meet; the quoins are then pricked 
 through on to the paper below : finally, these are to be joined by 
 dotted or interrupted lines. This method of course only applies 
 to diplohedral figures. In others the back-half of the crystal has 
 to be drawn by the same methods as the front-half. Where the 
 method of tracing can be applied it is of great advantage as 
 a test of the accuracy of the original drawing, since all the parts 
 of the one figure ought to join to and exactly meet with the 
 corresponding parts of the other. 
 
 It is necessary to use tracing paper for this and for transferring 
 a drawn figure to a plate on which it is to be engraved, since the 
 thickness of drawing paper is sufficient to displace points pricked 
 
Transference of the figure. 501 
 
 through it from their true position, owing to the difficulty of passing 
 a needle-point quite perpendicularly in every case. 
 
 For transference we may also use a coloured under-surface 
 to the tracing paper, on which a little rouge for instance has been 
 rubbed ; after laying the tracing on the sheet of paper or plate of 
 metal or stone to which it is to be transferred, the lines are gone 
 over with a hard point and a ruler, and are thus impressed on the 
 surface below. 
 
 Where the figure has to be reversed, as for direct engraving or 
 lithographing, the tracing has, of course, to be turned over and 
 the figure reversed on the surface that is to be engraved or litho- 
 graphed. 
 
DESCRIPTION OF THE PLATES. 
 
 THE Plates I to VIII represent in stereographic projection the poles of 
 a general independent form under all the varieties of mero-symmetry required 
 by the symmetrical conditions of the six crystallographic systems. The total 
 number of holo-symmetrical and mero- symmetrical types is thirty-two ; they 
 are numbered consecutively in the table given below. Some of the hemi- 
 morphous forms, as being readily deducible from those in the Plates, are not 
 represented in the projections, and the two types belonging to the Anorthic 
 system are too simple to require representation ; the omitted types are marked 
 with an asterisk in the table. The plan on which the projections are arranged 
 has been described in Article 149, p. 168. The merohedra corresponding to 
 those forms of which the poles lie in, and the faces of which are perpendicular 
 to, systematic planes of which the symmetry is in abeyance, in most cases reveal 
 their mero-symmetrical nature in the physical characters of their faces. 
 
 In the Plates the projections of proto-systematic planes are indicated by 
 darker lines than those of deutero- systematic planes. A trito-systematic plane 
 is taken as the plane of projection. 
 
 Subjoined is a list of the thirty-two types of symmetrical forms, and one or 
 two examples are given in each case of substances whose crystals present that 
 particular type of symmetry, where such are known to exist. 
 
 PLATE I. 
 
 CUBIC SYSTEM. 
 Holo-systematic forms. 
 
 1. Holo-symmetric (p. 201) . {hkl} fluor. 
 
 2. Haplohedral (p. 207) . a {hkl} cuprite. 
 
 3 (p. 212) . <r{hkl} blende; fahlore. 
 
 He m i-systematic forms . 
 
 4. Diplohedral (p. 215) . IT {hkl} pyrites; stannic iodide. 
 
 5. Haplohedral (p. 217) . air {hkl} sodium chlorate; sodium bromate ; 
 
 lead nitrate. 
 
Description of the plates. 503 
 
 PLATES II and III. 
 
 TETRAGONAL SYSTEM. 
 Holo-systematic forms. 
 
 6. Holo-symmetric (p. 247) . {hid} cassiterite. 
 
 TT i i , , / s r , , , ) ( strychnine sulphate ; ethylene- 
 
 7. Haplohedral (p. 254) . a {hkl} *. . F 
 
 1 J ( diamme sulphate. 
 
 8 ...... (p. 255) . sjhkl} 
 
 chalcopynte ; edmgtomte : urea. 
 (p. 257) . <r{hkl} i 
 
 9 ...... (hemimorphous) (p. 259) . p {hkl} iodo-succinimide. 
 
 Hemi- systematic forms. 
 
 10. Diplohedral (p. 259) . </>{hkl| scheelite, stoltzite. 
 
 11. Haplohedral (p. 262) . sa {hkl} (no example known). 
 
 12 ...... (hemimorphous) (p. 263) . pa {hklj wulfenite. 
 
 PLATES IV-VII. 
 
 HEXAGONAL SYSTEM. 
 Holo-systematic forms. 
 13. Holo-symmetric (p. 276) . {hkl, efg} beryl. 
 
 ,4. Haplohedral (p. ,87) . a {htl, efg} 
 
 5 ..... (P. 89). {hkl}, .{efg}, or f {hkl, efg} 
 
 \ r^ii ri i greenockite ; silver 
 *io ...... (hemimorphous) (p. 292) . p {hkl, efgj < . ... 
 
 Hemi-systematic forms. 
 17. Diplohedral (p. 293) . TT {hkl} 
 . . . .(p. 29 8) . ^ { hkl } 
 
 18 ..... (p. 299) . < {hkl, efg} apatite. 
 
 19. Haplohedral (p. 302) . aw {hkl} quartz; calcium dithionate. 
 20 ..... (p. 304) . x<p {hkl} (no example known). 
 
 *2i ...... (p. 306) . />rr{hkl} or p^ {hkl} tourmaline. 
 
 *22 ...... (hemimorphous) (p. 307) . p^ {hkl, efg} nepheline. 
 
 Tetarto-systematic forms. 
 
 23 Diplohedral (p. 307) .7r</>jhkl} or $$ {hkl, efg} dioptase, phenakite. 
 *24. Haplohedral (hemimorphous) (p. 308) . pir<j> {hkl} sodium meta-periodate. 
 
Description of the plates. 
 
 PLATE vm. 
 
 ORTHO-RHOMBIC SYSTEM. 
 Holo-systematic forms. 
 . Holo-symmetric (p. 333) . {hkl} barytes. 
 
 26. Haplohedral (p. 336) . .a {hkl} 
 
 27 (hemimorphous) (p. 337) . />{hkl}, ^{hkl} or {hkl} 
 
 hemimorphite, struvite. 
 
 MONO-SYMMETRIC SYSTEM. 
 Holo-systematic forms. 
 
 28. Holo-symmetric (p. 354) . {hkl} augite. 
 
 29. Haplohedral (p. 358) . . s{hkl} scolecite. 
 
 30 (hemimorphous) (p. 359) .a {hkl} lactose. 
 
 ANORTHIC SYSTEM. 
 
 *3i. The Holohedron (p. 372) . {hkl} axinite. 
 *32. The Hemihedron (p. 373) .a {hkl} calcium thiosulphate. 
 
 The holo-symmetric form of the Hexagonal system, not represented on 
 Plates IV- VII, is given in the adjoining figure on a scale somewhat larger 
 than that of the projections in the Plates. 
 
 Fig. 398. 
 
PLATE I." CUBIC SYSTEM. 
 
 505 
 
 The hexakis- 
 
 -octaliedran. 
 
 CTTTJkhl} 
 
 JTjJlklJ 
 
 Ll 
 
PLATE II. TETRAGONAL SYSTEM. 
 
 The scalene - 
 
 - dioeialiedinuL. 
 
PLATE III. TETRAGONAL SYSTEM. 
 uaoi 
 
 57 
 
 The scalene - 
 
 dio ciate dron. 
 
 cr|ldal{ Hemi-clioctahearon. 
 f 
 
 ILeminior ph ons 
 Hcmim.ojpphoTis - 
 
 scalenoKedron. 
 - dio c t aliedr oix. 
 
 Ll 2 
 
508 
 
 PLATE IV. HEXAGONAL SYSTEM. 
 For the Discalenohedron {hkl, efg} see Fig. 398, p. 504. 
 
 AM 
 
 ^ ttigonoJiedron. 
 - S-^rpramid. 
 
509 
 
 PLATE V. HEXAGONAL SYSTEM. 
 
 -trapexohcdron 
 s calenolie dr on . 
 
 JT{efgj 
 
PLATE VI. HEXAGONAL SYSTEM. 
 
PLATE VIL HEXAGONAL SYSTEM. 
 
 AM 
 
 scalenoliedrou.. 
 
PLATE VIII. ORTHO-RHOMBIC SYSTEM. 
 
 The scalene - 
 
 - octahedron. 
 
 The rhombic 
 
 sphenoid. 
 
 - henri- octahedron . 
 
 MONO-SYMMETRIC SYSTEM. 
 
INDEX. 
 
 The numbers refer to pages. 
 
 Abeyance of symmetry-planes, 165. 
 Abortive symmetry, 118. 
 Actual symmetry, 106. 
 Adjacent octants, 16. 
 Adjustment of crystal, 404. 
 Adjustment of goniometer, 394, 409. 
 yEolotropy, 13. 
 
 Albite, 176, 1 81, 374, 376, 379, 380. 
 Alexandrite, 185, 347. 
 Ambiguity of indices, 52. 
 Analysis of forms, 168. 
 Analytical investigation of the zone- 
 law, 65. 
 Anatase, 265. 
 
 Angle between two planes, 21. 
 Angles of axes of a crystal, 16. 
 
 of twins, 181. 
 
 Anharmonic ratio of four planes, 58, 71. 
 Anharmonic ratio, rationality of, 58, 7 1 . 
 Anhydrite, 346. 
 Anorthic system, 370. 
 
 calculation, 427. 
 
 combinations, 373. 
 
 twins, 375. 
 Anorthite, 375, 376, 387. 
 Antihemihedrism, 170. 
 Antistrophe, 99. 
 Apatite, 312. 
 Apophyllite, 172. 
 Aragonite, 184, 343, 347. 
 Argentite, 193, 200. 
 Arsenic, i. 
 
 Arsenious anhydride, 2. 
 
 Arzruni, 198. 
 
 Asymmetric semiforms, 207, 223, 
 
 254- 
 
 Attingent octants, 16. 
 Augite, 36, 365. 
 Axes are crystalloid lines, 91. 
 
 Change of, 80. 
 
 Axes of a crystal, 15. 
 
 of hexagonal system, 471. 
 
 of magnetic induction, 13. 
 
 of tetragonal crystals, 259. 
 
 Projection of, 479. 
 Axial-disphenoid, 255. 
 
 forms, 249. 
 
 isosceles octahedron, 250. 
 
 point, 27. 
 
 sphenoid, 256. 
 
 square prism, 251. 
 Axial-system, 27, 112. 
 
 for hexagonal type, 135. 
 Axinite, 90. 
 
 Axis of symmetry, 99. 
 
 morphological, 5. 
 
 Barium nitrate, 231. 
 
 Barytes, 343. 
 
 Basal pinakoid, 336, 358, 372. 
 
 Bauer, 377. 
 
 Baumhauer, 6, 266. 
 
 Baveno-twin, 369. 
 
 Ben-saude, 198. 
 
 Bertrand, 239. 
 
 Beryl, 298, 309. 
 
 Bevilment of edges, 70. 
 
 Bismuth, i, 320. 
 
 Blende, 166, 172, 179, 195, 197, 210, 
 
 211, 212, 224, 226, 227, 235. 
 Boracite, 2, 186, 226. 
 Bournonite, 343. 
 Bravais-symbols, 471. 
 Breithaupt, 381. 
 Brezina, 227, 380. 
 British Museum, 180, 222, 223, 226, 
 
 2 34> 2 4i> 3 2 5> 329, 34 T > 35, 3 62 , 
 
 366, 37* 
 Brochantite, 377. 
 Bronzite, 364. 
 
 m 
 
Index. 
 
 Calcite, 6, 9, 54, 172, 319, 320, 321, 
 
 34 2 . 
 
 Calculation of crystals, 417, 425. 
 Camphor, I. 
 Carlsbad-twin, 176, 366. 
 Cassiterite, 265, 268. 
 Cathrein, 185. 
 Centre of symmetry, 98, 99. 
 Centre-symmetry, 100. 
 
 normal, 17. 
 Cerussite, 351. 
 Chabazite, 322. 
 
 Change in solid condition, Crystallisa- 
 tion by, 2. 
 
 Change of parametral plane, 80. 
 Chromite, 193. 
 Chrysoberyl, 185. 
 Circle of projection, 29. 
 
 symmetry, 101. 
 Cleavage, 7. 
 Clinodome, 357. 
 Clinopinakoid, 358. 
 Clinorhombic system, 352. 
 Cobaltine, 215, 217, 227, 229. 
 Coefficients of elasticity, 5. 
 Cohesion, 5. 
 
 Coincidence of zone-axis and plane- 
 normal, 95, 112. 
 Combination-plane, 174. 
 Combinations, Anorthic system, 373. 
 
 Cubic system, 219. 
 
 Hexagonal system, 309. 
 
 Mono-symmetric system, 360. 
 
 Orthorhombic system, 339. 
 
 Tetragonal system, 264. 
 Composite crystals, 172. 
 
 symbol, 143. 
 
 Condition for mero-symmetry, 107. 
 
 several planes of symmetry,! 09, 117. 
 
 tautozonality, 49, 67, 68. 
 Conductivity, Thermal, IT. 
 Conformable symmetry, no. 
 Congruent heterozonal symmetry- 
 planes, 144. 
 
 Congruent symmetry, no. 
 Connellite, 310. 
 Contact-goniometer, 388. 
 Copper, 192, 193, 197, 221, 222, 234. 
 Copper glance, 184, 346, 351. 
 
 pyrites, 266, 270. 
 Correlative semiforms, 143. 
 Cromfordite, 265. 
 Crystalline condition, I. 
 Crystallisation after mutual decompo- 
 sition, 2. 
 
 from fusion, i . 
 
 Crystallisation from solution, I. 
 
 from sublimation, I. 
 
 in the solid condition, 2. 
 Crystallograph, 481. 
 Crystallographic calculation, 417. 
 
 elements, 4, 27. 
 
 systems, 4, 150. 
 Crystalloid plane-system, 25, 157. 
 
 symmetry, 101. 
 Crystallometric angles, 77, in. 
 Crystals, i. 
 
 as crystalloid polyhedra, 156. 
 
 Morphological characters of, 3, 103. 
 
 Physical characters of, 5. 
 Cube, 146, 191. 
 
 Angles of, 456. 
 Cube-pyramidion, 195. 
 Cubic system, 188. 
 
 calculation, 453. 
 
 holosymmetry, 188. 
 
 merosymmetry, 204. 
 
 Projection of, 479. 
 
 Relation of, to trigonal and hexa- 
 gonal, 151. 
 
 Cubo-octahedron, 28. 
 Cubo-octahedron symmetry, 146. 
 Cuprite, 180, 192, 193, 197, 198, 220, 
 223. 
 
 Dana, 227. 
 Datolite, 361. 
 Deformation of crystals, 6. 
 De 1'Isle, 228, 388. 
 Deltoid-dodecahedron, 210. 
 Derivation of determinants, 50. 
 Des Cloizeaux, 181, 366. 
 Designations for planes, 20. 
 Determinant-symbols, 50. 
 Determination of Crystallographic ele- 
 ments, 434. 
 
 twin-plane, 182. 
 Deutero-disphenoid, 257. 
 
 dome, 334. 
 
 hemioctahedron, 338. 
 
 pinakoid, 336. 
 
 prism, 283. 
 
 rhombohedron, 298. 
 
 scalenohedron, 298. 
 
 systematic planes, no. 
 Diagonal disphenoid, 257. 
 
 forms, 249. 
 
 isosceles octahedron, 251. 
 
 sphenoid, 258. 
 
 square prism, 252. 
 
 symmetry, 98, 99, 121. 
 Diamagnetism, 13. 
 
Index. 
 
 515 
 
 Diamond, 172, 179, 192, 193, 195, 
 
 221, 226, 232, 239. 
 Dihedral angle, 4, 21. 
 Dihexagonal prism, 278. 
 
 scalenohedron, 142. 
 Dilatation, Thermal, II, 112. 
 Di-n-gonal symmetry, 102. 
 Dioctahedron, 246, 259. 
 Diopside, 60, 361. 
 Dioptase, 314, 317. 
 Diplohedra, Hemitesseral, 213. 
 Diplohedral form, 105, 160. 
 Diplohedron, 213, 229. 
 
 Angles of, 462. 
 Direct forms, 141. 
 
 rhombohedra, 312. 
 Direction- cosines, 18. 
 Directions of planes, 17. 
 Dirhombohedron, 279, 280. 
 Discalenohedron, 276. 
 Disphenoid, Tetragonal, 255, 257. 
 Ditetragonal prism, 248. 
 
 scalenohedron, 247. 
 Ditrapezohedron, 287. 
 Ditrigonal prism, 294. 
 
 proto-prism, 290. 
 
 proto-pyramid, 289. 
 
 scalenohedron, 293. 
 
 symmetry, 133. 
 
 Dodecahedrid pyramidion, 201, 204. 
 Dodecahedron, 147. 
 
 Dolomite, 322. 
 Dome, 125. 
 Dome-forms, 356. 
 Drawing implements, 499. 
 
 of crystals, 488. 
 Ductility, 6. 
 Dyakis-dodecahedr on ,213. 
 
 Edges, 103. 
 
 Direction of, 22. 
 
 Relations of, to normals, 51. 
 Edingtonite, 266. 
 
 Elastic deformation, 6. 
 
 Elasticity, 5. 
 
 Electricity, 12. 
 
 Electrum, 195, 221. 
 
 Elements, The crystallographic, 4, 27. 
 
 Ellipsoid in crystals, 10, II, 13. 
 
 of magnetic induction, 13. 
 Embedded twin, 1 76. 
 Enantiomorphous figures, 169, 207, 
 
 217, 254, 261, 287,302, 308, 336, 
 
 359> 373- 
 
 Enstatite, 342, 364. 
 Epidote, 362, 366. 
 
 M 
 
 Equation of an edge, 65. 
 
 of a plane, 65. 
 Equilateral octahedron, 192. 
 Equipoised polyhedra, 100. 
 Erosion, 8. 
 
 Error of adjustment, 411. 
 
 of centering, 401. 
 Etching, 9. 
 
 Eulytine, 210, 225, 238. 
 Euthy-symmetrical division, 98. 
 Example of signs of indices, 53. 
 
 of tautozonal problems, 60, 69. 
 
 of transformation of axes, 88. 
 Expression for an edge, 22. 
 
 for anharmonic ratio, 57. 
 
 for a plane, 1 8. 
 
 for a pole, 41. 
 
 for a zone, 44. 
 
 for normal angle, 421. 
 
 for position of a face, 418. 
 
 for transformation of axes, 84. 
 Eyelets in projection, 41. 
 
 Face, 25, 103. 
 
 of union, 1 74. 
 Face-pole, 41. 
 Faces of a zone, 44. 
 
 Symmetry of, 151. 
 Factor-Ratios, 47. 
 
 Fahlore, 210, 211, 212, 224, 225, 226, 
 
 227, 235, 237. 
 Felspar, 61, 364. 
 Fergusonite, 267. 
 Fichtelite, 363. 
 Fletcher, 270. 
 Fluor, 192, 193, 197, 198, 199, 200, 
 
 220, 221, 222. 
 
 Form, 3, 101. 
 
 Forty-eight scalenohedron, 201. 
 
 Fourlings, 177. 
 
 Franklinite, 193, 200. 
 
 Friedel, 227. 
 
 Fuess-goniometer, 409. 
 
 Fundamental equation, The, 1 8. 
 
 laws of crystallography, 157, 161. 
 Fusion, Crystallisation from, i. 
 
 Galena, 176, 192, 198, 199, 200, 220, 
 
 221, 232, 235, 238. 
 
 Garnet, 195, 198, 200, 210, 220, 221, 
 
 222. 
 General properties of crystal, i. 
 
 scalenohedron, 123. 
 Gersdorffite, 227. 
 Glide-planes, 6. 
 Gnomonic Projection, 492. 
 
 rn 2 
 
516 
 
 Index. 
 
 Gold, 192, 195, 197, 221, 234. 
 
 Goniometer, 156, 388. 
 Gothite, 343. 
 Grassmann, 94. 
 Great circle, 27. 
 
 Pole of, 35. 
 
 Projection of, 36. 
 Greenockite, 314. 
 
 Groth, 181, 236, 238, 243, 267, 271, 
 
 273, 325, 33i, 471- 
 Gypsum, 361, 368. 
 Gyroidal forms, Hexagonal, 299. 
 
 hemihedrism, 207. 
 
 Habit, Merosymmetrical, 170. 
 Haematite, 178. 
 Haidinger, 174, 198, 227, 270. 
 Hand -goniometer, 388. 
 Haplohedral form, 105, 160. 
 
 hemisymmetry, 359, 373. 
 Hardness, 8. 
 
 Harmonic division of a zone, 75. 
 Harmotome, 366, 368. 
 Hauerite, 217, 227. 
 Haiiy, 136, 228, 388. 
 Haiiyne, 195. 
 Helvine, 210, 225. 
 Hemi-cyclical conformity, 1 10. 
 Hemi-deuteroprism, 307. 
 Hemi-deuteropyramid, 307. 
 Hemi-deuterorhombohedron, 307. 
 Hemidioctahedron, 259. 
 Hemidiprism, 260, 299. 
 Hemidisphenoid, 262. 
 Hemihedra, Holotesseral, 206. 
 Hemihedron, 159. 
 Hemi-icosi tetrahedron, 211. 
 
 Angles of, 460. 
 Hemimorphism, 161. 
 Hemimorphite, 344, 352. 
 Hemimorphous crystals, 267, 313, 
 
 3 1 ?' 344, 363- 
 
 dihexahedron, 306. 
 
 dioctahedron, 259. 
 
 ditrigonal prism, 306. 
 
 ditrigonal protopyramid, 306. 
 - forms, 301, 337, 359. 
 
 hemidioctahedron, 263. 
 
 hexagonal forms, 292. 
 
 rhombohedron, 306. 
 
 tetartohedral forms, 305. 
 
 trigonal pyramid, 306. 
 
 tritoprism, 307. 
 Hemi-octahedron, 209. 
 Hemiprotopyramid, 306. 
 Hemiscalenohedron, 307. 
 
 Hemispheres, positive and negative, 52. 
 Hemi-symmetrical combinations, 310. 
 
 twins, 318. 
 Hemi-symmetry, 159. 
 Hemi-systematic diplohedral forms, 
 
 213, 227, 259, 293, 359. 
 
 forms, 1 60, 338. 
 
 haplohedral forms, 217, 261, 302, 
 338. 
 
 mero-symmetry, 59. 
 Hemi-tesseral-semiforms, 213. 
 Hemitetragonal diplohedra, 259. 
 Hemi-triakis- octahedron, 210, 226. 
 
 Angles of, 458. 
 Hemi-tritopyramid, 307. 
 Hemitropic crystals, 173. 
 Hessenberg, 185, 225. 
 Heterozonal congruent symmetry 
 
 planes, 144. 
 
 poles, 45. 
 Hexagonal axes, 135. 
 
 deuteroprism, 283. 
 
 dihexahedron, 279, 281. 
 
 diprism, 278. 
 
 hemi-symmetry, 163. 
 
 prism, 281. 
 
 protoprism, 283. 
 
 pyramid, 279, 281. 
 
 symmetry, 98, 99, 134. 
 
 system, 273. 
 
 system, calculation, 463. 
 
 system, combinations, 309. 
 
 system, holosymmetry, 273, 
 
 system, mero-symmetry, 283. 
 
 system, Projection of, 482. 
 
 system, twins, 318. 
 
 type of symmetry, 1 34. 
 Hexahedron, 191. 
 Hexakis-octahedron, 201, 221. 
 
 Angles of, 460. 
 Hexakis-tetrahedron, 212, 226. 
 
 Angles of, 461. 
 Holohexagonal haplohedra, 287. 
 Holo-symmetrical combinations, 309, 
 
 339. 36o. 
 
 forms, 331, 352, 370. 
 
 twins, 318. 
 Holo-symmetry, 158. 
 Holo-systematic form, 160. 
 Holo-systematic haplohedral forms, 
 
 207, 254, 285, 337, 358. 
 
 hemihedra, 336. 
 
 mero-symmetry, 358. 
 Holo-tesseral hemihedra, 206. 
 Holotetragonal semiforms, 254. 
 Homogeneity, 13. 
 
Inaex. 
 
 517 
 
 Homologous poles and planes, 101. 
 
 positions, 324. 
 Hornblende, 361, 365. 
 Humite, 347. 
 
 Iceland-spar : see Calcite. 
 Icositetrahedron, 197, 220. 
 
 Angles of, 459. 
 Idocrase, 172. 
 Ilmenite, 314, 317. 
 Impact-figures, 7. 
 Independent plane and pole, 101. 
 Indicatrix, The optical, 10. 
 Indices, 19. 
 
 Rationality of, 3, 25, 157. 
 Induction, Magnetic, 12. 
 Integral indices, 25. 
 Intercepts, 17. 
 Interpenetrant twin, 176. 
 Inverse forms, 141. 
 
 rhombohedra, 310. 
 Iodine, i. 
 
 lodosuccinimide, 267. 
 Irrationality of parametral ratios, 112. 
 
 117. 
 
 Isogonal zones, 75, 77. 
 Iso-parametral axes, 136. 
 Isosceles dodecahedron, 279. 
 
 octahedra, 249. 
 Isothermal ellipsoid, n. 
 
 Juxtaposed twins, 175. 
 
 Kayser, 375, 381, 384. 
 Klein, 187, 406. 
 Koch, 198. 
 Kokscharow, 185. 
 Kopp, 238. 
 Kundt, 12. 
 Kupffer, 392. 
 Kyanite, 377, 380. 
 
 Labradorite, 176, 376. 
 Lang, 166, 167, 173, 393. 
 Lavizzari, 9. 
 Law of mero-symmetry, 161. 
 
 of rationality of indices, 3, 25, 
 Lead hyposulphate, 316. 
 
 nitrate, 210. 
 Leucite, 2, 186. 
 Leucitohedron, 186, 192. 
 Leucitoid, 198. 
 
 Levy, 380. 
 Lewis, 231. 
 Libethenite, 342. 
 Light, 9. 
 
 Line of Symmetry, 97. 
 Lustre, 3. 
 
 '57- 
 
 Macled crystals, 172. 
 
 Magnesium sulphate, 342, 344. 
 
 Magnetic induction, 12. 
 
 Magnetism, 12. 
 
 Magnetite, 180, 193, 199, 200, 220, 
 
 221, 234. 
 Mallard, 2, 186. 
 Malleability, 6. 
 Manebach-twin, 366. 
 Manganite, 347. 
 Marbach, 231. 
 
 Measurement of crystals, 388, 415. 
 Mercuric iodide, 2. 
 Merohedral, 159. 
 Mero-symmetrical combinations, 223, 
 
 266, 310, 343, 363. 
 
 crystals, Drawing of, 491. 
 
 forms, 204, 252, 283, 336, 358, 373. 
 Mero-symmetry, 156, 158. 
 Metastrophe, 99. 
 
 Microcline, 380 . 
 
 Miers, 223. 
 
 Milk-sugar, 345. 
 
 Miller, 80, 94, 136, 166, 382, 393, 475. 
 
 Mimetic crystals, 185. 
 
 Mispickel, 342, 347. 
 
 Mitscherlich, 392. 
 
 Models, 474. 
 
 Mohs, 228, 381, 384. 
 
 Molecules, Position of, 171. 
 
 Monosymmetric system, 352. 
 
 calculation, 436. 
 
 combinations, 360. 
 
 twins, 364. 
 Morphological axis, 5, 128. 
 Morphological characters of crystals, 
 
 3, 103- 
 
 Natrolite, 342. 
 Naumann, 228, 272. 
 Necessary conditions for plane of sym- 
 metry, 1 06. 
 Negative hemidome, 358. 
 
 hemiprism, 372. 
 
 hemispheres, 52. 
 
 parahemidome, 372. 
 
 prismatid, 354. 
 
 prohemidome, 372. 
 Neumann, 375, 380. 
 Nomenclature for planes of symmetry, 
 
 no. 
 Normals as axes of zone-planes, 92. 
 
 Relations of, to edges, 5 1 . 
 
Index. 
 
 Normals to planes, 17, ,21. 
 Notation for factor-ratios, 47. 
 
 for mero-symmetry, 166. 
 
 Oblique axes, 114. 
 
 systems, Projection of, 483. 
 Octahedrid-planes, 20. 
 
 pyramidion, 199, 220. 
 Octahedron, 20, 146, 192. 
 
 Angles of, 457. 
 Octaid-planes, 20. 
 Octakis-hexahedron, 201. 
 Octants, 1 6. 
 
 as quoins, 105. 
 Opposite octants, 16. 
 
 sides of a pole, 51. 
 Optical characters of crystals, 9. 
 
 indicatrix, 10. 
 Orientation of axes, 124. 
 Origin, 15. 
 Origin-edge, 25. 
 Origin-plane, 25. 
 
 Orthoclase, 176, 366, 369, 370, 378. 
 
 Ortho-dome, 356. 
 
 Orthogonal hemi-symmetry, 164. 
 
 projection, 475. 
 
 type of symmetry, 121. 
 Orthographic projection, 28. 
 Orthopinakoid, 358. 
 Ortho-prism, 355. 
 Ortho-rhombic system, 331. 
 
 calculation, 446. 
 
 combinations, 339. 
 
 twins, 345. 
 Ortho-symmetric systems, 331. 
 
 calculation, 441. 
 
 Projection of, 482. 
 Ortho-symmetrical division, 98, 121, 
 
 Pajsbergite, 342. 
 Parahemidome, 372. 
 Parallel projection, 474. 
 Parallelism of faces, 105. 
 Paramagnetic, 13. 
 Parameters, 19, 21. 
 
 Change of, 80. 
 Parametral hemidome, 358. 
 
 hemiprism, 372. 
 
 plane, 19. 
 - prism, 355. 
 
 Parapinakoid, 372. 
 
 Pentagonal icositetrahedron, 207. 
 
 Angles of, 462. 
 Pentagon-dodecahedron, 213, 227. 
 
 Angles of, 458. 
 Pentagonohedron, 207. 
 
 Periclase, 193. 
 Pericline-twins, 380. 
 Perowskite, 197. 
 Pharmacosiderite, 200, 224. 
 Phenakite, 314, 330. 
 Physical characters of crystals, 5, 104. 
 Piezo-electricity, 12. 
 Pinakoid, 283, 335, 358. 
 planes, 20. 
 Plagihedral forms, 315. 
 Plane, Cleavage-, 7. 
 
 figures, Symmetry of, 97. 
 
 Glide-, 7. 
 
 of projection, 29. 
 
 of union, 1 74. 
 
 Symmetry-, 99. 
 
 system, A, 15. 
 
 Planes, Mode of expressing, 15. 
 
 Normals of, 17, 21. 
 
 of symmetry inclined at 30, 1 34. 
 
 inclined at 45, 126. 
 
 inclined at 60, 130. 
 
 perpendicular, 121. 
 Point, Axial, 27. 
 Polarisation, 9. 
 
 Rotatory, 315, 344. 
 Pole of a great circle, 35. 
 
 of a plane, 27. 
 
 of symmetry, 98, 102. 
 Polyhedron, I, 25. 
 Polysynthetic twins, 1 76. 
 Position of a pole on the sphere, 41. 
 Positive hemidome, 358. 
 
 hemiprism, 372. 
 
 hemispheres, 52. 
 
 parahemidome, 372. 
 prismatid, 354. 
 
 prohemidome, 372. 
 Potential symmetry, 106. 
 Primitive circle, 29. 
 Prism, 125. 
 Prism-form, 355. 
 Prismatid forms, 334, 354. 
 Prismatoid planes, 20. 
 Problems of tautohedral zones, 62. 
 
 tautozonal planes, 58, 70. 
 Prohemidome, 372. 
 Projection, 474. 
 
 of a great circle, 36. 
 
 of an arc, 33. 
 
 of a pole, 35. 
 
 of circles, 31. 
 
 Sphere of, 27. 
 Propinakoid, 372. 
 Proto-disphenoid, 255. 
 
 dome, 334. 
 
Index. 
 
 519 
 
 Proto-hemioctahedron, 337. 
 
 octahedron, 249. 
 
 pinakoid, 336. 
 
 prism, 249, 283, 290, 291. 
 
 pyramid, 289, 291. 
 
 systematic planes, no. 
 Pseudo-cubic crystals, 186. 
 
 hexagonal crystals, 185, 341, 347, 
 
 348. 
 
 hexagonal twins, 184, 268. 
 
 rhombohedral crystals, 361. 
 
 symmetry, 184. 
 
 tetragonal crystals, 342. 
 Pyrites, 166, 192, 199, 215, 217, 227, 
 
 229, 243. 
 
 Pyritohedron, 213, 227. 
 Pyritoid minerals, 227. 
 Pyrochlore, 193. 
 Pyroelectricity, 12, 227, 318, 345. 
 
 Quartz, 12, 88, 172, 303, 314, 322. 
 Quoins, 16, 103. 
 
 Symmetry of, 153. 
 
 Rath, vom, 186, 318, 375, 376, 380, 
 
 382, 384, 386. 
 Rationality of anharmonic ratio, 58. 
 
 of indices, Law of, 3, 25, 157. 
 
 of symbols in twin crystal, 175. 
 
 of zone-symbol, 65. 
 Ratios, Factor-, 47. 
 
 Parametral, 19. 
 Ray-surface, 10. 
 Rays, 92. 
 
 Reciprocity of zone-system and plane- 
 system, 92. 
 Rectangular axes, 113. 
 
 in hexagonal type, 135. 
 Re-entrant angles, i, 177. 
 Reflection-goniometer, 390. 
 Regular octahedron, 192. 
 
 solids, 157. 
 
 Relation of three tautozonal planes, 46, 
 67. 
 
 of four tautozonal planes, 55, 70. 
 
 of edges and normals, 51. 
 Repetition, Symmetrical, 101. 
 Replacement of edges, 70. 
 Representation of crystals, 474. 
 
 Stereographic, 27. 
 Rhomb-dodecahedron, 193. 
 
 Angles of, 457. 
 Rhombic prism, 335, 354. 
 
 section, 382. 
 
 sphenoid, 336. 
 Rhombohedral system, 298, 309. 
 
 Rhombohedron, 295. 
 Rose,227, 229, 322,325,375,379,381, 
 Rotatory polarisation, 171, 315. 
 Rutile, 268. 
 
 Sadebeck, 179, 232, 237. 
 
 Sal-ammoniac, 223. 
 
 Salt, 192. 
 
 Sarcolite, 267. 
 
 Scalene dioctahedron, 246. 
 
 octahedron, 333. 
 Scalenohedron, 123. 
 Scheelite, 267, 269. 
 Scheererite, 363. 
 
 Schrauf, 135, 227, 380, 382, 386. 
 Sclerometer, 8. 
 Seignette salt, 352. 
 Semi-form, 159. 
 Senarmontite, 193. 
 Shear-plane, 7. 
 Sides of a pole, 51. 
 Signs of indices, 19, 51. 
 Silver, 192, 193, 197, 220, 221. 
 Similar edges, 103. 
 
 faces, 104. 
 
 planes of symmetry, no. 
 
 quoins, 104. 
 
 Simulation of higher symmetry, 184. 
 Skew rhombohedron, 307. 
 
 trigonohedron, 304. 
 Smith, Prof. H. J., 117. 
 Sodalite, 195, 235. 
 Sodium-ammonium tartrate, 343. 
 
 bromate, 231. 
 
 chlorate, 231, 243. 
 
 - periodate, 308, 314, 317, 330. 
 
 uranyl-acetate, 231, 267. 
 
 Solid condition, Crystallisation in, 2. 
 
 figures, Symmetry of, 99. 
 Solution, Crystallisation from, I. 
 
 of crystal-faces, 9. 
 
 of tautozonal problems, 58. 
 Sphene, 89, 362, 366. 
 Sphenoid, Tetragonal, 256, 258. 
 Sphenoidal hemi-symmetry, 337. 
 
 mero-symmetry, 255. 
 Sphere of projection, 27. 
 Spherical trigonometry, Use of, 27. 
 Spinel, 174, 193, 199, 200, 220. 
 Spinel-twin, 174, 233. 
 
 Square hexahedron, 191. 
 Staurolite, 346. 
 Steinmannite, 180. 
 Stephanite, 351. 
 Stereographic projection, 27. 
 Stolzite, 267. 
 
520 
 
 Index. 
 
 Striation, 3. 
 Stromeyerite, 346. 
 St diver, 228, 230. 
 Struvite, 344, 352. 
 Strychnine sulphate, 266. 
 Sublimation, Crystallisation from, i. 
 Sulphur, i, 341. 
 
 Supplementary planes of symmetry, 
 105. 
 
 twins, 181. 
 Sylvine, 192. 
 Symbol of a form, 101. 
 
 of a plane, 19. 
 
 of a zone, 45. 
 
 of cubo-octahedral forms, 147. 
 
 of hexagonal forms, 140. 
 
 of orthosymmetrical forms, 125. 
 
 of plane in two zones, 49. 
 
 of planes with changed axes, 81, 84. 
 
 of tetragonal forms, 129. 
 
 of truncating and bevilling planes, 
 
 of zones with changed axes, 81, 84. 
 Symmetral, 102. 
 
 Symmetrical twins, 181. 
 Symmetry, 97. 
 
 Axis of, 99. 
 
 Centre of, 98, 99. 
 
 Circle of, 102. 
 
 Crystalloid, 101. 
 
 Line of, 97. 
 
 of crystals, 4, 5. 
 
 effaces, 151. 
 
 of quoins, 153. 
 
 of twins, 181. 
 
 Plane of, 99. 
 
 Pole of, 98, 102. 
 
 Significance of, 171. 
 
 to one plane, 1 06. 
 Synthesis of forms, 168. 
 System of axes, 15. 
 
 - The planes of a, 15. 
 Systematic pinakoid, 358. 
 
 planes, no, 151. 
 
 projection, 475. 
 
 triangle, 123, 149. 
 Systems, The six, 4, 150. 
 
 Tartaric acid, 363. 
 Tartrates, 352. 
 Tautohedral zones, 45. 
 
 Relations of, 62. 
 
 Problems of four, 62. 
 Tautomorphous forms, 169. 
 Tautozonal planes, 46, 55. 
 
 planes, Problems of, 58. 
 
 Tautozonal planes, Relations of, 55. 
 - poles, 45. 
 Tennantite, 212. 
 Tesseral system, 188. 
 
 tetartohedron, 217. 
 Tetartohedrism of cubic system, 217, 
 
 230. 
 
 hexagonal system, 301. 
 
 tetragonal system, 261. 
 Tetartohedron, 159. 
 
 Tetarto - symmetrical combinations, 
 
 3I4- 
 
 twins, 322. 
 Tetarto-symmetry, 159. 
 
 hexagonal, 301. 
 Tetarto-systematic diplohedral forms, 
 
 307. 
 
 form, 1 60. 
 
 haplohedral forms, 308. 
 Tetragonal axes, 128. 
 
 axial prism, 251. 
 
 di-sphenoid, 255. 
 
 hemi-symmetry, 162. 
 
 isosceles octahedron, 259, 263. 
 
 pinakoid, 252. 
 
 scalenohedron, 127. 
 
 sphenoid, 256, 258. 
 
 symmetry, 98, 99, 126. 
 
 system, 245. 
 
 system, calculation, 448. 
 
 system, combinations, 264. 
 
 system, holosymmetry, 245. 
 
 system, merosymmetry, 252. 
 
 system, twins, 268. 
 tetartohedra, 261. 
 
 trapezohedron, 254. 
 
 type of symmetry, 126. 
 Tetrahedrid mero-symmetry, 208, 254. 
 pyramidion, 211, 224. 
 
 semi-forms, 208, 223. 
 
 twelve-pentagonohedron, 217. 
 Tetrahedron, 209. 
 Tetrakisdodecahedron, 201, 204. 
 Tetrakis-hexahedron, 195, 221. 
 
 Angles of, 457. 
 Tetrakonta- octahedron, 201. 
 Thermal conductivity, n. 
 
 dilatation, n, 112. 
 Thermo-electricity, 227. 
 Thickness of twin-crystals, 177. 
 Thomson, 399. 
 
 Topaz, 341. 
 
 Tourmaline, 12, 172, 317. 
 Transformation of axes, 80. 
 Transverse pole, 141. 
 
 rhombohedra, 312. 
 
Index. 
 
 Trapezohedron, 302, 314. 
 Triakis-octahedron, 199. 
 
 Angles of, 458. 
 
 tetrahedron, 211, 224. 
 Trigonal deutero-trapezohedron, 303. 
 
 clihexahedron, 297. 
 
 dodecahedron, 211. 
 
 proto-prism, 291. 
 
 proto-pyramid, 291. 
 
 symmetry, 98, 99, 130. 
 
 trapezohedron, 302. 
 
 type of symmetry, 130. 
 Triplings, 177. 
 Trito-diprism, 300. 
 
 hemioctahedron, 337. 
 
 pinakoid, 336. 
 
 pyramid, 299. 
 
 systematic planes, no. 
 Truncation of edges, 70. 
 Tschermak, 223. 
 Turnerite, 362. 
 Twelve-deltohedron, 210. 
 
 icoscelohedron, 211. 
 
 pentagon ohedron, 213. 
 Twenty-four-deltohedron, 197. 
 
 pentagonohedron, 207. 
 
 trapezohedron, 213. 
 Twin-axis, 173. 
 Twin-crystals, 173. 
 
 Projection of, 485. 
 Twin-face, 173. 
 Twin-plane, 173. 
 
 Determination of, 182. 
 
 parallel to face of union, 377. 
 
 perpendicular to plane of union, 
 380. 
 
 Twinning on a systematic plane, 1 79. 
 
 on octahedron-face, 233. 
 
 on rhomb-dodecahedron-face, 238. 
 
 on tetrahedron-face, 235. 
 
 Secondary, 6. 
 Twins, Angles of, 181. 
 
 anorthic system, 375. 
 
 cubic system, 232. 
 
 Twins, hexagonal system, 318. 
 
 Investigation of, 183. 
 
 mono-symmetric system, 364. 
 
 of hemi-symmetrical crystals, 1 78. 
 
 of quartz, 322. 
 
 on dome-face, 346. 
 
 on pinakoid, 321, 351. 
 
 on prism, 347. 
 
 on rhombohedron, 319. 
 
 on scalene octahedron, 346. 
 
 orthorhombic system, 345. 
 
 Symmetry of, 181. 
 
 tetragonal system, 268. 
 
 - Thickness of, 177. 
 
 Types of crystalloid symmetry, 121. 
 
 Unconformable symmetry, no. 
 Urea, 263. 
 
 Vertical prism, 355. 
 
 Water, i. 
 Wave-surface, 10. 
 Websky, 393, 409. 
 Weiss, 326. 
 Werther, 263. 
 Witherite, 184, 341. 
 Wolfram, 369. 
 Wollaston, 390. 
 Wulfenite, 267. 
 Wurtzite, 314. 
 
 Zero-index, 19. 
 Zinc oxide, 318. 
 Zippe, 229. 
 Zircon, 268. 
 Zone-axis, 25, 45. 
 
 circle, 44. 
 
 law, 46, 55, 65. 
 line, 25. 
 
 lines and normals, 95. 
 
 - plane, 44. 
 
 symbol, 45. 
 Zones, 44. 
 
 END OF PART I. 
 
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