UC-NRLF 
 
m. 
 
 REESE LIBRARY 
 
 OF THE 
 
 UNIVERSITY OF CALIFORNIA. 
 
 Deceived 
 ccession No. 7 / < )/ / Class No. 
 
THE 
 
 CHARACTERS OF CRYSTALS 
 
 AN INTRODUCTION 
 
 TO 
 
 PHYSICAL CRYSTALLOGRAPHY 
 
 BY 
 
 ALFRED J. MOSES, E.M., PH.D. 
 
 Professor of Mineralogy 
 Columbia Unvver$ity, New York City 
 
 NEW YORK 
 
 D. VAN NOSTRAND COMPANY 
 1*9* 
 

 COPYRIGHT 1899 
 
 BY 
 ALFRED J. MOSES 
 
 77.2-4 
 
 PRESS OF 
 
 THE NEW ERA PRINTING COMPANY, 
 LANCASTER, PA . 
 
PREFACE. 
 
 I have attempted, in this book, to describe, simply and con- 
 cisely, the methods and apparatus used in studying the physical 
 characters of crystals, and to record and explain the observed phe- 
 nomena without complex mathematical discussions. In the last 
 chapter the graduate course in Physical Crystallography given in 
 Columbia University has been outlined. 
 
 The works most consulted have been Grundriss der Physikalis- 
 chcn Krystallographie, 1 896, by T. Liebisch ; Physikalische Krys- 
 tallographie , 1895, by P. Groth ; Traite de Crystallographie, 
 1884, by E. Mallard; Treatise oil Crystallography, 1839, by W. 
 H. Miller; Crystallography, 1895, by N. Story-Maskelyne. 
 Footnote references are also given to many important articles. 
 
 It is hoped the book will be found useful to organic chemists, 
 geologists, mineralogists and others interested in the study of 
 crystals. 
 
 A. J. M. 
 
 MlNERALOGICAL DEPARTMENT, 
 COLUMBIA UNIVERSITY, NEW YORK CITY, 
 MARCH, 1899. 
 
CONTENTS. 
 
 PART I. GEOMETRICAL CHARACTERS. 
 
 CHAPTER I. INTRODUCTORY 1-9 
 
 CHAPTER II. THE GENERAL GEOMETRIC PROPERTIES OF CRYS- 
 
 
 ERRATA. 
 
 In Figs. 41, 89, 190 the axis of composite symmetry should be dotted 
 rig. 1 80 is upside down. 
 P. 50 for \h h 2 h i\ read \h h 2 hi}, 
 P- 53> line 7, for dehexagonal read dihexagonal. 
 P- 56, line 4, for Fig. m read Fig. 162. 
 P. 82, line 20, forOA=i. 73 2readOAXi. 7 32. 
 P. 84, line 5, for O read O'. 
 P. 96, last line, for Weinschenk read Weinshenk. 
 '114 Hne next to last for ^ = .009 read A = 530 /*/*. 
 P. 123, line 28, for two circular read two equal circular 
 P. 142, line 5 from the bottom, for acute read obtuse 
 P. 145, lines 3 and 4, for 146 and 147 read 118 and 119. 
 P. 146, lines 14 and 22, for 146 read 118. 
 
 dron, 47; C. 19, of 3 ord. Trigonal Bipyramid, 4&]~U^2o^~oi Jji- 
 trigonal Pyramid, 48; C. 21, of Ditrigonal Scalenohedron, 48; C. 22, 
 of Ditrigonal Bipyramid, 49; C. 23, of 3 ord. Hexagonal Pyramid, 52 \ 
 C. 24, of Hexagonal Trapezohedron, 52 ; C. 25, of 3 ord. Hexagonal 
 Bipyramid, 52; C. 26, of Dihexagonal Pyramid, 53; C. 27, of Dihex- 
 agonal Bipyramid, 53; Projection and Calculation, 55. ISOMETRIC: C. 
 28, of Tetartoid, 57; C. 29, of Gyroid, 57; C. 30, of Diploid, 58; 
 C. 31, of Hextetrahedron, 58; C. 32, of Hexoctahedron, 58; Projec- 
 tion and Calculation, 61. 
 
OF THE 
 
 UNIVERSITY 
 
 CONTENTS. 
 
 PART I. GEOMETRICAL CHARACTERS. 
 
 CHAPTER I. INTRODUCTORY 1-9 
 
 CHAPTER II. THE GENERAL GEOMETRIC PROPERTIES OF CRYS- 
 TALS 10-15 
 
 Symmetry, 10; Axes, n; Law of Rational Indices, n; Param- 
 eters, 12; Indices, 12; Determination of Elements of a Crystal, 13. 
 
 CHAPTER III. SPHERICAL PROJECTION 16-24 
 
 Zones, 1 6 ; Symbol for Zone Axis, 17 ; Equation for Zone Control, 
 17 ; Face in Two Zones, i 7 ; Fourth Face in a Zone, 18 ; Zone of Two 
 Pinacoids, 19; Zone through one Pinacoid, 19; Zones in which two In- 
 dices are Constant, 19; Changing Axes, 19; Changing Parameters, 20; 
 Stereographic Projection, 20; Problems in Stereographic Projection, 
 21-24. 
 
 CHAPTER IV. THE THIRTY-TWO CLASSES OF CRYSTALS. . . 25-62 
 TRICLINIC: C. i, Unsymmetrical, 26; C. 2, of Pinacoids, 26; 
 Projection and Calculation, 28. MONOCLINIC : C. 3, of Sphenoid. 31; 
 G. 4, of Dome, 31; C. 5, of Prism, 31; Projection and Calculation, 
 33. ORTHORHOMBIC : C. 6, of Bisphenoid, 35 ; C. 7, of Pyramid, 35 ; 
 C. 8, of Bipyramid, 36; Projection and Calculation, 38. TETRAGONAL: 
 C. 9, of 3 ord. Bisphenoid, 40;. C. 10, of 3 ord. Pyramid, 40; 
 C. n, of Scalenohedron, 41: C. 12, of Trapezohedron, 41; C. 
 13, of 3 ord. Bipyramid, 41 ; C. 14, of Ditetragonal Pyramid, 
 42; C. 15, of Ditetragonal Bipyramid, 42; Projection and Cal- 
 culation, 45. HEXAGONAL: C. 16, of 3 ord. Trigonal Pyramid, 47; 
 C. 17, of 3 ord. Rhombohedron, 47; C. 18, of Trigonal Trapezohe- 
 dron, 47; C. 19, of 3 ord. Trigonal Bipyramid, 48; C. 20, of Di- 
 trigonal Pyramid, 48; C. 21, of Ditrigonal Scalenohedron, 48; C. 22, 
 of Ditrigonal Bipyramid, 49; C. 23, of 3 ord. Hexagonal Pyramid, 52 ; 
 C. 24, of Hexagonal Trapezohedron, 52 ; C. 25, of 3 ord. Hexagonal 
 Bipyramid, 52; C. 26, of Dihexagonal Pyramid, 53; C. 27, of Dihex- 
 agonal Bipyramid, 53; Projection and Calculation, 55. ISOMETRIC: C. 
 28, of Tetartoid, 57; C. 29, of Gyroid, 57; C. 30, of Diploid, 58; 
 C. 31, of Hextetrahedron, 58; C. 32, of Hexoctahedron, 58; Projec- 
 tion and Calculation, 61. 
 
vi CONTENTS. 
 
 CHAPTER V. MEASUREMENT OF CRYSTAL ANGLES. ... . . 63-75 
 
 Application Goniometers, 63 ; Goniometers with Horizontal Axes, 
 64 ; Goniometers with Vertical Axes, (56 ; Errors due to Imperfect Cen- 
 tering, 7 1 ; Special Cases in Measurement, 7 2 ; Theodolite or Two- 
 Circle Goniometers, 74. 
 
 CHAPTER VI. CRYSTAL PROJECTION OR DRAWING 76-84 
 
 Linear Projections, 76; Orthographic Parallel Perspective, 77; 
 Clinographic Parallel Perspective, 79. 
 
 PART II. THE OPTICAL CHARACTERS. 
 
 CHAPTER VII. THE OPTICALLY ISOTROPIC CRYSTALS 85-96 
 
 Light Rays, 85 ; Ray Surfaces, 85 ; Ray Front and Front Normal, 
 85. CRYSTALS WHICH ARE SINGLY REFRACTING: Ray Surface, 86; 
 Index of Refraction, 86 ; Determination by Prism Method, 88 ; by 
 Total Reflection, 90. CRYSTALS WHICH ARE CIRCULARLY POLARIZING, 
 95. Absorption, 96. 
 
 CHAPTER VIII. THE OPTICALLY UNIAXIAL CRYSTALS . . . .97-121 
 CRYSTALS IN WHICH THE OPTIC Axis is A DIRECTION OF SINGLE RE- 
 FRACTION : Double Refraction, 97; Plane of Vibration, 98; Plane of 
 Polarization, 100 ; Ray Surface, 100; Optical Indicatrix, 101 ; Deriva- 
 tion of Positive Ray Surface, 102 ; Direct Determination of Principal 
 Indices of Refraction, 103. INDIRECT DETERMINATIONS WITH PLANE 
 POLARIZED LIGHT: Interference, 106; Poiariscopes, 106; Phenomena 
 with Parallel Monochromatic Light and Crossed Nicols, no; With 
 Parallel White Light, 112; Interference Colors, 113; Phenomena with 
 Convergent Light and Crossed Nicols, 115; With Parallel Nicols, 116; 
 Determination of Planes of Vibration or Extinction, 117 ; of Vibration 
 Directions of Faster and Slower Rays, 118 : of the Retardation A, 118; 
 of the Strength of the Double Refraction, 119; of Thickness of Section, 
 119; Approximate determination of Principal Indices, 120; Determina- 
 tion of Character of Ray Surface ,120. 
 
 CHAPTER IX. THE OPTICALLY UNIAXIAL CRYSTALS 
 
 (Continued). 122-131 
 
 CRYSTALS IN WHICH THE OPTIC Axis is A DIRECTION OF CIRCULAR 
 POLARIZATION: Plane, Circular and Elliptical Polarization, 122; Rota- 
 tion of Plane of Polarization, 1^3; Optic Axis a Direction of Double Re- 
 fraction, 124; Ray Surface, 125; Phenomena with Parallel Monochro- 
 matic Light and Crossed Nicols, 126; With Parallel White Light, 127; 
 With Convergent Light, 127 ; Determination of Direction of Rotation, 
 128; of Angle of Rotation, 129; Absorption and Pleochroism, 130. 
 
CONTENTS. vii 
 
 CHAPTER X. THE OPTICALLY BIAXIAL CRYSTALS 132-144 
 
 Optical Indicatrix, 133; Ray Surface, 133; Positive and Negative 
 Ray Surfaces, 136 ; Refraction, 136 ; Phenomena with Parallel Monochro- 
 matic Light and Crossed Nicols, 137 ; With Parallel White Light, 137 ; 
 With Convergent Monochromatic Light, 138; With Convergent White 
 Light, 140; Distinctions between Orthorhombic, Monoclinic and Tri- 
 clinic Crystals, 140. 
 
 CHAPTER XI. DETERMINATION OF THE OPTICAL CHARACTERS OF BI- 
 AXIAL CRYSTALS 145-162 
 
 Orientation and Determination of the Principal Vibration Directions, 
 145 ; Measurement of the Principal Indices, 146 ; Determination of 
 Angle between Optic Axes, 148 ; of True Axial Angle, 153 ; Calculation 
 of Axial Angle from Indices, 153 ; Determination of Character of Ray Sur- 
 face, 154; Crystals in Thin Rock Sections, 154; Absorption and Pleo- 
 chroism, 155. 
 
 Absorption Tufts, 158; Metallic Refraction or Metallic Lustre, 
 159; Surface Colors, 159; Fluorescence, 160; Phosphorescence, 161 ; 
 Norremberg and Reusch Combinations of Mica Plates, 161. 
 
 PART III. THE THERMAL, MAGNETIC AND ELEC- 
 TRICAL CHARACTERS, AND THE CHARAC- 
 TERS DEPENDENT UPON ELASTIC- 
 ITY AND COHESION. 
 
 CHAPTER XII. THE THERMAL CHARACTERS 163-171 
 
 Heat Conductivity, 164; Expansion by Heat, 166; Direct Meas- 
 urement of Linear Expansion, 167; Measurement of Expansion by 
 Change of Diedral Angles, 168; Determination of Expansion by 
 Changes in the Optical Character, 169. 
 
 CHAPTER XIII. THE MAGNETIC AND ELECTRICAL CHARACTERS OF 
 
 CRYSTALS 172-184 
 
 The Magnetic Induction of Crystals, 172 ; Strength of Magnetization 
 in Different Directions in a Crystal, 173; Transmission of Electric Rays, 
 175; Electrical Conductivity, 175; Thermoelectric Currents, 176; 
 Dielectric Induction in Crystals, 177; Pyro-Electricity, 180; Piezo- 
 Electricity, 182 ; Theory of Pyro- and Piezo-Electricity, 183. 
 
 CHAPTER XIV. ELASTIC AND PERMANENT DEFORMATION OF CRYS- 
 TALS '... 185-198 
 
 Homogeneous Elastic Deformation, 185 \ Elastic Deformation Due 
 to Pressure in One Direction, 185 ; Surface of Extension Coefficients, 
 
viii CONTENTS. 
 
 1 86; Effect of Pressure in One Direction Upon the Optical Characters, 
 187 ; Cleavage, 188 ; Gliding Planes, 190 ; Parting, igi ; Percussion Fig- 
 ures, 191; Etch Figures, 192; Corrosion Faces, 196; Hardness, 196; 
 The Methods of Static Pressure, 198. 
 
 APPENDIX. 
 
 SUGGESTED OUTLINE OF A COURSE IN PHYSICAL CRYSTAL- 
 LOGRAPHY 199-206 
 
 Preliminary Experiments, 199 ; Systematic Examination of the 
 Crystals of Any Substance, 203. 
 
PART I. GEOMETRICAL CHARACTERS. 
 
 CHAPTER I. 
 
 INTRODUCTORY. 
 
 It s a general property of definite chemical substances to as- 
 sume at solidification regular forms bounded by planes and obser- 
 vation has proved that the forms which occur are related to each 
 other and characteristic of the substance. 
 
 The geometric study of crystals has for its purpose the group- 
 ing of crystals into series, each of which shall consist of the forms 
 in which one substance can appear ; the determination of the angles 
 between the faces of crystals, and from these the elements and sym- 
 
 FlG. I. 
 
 FIG. 2. 
 
 bols of the faces, or conversely the determination of the form and 
 angles from the elements and the face symbols. 
 
 The term crystal originally meant the angular forms of the sub- 
 stance called rock crystal, Figs. I and 2, which, according to 
 PLINY,* was only "water frozen by the most excessive cold and 
 found only in places where snow is changed into ice." The an- 
 
 * Quoted by Rome Delisle, Essai de Crystallographie (1772), p. 3. 
 
2 CHARACTERS OF CRYSTALS. 
 
 gular shapes of this substance and of garnet,* beryl and possibly 
 diamond, were known to the ancients, but were regarded as acci- 
 dents and no general property was suspected. 
 
 In 1568 WENTZEL jAMiTZER,f a Nuremburg .goldsmith, devel- 
 oped in perspective over one hundred and forty simple and 
 complex shapes from the geometric tetrahedron, octahedron, 
 cube, dodecahedron and icosahedron by replacing all similar edges 
 and angles by one. or more planes, and the famous astronomer 
 KEPLER, in 1619, developed a similar series of figures. Many of 
 these shapes correspond to crystals, and the method of modifica- 
 tion may have suggested to Rome Delisle the method used by 
 him more than a century later. 
 
 The development of chemistry from alchemy was accompanied 
 by the study of many salts, the solutions of which, on evaporation, 
 yielded regular and constant shapes which were suspected to be to 
 
 FIG. 3. 
 
 some extent at least characteristic of the salt, for LIBAVIUS,^: in 
 1597, stated that the nature of the saline components of a mineral 
 water could be ascertained by an examination of the crystalline 
 form of the salts left on the evaporation of the water. 
 
 In 1669 NICOLAS STENO, a Danish anatomist, announced that if 
 
 * " There is also an incombustible stone found about Miletum which is of an angu- 
 lar shape, and sometimes regularly hexangular ; they call this also a carbuncle." The- 
 ophrastus's History of Stones. Trans, by Sir John Hill, p. 77. 
 
 f-Perspectiva Corporum Regularum, Quoted in Marx's Geschichte der Krystalkunde. 
 
 ^Roscoe Schorlemmer's Treatise on Chemistry, I. 705. 
 
 $De solido intra solidum naturaliter contento dissertationis prodromus. Florentiae, 
 1669: English translation, London, 1671. 
 
GEOMETRICAL CHARACTERS. 
 
 different specimens of rock crystal were examined, it would be 
 found that in spite of the variation in the relative size of the faces 
 and in the shape of the crystal there was no variation in the angles 
 between the faces. This he illustrated by figures of sections at right- 
 angles to a prism edge and others of sections at right angles to an 
 edge between a prismatic and a pyramidal face, as shown in Fig. 3. 
 A statement showing a very considerable, advance from this 
 was made by DOMINICO GULIELMINI in 1704, who asserted that 
 every salt had its peculiar shape which .never changed .--and that 
 even in imperfect and broken crystals the angles were constant. 
 
 T.ORBERN BERGMAN* records that his pupil GAHN having broken 
 a piece of Dogtooth Spar observed that it divided into little rhom- 
 bohedra. In studying this, Bergman found that by laying together 
 these rhombohedra in certain ways he could build up either a hexa- 
 gonal prism terminated by this form or the common scalenohedron 
 of calcite, or a form like the rhombic dodecahedron, all of which 
 are shown in Fig. 4. 
 
 From this he deduced that there was a relation be- 
 tween outer form and inner structure and that the con- 
 stituent parts of all crystals could be referred to a very 
 small number of primitive forms which could be found 
 by breaking the crystals. 
 
 In the preparation of models of crystal forms ROME 
 DELISLE, about 1783, measuredf the interfacial angles 
 directly with an application goniometer devised by 
 Carangeot for this purpose. He described over four 
 hundred crystal forms and formulated the now univer- 
 sally accepted law of constancy of interfacial angles 
 FIG. 4. as f u ows : j 4</ 7/ s pft e O f t j ie numberless variations of 
 which the primitive form of a salt or a crystal is capable, one thing 
 never vaties, but is always constant in each species, namely : The angle 
 of incidence or the respective inclination of the faces to each other." 
 
 The law of symmetry may also be said to be based on a law of 
 Delisle. " Every face has a similar face parallel to it." 
 
 * De Formis Crystallorum cited in von Kobell's Geschichte der Mineralogie. 
 p. 81. 
 
 \ The few measurements of crystals which had thus far been made were measure- 
 ments of plane angles only, which varied with differences in development so that the 
 interfacial angles calculated from them were often incorrect. 
 
 \ Cristallographie, ou description der formes propres & tous les corps du regne- 
 mineral. Paris, 1783. p. 70. 
 
CHARACTERS OF CRYSTALS. 
 
 Delisle also developed Bergman's idea of primitive form and by 
 a method similar to that of Jamitzer derived a series of secondary 
 forms from each primitive by replacing the edges or angles of the 
 primitive form by single planes or groups of planes in such a way 
 that equivalent geometric parts were similarly treated. He re- 
 ferred all forms to six classes of primitive form : The regular tet- 
 lahedon, Fig. 5, and the cube, Fig. 6, which permit of no varieties, 
 and the rectangular octahedron, Figs. 7, 8, 9 ; the rhomboidal paral- 
 lelopipedon Figs. 10, II, 12 ; the rhomboidal octahedron Figs. 13, 14, 
 15, and the dodecahedron with triangular faces, Fig. 1 6, all of which 
 were susceptible of many varietions according to their angles. 
 
 13 
 
 15 
 
 16 
 
 As an example of the method of derivation there could be de- 
 rived from the cube, Fig. 17, Fig. 18 by truncating each edge, 
 Fig. 19 by bevelling each edge, Fig. 20 by truncating each solid 
 angle, Fig. 21 by replacing each solid angle by three planes, each 
 cutting equal lengths from two edges and a different length from 
 the third, Fig. 22 by replacing each solid angle by six planes, each 
 
GEOMETRICAL CHARACTERS. 
 
 5 
 
 cutting three unequal distances on the three edges, and many 
 others. Similarly all the secondary forms of any compound could 
 be derived from a primitive form, and thus connecting them in a 
 definite series. Any series could, however, be derived from almost 
 any member of the series in the manner described and the primi- 
 tive forms could only be chosen arbitrarily. 
 
 18 
 
 20 
 
 The establishment of crystallography upon a mathematical basis 
 is chiefly due to the ABBE HAOY, who, apparently in ignorance of 
 the discoveries of Bergman, had his attention directed to the inter- 
 nal structure of crystals by the accidental dropping of a six-sided 
 prism of calcite which broke into rhombohedral fragments. He 
 found that this property of " cleavage " was a general one and that* 
 the crystals were built up of molecules of the shape of the cleavage 
 forms. Although in many instances no cleavage was found or 
 cleavage only in one direction, which yielded no solid, he assumed 
 that such solids did exist in all crystals and were the true primi- 
 tive forms upon which the other secondary forms depended and 
 that the shape in such cases could be determined by striations 
 and other markings on the faces or by analogy between the shapes 
 of the secondary forms and similar forms of other crystals which 
 did show cleavage. 
 
 Like Delisle he made six groups of primitive forms, of which 
 only two were identical with those of Delisle namely the tetrahe- 
 dron, Fig. 5, and the dodecahedron with triangular faces, Fig. 1 6. 
 
 The forms, i. PARALLEOPIPEDON. Including cube; right prisms 
 
 * Essai d'une theorie sur la structure des cristaux 1784. Journal de Physique 
 XLIII, p. 103. 
 
 Also in Gren's Neues Journal der Physik, 1795, VII., p. 418. 
 
6 CHARACTERS OF CRYSTALS. 
 
 with bases square, rectangular, rhombic, oblique ; oblique prisms 
 with bases rhombic, Fig. 28; rectangular, oblique; rhomboktdron 
 obtuse; acute. 2, REGULAR TETRAHEDRON. 3. OCTAHEDRON WITH 
 TRIANGULAR FACES. Including the regular Octahedron and the 
 octahedra with bases rectangular, square, rhombic. 4. SIX-SIDED 
 PRISM. 5. DODECAHEDRON WITH RHOMBIC FACES. 6. DODECAHE- 
 DRON WITH < TRIANGULAR FACES. 
 
 The existence of cleavages other than those producing the prim- 
 itive form led Haiiy to assume the existence of still simpler shapes 
 which he called integrant molecules. These he limited to three 
 kinds. The tetrahedron, the trigonal prism and the parallelopip- 
 edon. 
 
 With crystals of any substance Haiiy discovered that all forms 
 other than the primitive form could be exactly imitated by build- 
 ing on the faces of the primitive form successive plates or layers 
 of integrant molecules each successive layer regularly diminishing 
 by the abstraction of one or more rows, either parallel to each 
 edge, or to the diagonals of the faces of the primitive form, or 
 parallel to some intermediate line. A rhombic dodecahedron 
 might, for instance, be found to cleave with equal ease in three 
 directions at right angles to each other. The integrant molecules 
 would then, according to Haiiy, be cubes and the kernel or prim- 
 
 FIG. 23. FIG. 24. 
 
 itive form also a cube, Fig. 23 ; then, as perfect cleavage requires 
 that the minute integrant molecules be so piled as not to break 
 joints, the structure would be similar to that shown in Fig. 24, in 
 which the successive layers piled on top of any face of the cubic 
 primitive form regularly diminish one row at a time on each edge. 
 
GEOMETRICAL CHARACTERS. 7 
 
 For only with such a rate of decretion can the pyramidal planes 
 sOI and tOI unite to one plane sOIt and the diedral angle between 
 alternate planes be ninety degrees. As the number of plates added 
 would be very large and the little cubic molecules too small to be 
 separately visible the steps would appear to lie in the planes.' 
 
 Another example may be given to show that this method of 
 building yields forms corresponding to actual crystals. Fig. 25 
 shows a form observed on cobaltite, in which the diedral angle 
 at the edge,/*?, is 126 52'. Assuming a cubic kernel, Fig. 26. Fig. 
 27 shows the structure according to Haliy, in which the decretions 
 
 FIG. 25. FIG. 26. FIG. 27. 
 
 correspond to a triangle with a base of 2 and a perpendicular of 7, 
 that is, the angle at the base has a tangent equal to 0.5, or is 26 
 <34 r . The angle at p q is twice the complement of this ; that is, 
 126 52'. 
 
 The decretion* was symmetrical that is it was repeated on all 
 similar parts of the kernel and Hauy's experiments showed that the 
 secondary planes usually resulted from the subtraction of one or of 
 two rows, and were always according to some simple rational num- 
 ber never to his knowledge exceeding four. This is essentially the 
 basis of the law of rational indices, the fundamental law of crystal- 
 lography. 
 
 PROF. BERNHARDI, of Erfurt, pointed out| that the primitive 
 form should be chosen, not as with Haliy, from molecules 
 of nature, but according to convenience and fitness, and that 
 
 *The law of symmetry was stated by Haiiy as follows: "It consists in this, that 
 any one method of decretion is repeated on all those parts of the nucleus of which the 
 resemblance is such, that one can be substituted for the other by changing the position 
 of this nucleus with respect to the eye, without it (the nucleus) ceasing to be pre- 
 sented in the same aspect." Memoire sur une loi de Cristallisation, 1815. 
 
 f Von Kobell's Geschichte der Mineralogie, p, 197. 
 
8 CHARACTERS OF CRYSTALS. 
 
 the paralellopipedon was not a satisfactory form for calculation 
 in many cases, as it was often an open form the height of which 
 could only be determined from some secondary face, and he recom- 
 mended the choice of closed primitive forms only and suggested 
 seven, six of which are still regarded as the simplest forms in the 
 crystal systems, namely : Cube, Rhombohedron, Square Octahedron, 
 Rlwmbic Octahedron, Rhomboidal Octahedron, Triple Rhomboidal 
 Octahedron, the seventh form was a rectangular octahedron which 
 can be referred to the rhombic octaJiedron. 
 
 PROF. WEISS, of Berlin, devised a purely geometric mode of treat- 
 ment in which he discarded entirely the idea of primitive form, say- 
 ying * " that Haiiy's hypotheses of decreted rows entangled the 
 problem with self-created difficulties " and that the " mechanical- 
 atomic presentation which Hauy deduced should be stripped away 
 so that the acquired knowledge of the mathematical relation should 
 be more clearly seen." Weiss assumed the existence of certain fun- 
 damental lines or axes passing through a common center and 
 made four groups: 1st, with three equal axes at right angles to 
 each other; 2d, with three axes at right angles to each other, of 
 which two were equal; 3d, with three axes at right angles, and all dif- 
 ferent lengths ; 4th, with three equal axes at sixty degrees to each 
 other in one plane and one at right angles to these, but of different 
 length. 
 
 He deduced all the primitive forms of Hauy by constructing 
 planes which passed : First Through ends of three lines. Second 
 Through ends of two of the lines and parallel to a third. Third 
 Through an end of one of the lines and parallel to two of them. 
 By taking points along each of these lines at twice, three times 
 and four times, etc., the original length, and constructing planes 
 in the same way as before, he obtained all the secondary forms. 
 
 The lengths of the semi-axes of the primitive forms were called 
 a, b and c, and the position of any face was denoted by the ratio 
 of the intercepts in terms of a, b and c ; thus, 2a : b : ^c or a : ^b : 2c, 
 and so on. This method is still the simplest and most satisfactory 
 if only an elementary knowledge of type forms is desired, but is 
 cumbersome and tedious in calculation. 
 
 HAUSMANN applied spherical trigonometry to crystallographic 
 calculations in 1803 and BERNHARDif in 1808 pointed out that it 
 
 * Uebersichtliche Darstellung der verschiedene Abtheilung der Krystallisations- 
 Systeme. Denkschrift der Berliner Akad. der Wissen. 1814-15. p. 298. 
 \ Gehlen's Journal, 1808, 2, 378. 
 
GEOMETRICAL CHARACTERS. 9 
 
 would be better to determine trigonometrically the relations be- 
 tween the lines from a common point, normal to each face, as these 
 were the directions of attraction and growth. 
 
 PROF. MOHS, of Freiberg, grouped* all forms in four SYSTEMS, the 
 Cubic, Pyramidal, Rhombohedral and Prismatic, each consisting of 
 a series of forms geometrically derived from a " fundamental form " 
 by modifying planes which cut distances from certain lines in the 
 ratio of whole numbers. The fundamental forms were the simplest 
 closed forms and were not selected from cleavage or structure, but 
 purely for geometric simplicity. In 1822 he proved the existence 
 of two more Systems, the Monoclinic and Triclinic, the forms of 
 which had previously been considered as partial forms of the other 
 four systems. 
 
 F. C. NEUMANN in 1823 proposed a graphic method in which 
 the crystal faces were indicated by the points in which radii drawn 
 normal to the faces met the surface of a circumscribing sphere. 
 
 In 1825 WHEWELLf showed that if a solid angle of the primitive 
 form was taken as the origin, and the edges as cordinates, with 
 lengths x, y, z, then any secondary face would be expressed by the 
 
 A Al & 
 
 equation 7 - *y. T = m, in which m was not dependent upon the in- 
 
 rl K / 
 
 clination of the face, and, therefore, the quantities \ T' T' j [ or 
 
 say { /, q, r } could represent the face. These are essentially 
 the reciprocals of the Weiss intercepts, and are the indices used by 
 Miller. 
 
 By far the most important advance since Haiiy was the method 
 developed J by PROFESSOR W. H. MILLER, of Cambridge Univer- 
 sity, in which, by spherical trigonometry, a series of simple sym- 
 metrical expressions were deduced for the normal angles in terms 
 of the reciprocal intercepts of Whewell, and the positions of faces 
 were indicated by stereographic projection of the points in which 
 radii perpendicular to the faces meet the surface of the sphere. 
 The method is still the best, and is gradually supplanting all 
 others except for elementary work. 
 
 * Treatise on Mineralogy, or the Natural History of the Mineral Kingdom. (Haid- 
 inger's translation), Edinburgh, 1825. 
 
 f A general method of calculating the angles made by any planes of crystals. Rev. 
 W. Whewell, Transactions Royal Society, 1825, CXV., 87-180. 
 
 \ A treatise on Crystallography, 1839. 
 
CHAPTER II. 
 
 THE GENERAL GEOMETRIC PROPERTIES OF 
 CRYSTALS. 
 
 A PLANE OF SYMMETRY is a plane through the centre which 
 divides the crystal so that either half is the mirrored reflection of 
 the other, and every line perpendicular to the plane (and within 
 the solid) connects corresponding points of the solid and is bisected 
 by the plane. Every plane of symmetry is necessarily parallel to 
 a possible crystal face, for if two faces are symmetrical to it, it 
 must pass through their edge, that is to be in the same zone. So, 
 also, with a second pair of faces. But any plane lying in two 
 known zones has rational indices (p. 11) and is a possible crystal 
 face, as will be shown later. 
 
 One plane of symmetry may occur alone. If two planes occur 
 they must be at right angles, and they make necessary a third at 
 right angles to both. 
 
 AN Axis OF SYMMETRY is a line through the centre such that a 
 revolution of 180 or less around it will bring a face into coinci- 
 dence with the original position of an equivalent face. 
 
 The intersection of two planes of symmetry is necessarily an 
 axis of symmetry. Every axis of symmetry is either parallel to 
 an edge between two possible faces or normal to a possible face. 
 It has been proved* that that there can be only four kinds of axes 
 of symmetry : Binary, Ternary, Quaternary and Senary in which 
 equivalent faces become coincident by revolutions of 180, 120, 
 90 and 60 respectively. These may occur alone or in combi- 
 nations or with planes of symmetry. 
 
 There may be distinguished three methods by which equivalent 
 planes equally distant from the centre may be made to coincide : 
 
 1st. By reflection in a plane of symmetry. 
 
 2d. By rotation around an axis of symmetry. 
 
 * Groth's Physikalische Krystallographie, p. 313, III. edition. 
 
GEOMETRICAL CHARACTERS. n 
 
 3d. By composite symmetry or simultaneous rotation around an 
 axis and reflection in a plane. 
 
 Composite symmetry can occur with a binary axis, and plane 
 perpendicular thereto ; it is impossible with a ternary axis, and with 
 quarternary or senary axes is equivalent to simple symmetry with 
 binary or ternary axes respectively. 
 
 AXES. The faces of crystals are defined in position by referring 
 them to imaginary lines called axes.* The selection is largely ar- 
 bitrary, but with the object of securing the simplest indices. For 
 this purpose they must be lines parallel to the edges between pos- 
 sible planes and lines as important to the symmetry as possible. 
 Generally three directions are selected which are parallel to the in- 
 tersections of three occurring or possible faces. 
 
 Let XX Y^T ZZ, Fig. 28, be three such 
 axes and let all distances be measured on 
 these lines from their origin or common in- 
 tersection O, distances in the direction OX, 
 OY or OZ being regarded as positive and 
 those in the directions OX, OY or OZ 
 as negative. 
 
 Any plane as A B C would be determined 
 in space by its intercepts OA, OB, OC, on 
 * these axes, or in angular position by the 
 
 ratios of these intercepts, and as in crystals 
 
 there is constancy in inclinations of planes to each other and there- 
 fore to the axes, but no constancy in the linear dimensions, some 
 expression for the relative instead of the absolute values of the in- 
 tercepts is used as a symbol, which may be the intercepts expressed 
 as a proportion or some condensed expression derived from the 
 ratios of the intercepts. 
 
 The Fundamental Law or Laiv of Rational Indices. 
 
 Experience teaches that a simple relation exists between the in- 
 tercepts of different planes of the crystals of any chemical sub- 
 stance, and that if the axial intercepts of any face be divided by the 
 corresponding intercepts of any other face the quotients will be only sim- 
 ple rational numbers. 
 
 * Equivalent axes are necessarily surrounded by the same number of faces placed 
 in the same way. 
 
12 CHARACTERS OF CRYSTALS. 
 
 For instance, if the intercepts of some face ABC are 
 OA: OB:OC = a: b\c t 
 
 then these values divided by the intercepts of any other face H K L 
 will yield quotients 
 
 _^ _b_ *"./.*./ 
 OH : OK* OL~ 
 
 in which h, k and / are simple rational numbers such as - , , i ,2,3. 
 PARAMETERS. 
 
 Some selected plane A B C is called the parametral face or unit 
 face, and the values a b c, which express the simplest ratios of its 
 intercepts, are called the parameters of the crystal. 
 
 INDICES. 
 
 It is always possible to express the ratios between h, k and / by 
 three whole numbers because, as just stated they are simple 
 rational numbers, and if fractional they may always be cleared of 
 fractions without changing the ratios, for evidently 
 
 h : k : /= mh : mk : ml 
 
 The simplest* whole numbers which express the ratios of h, k 
 and / are called the indices .of the face, and are always written in 
 the same order, the first referring to the intercept on the axis X X, 
 the second to that on Y Y, the third to that on Z Z. A bar over 
 an index, as k, indicates that the intercept is negative. As in- 
 dicating a face they should be written (hkl), though frequently 
 the parenthesis is omitted ; but if used as a symbol of a form they 
 should be written \hkl \. 
 
 Indices and intercepts are inversely proportional, for from the 
 equation above we obtain : 
 
 in which a, b and c are constant for all faces of a crystal. 
 
 If a face is parallel to an axis its intercept on that axis is infinite 
 
 and its index is zero; for example, if j =<x> , then h =o and the 
 symbol is (o//). 
 
 * Experience shows that the faces which most frequently occur will, with proper 
 selection of the parametral plane, have as indices o or I or rarely 2. 
 
GEOMETRICAL CHARACTERS. 
 
 13 
 
 The values of the intercepts OH, OK, OL, become the para- 
 metral values only when kkl=ni. For with these values only 
 can we obtain : 
 
 DETERMINATION OF THE ELEMENTS OF A CRYSTAL, 
 
 The three axial planes and the " parametral face " constitute the 
 elementary planes. These are chosen to yield the simplest indices 
 and are usually planes of cleavage, or twinning or gliding planes, 
 and are preferably planes of symmetry. 
 
 In the most general case there are five undetermined elements, 
 namely : 
 
 The angles between the axes, Y Z or a, X Z or /?, X Y or y. 
 The ratio between the parameters a^ b,c, in which b = I. 
 
 I Determination of the angles between the axes. 
 These may be determined most simply from the measured angles 
 between the axial planes, as follows : 
 
 Let the surface of a sphere, described around 
 O, the centre of the crystal, meet the axes in 
 X, Y and Z, Fig. 29. 
 
 Construct A B C the polar triangle of X Y Z, 
 then will A be the pole of the axial plane 
 YOZ, B the pole of XOZ, and C the pole 
 of XOY. 
 
 The sides of A B C therefore measure the 
 normal angles between the axial planes, 
 FIG. 29. that is between the planes (oio), (100), (ooi) 
 
 and these angles are determined by measurement 
 
 A B = (loo) (oio), B C= (oio) (ooi), A C=(ioo) (ooi). 
 The angles of the triangle can therefore be calculated by formula 
 
 - cos BC 
 for instance : cos CAB = 
 
 cosABcos AC 
 
 : r - o : ^-~ 
 sin A B sin A C 
 
 Since ABC and X Y Z are polar triangles 
 
 a= YZ=i8o CAB, /9=XZ= 180 CBA, 
 =XY= 180 ACB. 
 
14 CHARACTERS OF CRYSTALS. 
 
 The angles between the axes are therefore determined. 
 EXAMPLE. In Axinite by measurement it is found that 
 
 log. cos. log. sin. 
 
 ABor(ioo oio) = 482i' 8" whence 9.82253 9.873465 
 BC or(oio^ooi)=9750 / 8" " 9-13459 9-99593 
 AC or (loo ooi)= 93 48' 56" " 8.82311 9.99904 
 
 /cos C A B= 9.09181 
 CAB=82 54' 1 3" or 97 5' 47" 
 a = 97 5' 47" or 82 54' 1 3" 
 
 Similarity /? and Y can be determined 
 
 2 Determination of the parameter ratios . 
 
 In Fig. 29 let HKL be any plane cutting the three axes and 
 with known indices likl. 
 
 The spherical triangle rst described from L as a centre can be 
 solved because its angles are 
 
 /=(ioo (oio) s= (100) (Jikl] and r= (oio) (hkl), all of 
 which may be measured. 
 
 The sides, tr and ts t may be calculated by formula, for instance : 
 
 cos .y+cos t cos r. 
 
 cos tr= : 
 
 sin t sin r 
 
 These sides measure angles in the plane triangles H O L and 
 K O L, respectively, and in each of these one other angle has al- 
 ready been determined. The triangles may, therefore, be solved 
 for the relative lengths of their sides that is, for the intercepts of 
 the plane. 
 
 For instance, in K O L, the angle at L is measured by ts, and 
 the angle atO by Y Z or , hence the angle at K is 180 (a + ts), 
 and the sides are given by the formula, 
 
 O L : O K=sin (180 ats ) : sin ts 
 Similarly, 
 
 O L : O H = sin (180 /? tr ) : sin tr 
 
 EXAMPLE : 
 
 Given /= 100 41', ^=59 10', r=76 33', =822i / , /5=73 n f . 
 Required a, b and c. 
 
GEOMETRICAL CHARACTERS. 15 
 
 cos J+CQS / cos r_. 5125 + . I854X. 2335 _ 
 
 sin / sin r .98215 x .9726 t9 ' 
 
 ^-54 35'- 
 
 cos ? + cos / cos s .2335 + . I854X. 5125 
 COS ts = - sin t sins -. 98215 X. 8587 
 
 ^=67 5'. 
 
 In triangle K O L O L: O K==sin (180 82 21' 67 5'): 
 sin 67 5',=sin 30 34 r : sin 67 5^=0.5085 : .9205 = ^525 : I. 
 
 In triangle HOL OL:OH=sin (180 73 u'_54 35') : 
 sin 5435 r =sin 52 14': sin 54 35^=0.7905 : .8150=. 5525 : .5696 
 Hence O H : O K: O 1^.5696: I : .5525= a-.b'.c. 
 
CHAPTER III. 
 
 SPHERICAL PROJECTION. 
 
 Imagine a sphere described around the centre of a crystal 
 with any radius and radii drawn normal to each face. 
 
 The point in which the radius normal to any crystal face meets 
 the surface of the sphere, is called the pole of the face, and is de- 
 noted by the symbol of the face. 
 
 Planes which intersect in parallel edges will evidently have their 
 normals in one plane and their poles in the circle which it cuts 
 from the sphere of projection. Such a series of planes constitute 
 a Zone, the plane of the normals is the Zone Plane, the circle is 
 the Zone Circle, and the line through the centre parallel to the face 
 and edges of the zone is the Zone Axis. 
 
 Fig. 30 represents the section made 
 by a zone plane, the normals to the 
 faces A, B, C, etc., meet the zone circle 
 in the poles A p ~B V C lt etc. 
 
 The same radii are evidently nor- 
 c< mal to the crystal faces of the enclosed 
 ideal form in which equivalent faces 
 are equally distant from the centre. 
 Hence the poles on the surface of the 
 sphere, in their arrangement will reveal 
 the hidden regularity of unequally de- 
 veloped crystals. 
 Fig. 30 shows, also, that the arc of the zone circle between 
 any two poles measures the normal angles, the supplements of the 
 angles between the corresponding faces. Different intersecting 
 zone circles give spherical triangles, the sides of which are normal 
 angles, which can be solved by simple formulae, provided certain 
 parts have been determined by measurement or previous calcula- 
 tion. 
 
GEOMETRICAL CHARACTERS. 17 
 
 A number of simple relations between the faces in zones have 
 been deduced by means of which it is possible to greatly reduce 
 the number of necessary direct measurements and to simplify the 
 calculations. 
 
 SYMBOL FOR A ZONE Axis. 
 
 The direction of intersection* of two crystal faces (Jikt) and 
 (li 1 k' I 1 ) is [uvw] in which u = kl' Ik', v = Ik' hi 1 , vr^htfMi'. 
 
 These values may be obtained by cross multiplication of the 
 twice written indices, striking off end terms and reading down 
 alternately from left to right and from right to left, thus : 
 
 k I k k 
 
 XXX 
 k' I' h' - k' 
 
 As all the terms are whole numbers the values of u, v and w will 
 be also. A zone is designated by this symbol [uvw] or by the 
 symbols of two of its planes \Jikl, pqr^ y or by letters designating 
 the faces [P, Q], always enclosing with the parallel bars. 
 
 EQUATION FOR ZONE CONTROL OR CONDITION THAT A FACE MAY 
 BELONG TO A ZONE. 
 
 If a face (pqr) lies in a zone [uvw] its indices must satisfyf the 
 equation /u + qv -\- wr = o. 
 
 If two indices of a face are known, and the zone is known, the 
 third index may therefore sometimes be found. 
 
 Example. By test with a reflection goniometer the face (3/i) is 
 found to be in the zone [mj; substituting in the above equation 
 ^iX i = o whence 3 k i = o and k= 2. 
 
 FACE IN TWO ZONES. 
 
 The indices of a facej which lies in the zones [uvw] and [u'v'w'] 
 will be h = uv' vu', k = vw' wv', /= wu' uw'. 
 
 These values may be obtained by cross multiplication. 
 
 Example. By test a face (kkl) is found to lie in the zones 
 [ 2! o ] and [ o i 2 ], required the values of //, k and /. 
 
 * Miller's Treatise on Crystallography , p. 7. 
 j- Ibid p. 10. 
 J Ibid, p. 8. 
 
i8 
 
 CHARACTERS OF CRYSTALS. 
 o 
 
 I O 2 I 
 
 XXX 
 
 = 2 O, =0-1-4, / = 2 O, 
 I 2 O I 
 
 that is, (hkl) = (242)= (121). 
 
 To FIND A FOURTH FACE IN A ZONE.* 
 
 Let A = (efg), B = (///) and C = (pqr} be known faces 
 of a zone, the poles of which lie in the order named. 
 
 To find the position of any fourth face, D =(mno}, if its in- 
 dices are known. 
 
 AC cot A D== 
 
 AB cot AD). 
 
 AC A B 
 
 The values of ^-^ and ^-=? are such as result by cross multi- 
 
 (^f .L/ -D J_/ 
 
 plication of pairs of corresponding indices : 
 
 V / 
 X 
 
 / g 
 X 
 
 p q eqfp . q r frgq . 
 
 V* = > 
 
 CD p q pn qm 
 
 X 
 m n 
 
 ' S 
 X 
 
 or <- 
 
 X 
 
 qo rn 
 
 p r p o rin 
 
 X 
 m o 
 
 e f 
 
 X 
 AB h k ekfh 
 
 f g 
 
 X 
 k I _flgk 
 
 IT^l^ ko ln ; 
 
 X 
 n o 
 
 e g 
 
 X 
 h I _ elgh 
 
 h I ho Im 
 
 X 
 m o 
 
 Very frequently two of the three identical ratios are indeter- 
 minate. 
 
 BD h k~ hn fan 
 X 
 
 or 
 
 * Groth's Fhysikalische Krystallographie^ III. ed., p. 584. 
 
GEOMETRICAL CHARACTERS. 19 
 
 EXAMPLE. 
 
 In a crystal of pyroxene, given: A= (efg] = (100) , B = (likl) 
 = (101), C=(/^=(ooi), D=(w;w)=(3~oi)., AB=49 39', 
 A C = 73 59'. Required A D. 
 
 By trial the first and third ratios are found to be indeterminate, 
 but from the third we obtain 
 
 AC i.i o.o i AB i.i o.i I 
 
 CD o.i 1.3 3 B D i.i 1.3 4 
 
 substituting in the equation 
 
 i (cot 73 59' cot AD)=-(cot49 39' cot AD) 
 
 1.1504 4 cot AD 2.54873 cot AD. cot AD= 1.3983 
 
 AD= 144 26' 
 ZONE OF TWO PINACOIDS. 
 
 Every face in the zone will have that index zero which is zero 
 in both pinacoids. For example if a face lies in the zone of 
 [100, oio] its third index must be zero for the zone symbol is 
 [ooi] p. 17 and h. o -f k. o -f /. I = o can be true only if / o. 
 
 ZONE THROUGH ONE PlNACOID. 
 
 The ratio of the two indices which are zero for the pinacoid, is 
 constant for all faces of the zone. For example the symbol of the 
 zone [\oo,hkl~\ is \plk~\. Any third face (pqr) must satisfy the 
 
 k a 
 equation p& -\- q I r = o, that is q 1= rk or -=- = ~ 
 
 ZONES IN WHICH TWO INDICES ARE CONSTANT. 
 
 If two faces have two corresponding indices in each with the 
 same ratio, all faces in their zone will have those two indices in 
 that ratio. For example, the symbol of the zone of the faces (123) 
 and (245) is [210] A face (hkl) to be in this zone must satisfy 
 the equation 2/*-f/-fo/=o, that is 2 h = k. 
 
 CHANGING AXES.* 
 
 If three edges are preferred to those originally selected as axes 
 directions proceed as follows : 
 
 From the original indices of the new axial planes determine the 
 symbols of their intersections (that is, their zone symbols), [uvw], 
 [u lVl wJ [u 2 v 2 w 2 ]. ______ 
 
 * Miller Treatise on Crystallography, p. 17. 
 
20 CHARACTERS OF CRYSTALS. 
 
 Then, if the indices of any face referred to the old axes is (hki), 
 its new indices h^k^ and ^ will have the following values: 
 
 h v /m -f hv -f //w 
 k l ku v -f- ^Vj + / 7 w 1 
 / t = /u 2 + /v 2 + /w 2 . 
 
 CHANGING PARAMETERS. 
 
 hkl the indices of a face referred to parameters <z, , <:, 
 
 " " "" " " 
 
 
 JL=.fL, ^J_ = , _J ! = _ then if this face is chosen as the parame- 
 //j h k^ /! / 
 
 tral plane a l= ~, b l= , ^ = -^ and any other face, the indices 
 
 It K I 
 
 of which were/^r, will receive new indices p^q^ in which 
 
 STEREOGRAPHIC PROJECTION. 
 
 The sphere containing the poles of the crystal faces is most 
 conveniently represented in Stereographic Projection. 
 
 Some important plane through the centre of the sphere is se- 
 lected as a plane of projection, and all poles of the sphere are 
 projected in this plane and fall within the limits of the so-called 
 " PRIMITIVE CIRCLE." 
 
 For instance, let Fig. 31 represent the upper half of a crystal of 
 cassiterite within a sphere of projection. The crystal faces a, d, 
 g, k, m., etc., will have their poles in the points A, D, G, k lf m lt 
 etc., where the normal radii intersect the spherical surface. 
 
 Let the equatorial plane be the plane of projection, let the south 
 pole be the point of sight, then if lines are drawn from the south 
 pole to the various poles in the northern hemisphere, they will all 
 pierce the equatorial plane at points A, D, G, K, M, etc., in the 
 primitive circle which are the projections of the poles. 
 
 PRINCIPAL CHARACTERISTICS OF STEREOGRAPHIC PROJECTION. 
 1 All zone circles at right angles to the primitive circle are 
 
GEOMETRICAL CHARACTERS. 
 
 21 
 
 FIG. 31. 
 
 projected as diameters, all others, as arcs of circles,* cutting the 
 primitive circle in the extremities of a diameter. 
 
 2 If the pole (F. Fig. jj) of a zone circle CPC V is united with the 
 poles of two faces (P and Q) of the zone by straight lines, and these 
 prolonged to the circumference, they cut from it an arc equal to the 
 normal angle between the two faces. "\ 
 
 Free-hand construction in many instances serves to give a 
 comprehensive view of the symmetry and relations of the faces 
 and zones. For accurate construction a few rules are needed, 
 which are based upon characteristics I and 2. 
 
 PROBLEM i. 
 
 Given the projection of a zone circle, to find tJiat of its pole. \ 
 
 *For demonstration, Story-Maskelyne's Crystallography, p. 30. 
 
 f Ibid, p. 33. 
 
 \ Miller's Treatise on Crystallography, p. 133. 
 
22 
 
 CHARACTERS OF CRYSTALS. 
 
 FIG. 32. 
 
 In Fig. 32. Given C P Q to find F. 
 
 C P Q intersects the primitive circle in 
 the diameter C C r Its pole must lie on 
 that circle which is projected as the 
 diameter D D 4 normal to C Q and at 90 
 from the point T where the two circles 
 'intersect. Draw C E through T, layoff 
 EE^ 90, draw , then is F the re- 
 quired pole because it is a point on D D l 
 and by the second characteristic it lies at 
 a quadrants distance from T. . 
 
 PROBLEM 2. 
 
 Given the projection of the pole of a great circle to draw the circle. 
 
 In Fig. 32 given F to find C P Q. 
 
 Draw a diameter D D t through F and another C Q normal to 
 this. Through F draw CEj, make EEj 90, draw EC, then are 
 T, C and C lf three points of the zone desired. Through these pass 
 the arc of a circle. 
 
 PROBLEM 3. 
 
 In any zone given the angle between two faces and the projection of 
 one of them to find that of the other. 
 
 In Fig. 33, given the zone CPQ, the face P, and the angle P to 
 Q=a. To find Q, 
 
 FIG. 33. 
 
 FIG. 34. 
 
 Find F the pole of the zone C PCj, as in problem I. Draw FE 
 through P, make EEj equal a, and draw EjF. Then by the second 
 characteristic, Q is the desired projection. 
 
 In Fig 34, is represented the special case in which the given 
 
GEOMETRICAL CHARACTERS. 
 
 zone C P Qis normal to the primitive circle. In this case the pole 
 of the zone falls at D, the extremity of the diameter normal to C Q. 
 If a=i8o, the second face will lie outside the primitive circle. 
 The construction is not changed. 
 
 PROBLEM 4. 
 
 Given the projections of tivo faces 
 tn a zone to find the projection of the 
 zone circle* 
 
 In Fig. 35, given P and Q to 
 find C P Q. 
 
 Draw a diameter through P. The 
 face P! opposite P, will be on it. 
 
 Draw O A perpendicular to this 
 diameter, draw P A, and from A, a FlG ' 35< 
 
 line perpendicular to P A. The intersection of this perpendicular 
 with the diameter through P, will be P x ; the projection of the pole 
 of a face opposite P, that is 1 80 from P, in the same zone. 
 
 The arc of a circle through the three points, P, Q and P lf will be 
 the desired circle. 
 
 EXAMPLES. Application of preceding constructions to the pro- 
 jection (Fig. 36) of the cassiterite crystal (Fig. 31). 
 
 The poles lying in the primitive circle are separated by arcs 
 
 *Miller's Treatise on Crystallography, p. 133. 
 
2 4 
 
 CHARACTERS OF CRYSTALS. 
 
 equal to the true normal angles. The poles of all other faces in 
 this instance lie in zone circles normal to the primitive circle, 
 therefore projected as diameters. The true position of any one of 
 these is found by Problem 3. For example: M J or (no) 
 (Hi) = 46 27'. Lay off J x = 46 27', draw x I to the pole I 
 
 FIG. 37. 
 
 FIG. 38. 
 
 of the circle J H cutting the latter at M, the required projection. 
 The pole R is in the zone of M and J, and being an equivalent face 
 O R = O M. 
 
 Draw zone circles through C M D, B M A, C R D and B R A. 
 
 The poles of N and S will lie at the intersections as shown. 
 For the second order forms K lies in the zone [A B] and in the zone 
 [M N], hence is at K. Similarly the faces e, I and/ have poles E, 
 L and F. 
 
 A crystal of barium chlorate shown in Fig. 37 yields the pro- 
 jection Fig. 38. The plane of projection is taken normal to the 
 zone [m, a]. 
 
 Given;// a = 48 53j',/ ^=41 51^', ' mp= 56 39' 
 In Fig. 37 lay off arcs equal 48 53^' each side of a and a' de- 
 termining m, m', m", m" ' . Lay off an angle equal 41 51^' and 
 draw a line (not shown) to pole of the zone \a a'~\, intersecting the 
 latter at r. By trial / lies in the zone \_m" ' r~\ f construct this zone 
 circle, find its pole, F by Problem. Lay off m f s = 56 39', draw 
 s F, then, by Problem 3,/ is the desired pole; p' is in the zone \in r\ 
 symmetrical to /. 
 
CHAPTER IV. 
 
 THE THIRTY-TWO CLASSES OF CRYSTALS. 
 
 In this classification, following Professor Groth, the conception 
 of hemihedral and tetartohedral forms is abandoned because the 
 geometrically connected whole and partial forms have no structural 
 connection and are incapable of occurrence upon crystals of the same 
 substance. 
 
 Each occurring form is, therefore, considered to be complete and 
 independent, and with a grade of symmetry which is not in every 
 case determinable from a consideration of the grouping of the faces, 
 but involves the far wider conception that two directions are not 
 structurally equivalent unless they are equivalent with respect to all 
 properties, and that two forms geometrically identical are not struc- 
 turally so if in corresponding directions there is revealed an essen- 
 tial difference in behavior with polarized light or etching or pyro- 
 electricity or any other test, the results of which depend upon the 
 manner the crystal molecules are built together. 
 
 With this conception the same geometric form may occur on 
 crystals structurally different, and is to be regarded as in each case 
 a limit form of geometrically distinct general forms. If a class be 
 made of each conceivable variation of general form and its geo- 
 metric limit forms, two great essentials will be fulfilled : 
 
 1. All known and some unknown forms will be classed. 
 
 2. Each class will consist of forms capable of occurring upon- 
 crystals of the same substance, and will contain all the forms which 
 can occur on these crystals. 
 
 There have been distinguished thirty-two grades or classes of 
 symmetry which may be united into six systems by grouping to- 
 gether classes in which the selected axes of reference are geo- 
 metrically similar. 
 
26 
 
 CHARACTERS OF CRYSTALS. 
 
 In the stereographic projection of the general form given under 
 each class all full lines, whether diameters, circles or arcs of cir- 
 cles represent planes of symmetry and the small black ellipses, 
 triangles, squares and hexagons represent the axes of binary, 
 ternary quaternary and senary symmetry. 
 
 TRICLINIC SYSTEM. 
 
 This system must include all crystallographic forms which can 
 only be referred to three non-equivalent axes, Fig. 39, at oblique 
 angles, a, ft, and f y to each other. 
 
 The selection of axes is arbitrary, two edges are chosen as direc- 
 tions of the basal axes, a and b. Two planes from the zones of 
 these edges are chosen as Jiooj and Joioj and their intersections 
 are the vertical axes c. 
 
 c 
 
 X/ X 
 
 FIG. 39. 
 
 X. 
 
 / \j 
 
 \ 
 
 a 
 FIG. 40. 
 
 i. UNSYMMETRICAL CLASS. 
 
 Without either planes or axes of symmetry. The projection, 
 Fig. 40, shows that the symmetry of the class is satisfied with one 
 face for the most general form. EXAMPLE. Calcium thiosulphate, 
 CaS 2 O 3 - 6 H 2 O. 
 
 2. CLASS OF PINACOIDS. 
 
 With composite symmetry to a binary axis and a plane normal 
 thereto. 
 
 As shown in projection, Fig. 41, any upper face indicated by X 
 would by simultaneous rotation around the axis and reflection in 
 the plane coincide with the diametrically opposite lower face, that 
 is, the symmetry of the class is satisfied by two faces for the most 
 
GEOMETRICAL CHARACTERS. 
 
 27 
 
 general form or PINACOID (Tetarto Pyramid), Fig. 42. EXAMPLES. 
 Albite, cyanite and chalcanthite. 
 
 o 
 
 a 
 FIG. 41. 
 
 FIG. 42. 
 
 THE Six LIMIT FORMS. 
 
 In each class there are six limit forms corresponding to six 
 special positions of the faces of the general form. In Class 2 
 these will be pairs of parallel planes, and in Class I single planes 
 only. 
 
 010 
 
 Position of any Face. Symbol. 
 
 1. Parallel to a and b. JooiJ 
 
 2. Parallel to a and c. 
 
 3. Parallel to b and c. 
 
 4. Parallel to a. 
 
 5. Parallel to b. 
 
 6. Parallel to c. 
 
 Class i. 
 
 PLANE of Fig. 43. 
 PLANE of Fig. 44. 
 PLANE of Fig. 45. 
 PLANE of Fig. 46. 
 PLANE of Fig. 47. 
 PLANE of Fig. 48. 
 
 Class 2. 
 
 PINACOID Fig. 43. 
 PINACOID Fig. 44. 
 PINACOID Fig. 45. 
 PINACOID Fig. 46. 
 PINACOID Fig. 47. 
 PINACOID Fig. 48. 
 
 FIG. 43. 
 
 FIG. 44. 
 
 FIG. 45. 
 
28 
 
 CHARACTERS OF CRYSTALS. 
 
 FIG. 46. 
 
 FIG. 47. 
 
 FIG. 48. 
 
 PROJECTION AND CALCULATION OF TKICLINIC FORMS. 
 
 THE PROJECTION is conveniently made upon a plane normal to 
 the vertical axis c. All planes in the zone of c as (100), (oio); 
 (no), being projected in the primitive circle at the measured 
 angles apart. 
 
 The pole of any face P will lie at the intersection of two 
 circles found as follows : * Their centers are on prolongations 
 of diameters through ^(100) and J5(oio) and distant from the 
 
 Y ' 
 
 center of the primitive circle respectively OK=-, OL = -^ 
 
 and their radii are respectively Kp^rtanPA, Lp = rtenPB. 
 
 Most of the poles are found by zones and problem 3, p. 22. 
 
 THE CALCULATIONS are principally solutions of spherical tri- 
 angles t and of equations which connect measured angles with 
 
 *Groth, Physikalische Krystallographie, p. 580, III. ed. 
 \ For right-angled spherical triangles 
 
 C= 90 a, b, c = sides opposite angles A, B, C. 
 
 sin a cos B tan b 
 
 cos A = -7 =cos a sin B, 
 
 sin A 
 
 cos b ' 
 
 tanr 
 
 tan a 
 tan A = - T- cos c = cos a cos b = cot A cot B 
 
 In oblique angled spherical triangles 
 A, B, C denote angles , a, b, c, opposite sides, _IL _ J _ = $ t an & 
 
 
 sin A 
 sin a 
 
 sin B sin C 
 
 sin b sin c 
 cos A = cos B cos C -f- sin B sin C cos # 
 cos a = cos <5 cos c -f- sin sin r cos ^4 
 
 2 
 
 A 
 
 sin b sin c 
 
 sin s sm (s a] 
 sin b sin r 
 
 4 
 
 *L. = -t / cos ^ cos ( s ^ ) 
 
 2 
 
 sin ^ sin 6' 
 
GEOMETRICAL CHARACTERS. 
 
 29 
 
 indices and elements. Wherever possible the zonal equations, pp. 
 16-20, are used to simplify the calculations. 
 
 DETERMINATION OF ELEMENTS. 
 
 The general determination of pp. 13-15 requiring the meas- 
 urement of five angles between four planes, no three of which lie 
 in the same zone, is followed as in the example there given. 
 
 GENERAL EQUATION BETWEEN AXES AND INDICES. 
 
 Let any plane the indices of which are hkl meet the axes in 
 HKL, let the axes meet the surface of a sphere described around 
 O, in XYZand let the normal Op to HKL meet the sphere in P. 
 Fig. 49. 
 
 From a section through OHp it is evident that Op=OH 
 cos PX, and, similarly, Op=OK cos PFand Op=OL cos PZ. Equa- 
 ting, OH cos PX=OKcos PY=OL cos PZ. 
 
 From page 13, we have 
 
 ffl-, 
 
 Substituting, 
 
 PZ 
 
 FIG. 49. 
 
 To DETERMINE POSITION OF ANY POLE P. 
 
 Let Fig. 50 show the poles in projection. From the following 
 equations the position of P will result if the indices and elements 
 
. 30 CHARACTERS OF CRYSTALS. 
 
 are known, or, conversely, the indices will result* if the position is 
 known in one zonal circle. 
 
 sin AB sin CH __ sin CAP _ck 
 sin CA smBH~ sinAP ~~bl 
 
 sin BCsm AK_sm ABP_ al 
 sin AB sin CK~'Sri~CBP~~ck 
 
 sin CA sin BL__sm BCP_ bh 
 sin JSCsin AL~~s'm ACP~ak 
 
 If any five of the arcs in the following are known, the sixth 
 results 
 
 sin BHsm ATsin AL=sm CHsm AKsm BL. 
 
 IF THE ELEMENTS ARE UNKNOWN the indices of a face lying in a 
 known zone may be determined by measuring the angle to one 
 face in that zone, and as three faces must already be known, sub- 
 stituting in the formula to find fourth face in a zone, p. 18. 
 Conversely, if the indices of such ajplane are known, its position 
 may be calculated. 
 
 To FIND t THE ARC JOINING ANY TWO POLES P AND P' . 
 
 Calculate by preceding equations the distances of P and 
 P 1 from one of the poles A, B or^C'and the angles that these 
 distances make with one of the adjacent sides of ABC, therefore, 
 the angle that they make with each'other. From two sides and 
 included angle calculate the third side PP. 
 
 MONOCLINIC SYSTEM. 
 
 All forms in this system must be referable to three non-equiva- 
 lent axes, Fig. 51, two oblique to each other, the third, normal to 
 their plane. 
 
 Conventionally the normal or 
 ortho axis b extends from right 
 to left, either of the other axes 
 is made the vertical, c, and the 
 third the clino, a, dips down- 
 ward from back to front. The 
 acute angle between the ver- 
 tical and clino axes is /9. FIG. 51. 
 
 * Story-Maskelyne's Crystallography, p. 430. 
 f Miller's Crystallography^. 98. 
 
GEOMETRICAL CHARACTERS. 
 
 The system comprises three classes, in which are possible many 
 series each including all the forms which can be referred to the 
 same value for /? and to the same parameters a, b % c. 
 
 3. CLASS OF THE MONOCLINIC SPHENOID. 
 
 With one axis of binary symmetry. As shown in the projection 
 upon (oio) Fig. 52, the pole of any face by rotation 1 80 around 
 the binary axis must reach the pole of an equivalent face. The 
 two faces satisfy the symmetry for the most general form or SPHE- 
 NOID, Fig. 53. EXAMPLES. Tartaric acid, milk sugar. 
 
 FIG. 52. 
 
 FIG. 53. 
 
 4. CLASS OF THE MONOCLINIC DOME. 
 
 With one plane of symmetry. As shown in the projection of the 
 general form, Fig. 54 any face reflected in the plane of symmetry 
 coincides with an equivalent opposite face. The two faces satisfy 
 the symmetry of the class for the most general form or DOME, Fig. 
 55. EXAMPLE. Potassic tetrathionate K 2 S 4 O 6 . 
 
 FIG. 54. 
 
 FIG. 55. 
 
 5. PRISMATIC CLASS. 
 
 With one plane of symmetry, Fig. 56, at right angles to an axis 
 of binary symmetry. As shown in projection, Fig. 57, any face 
 
32 CHARACTERS OF CRYSTALS. 
 
 the pole of which is marked X by rotation of 180 around the bi- 
 nary axis coincides with an equivalent face in the alternate octant, 
 these reflected in the plane of symmetry coincide with faces the 
 poles of which are marked by a circle. 
 
 FIG. 56. 
 
 FIG. 57. 
 
 FIG. 58. 
 
 That is four planes satisfy the symmetry for the most general 
 form or PRISM, Fig. 58. EXAMPLES. Pyroxene, orthoclase, gyp. 
 sum. 
 
 THE Six LIMIT FORMS. 
 
 In each class there are six limit forms corresponding to special 
 positions of the faces of the general form. These may be tabu- 
 lated as follows : 
 
 Position of Any Face and its Pole. 
 
 1. Parallel to a and b 
 
 Poles projected* on the 
 
 primitive circle 
 
 2. Parallel to a and c . . . . 
 Poles projected at centre . . 
 
 3. Parallel to b and c . . . . 
 Poles at intersections of 
 
 primitive circle and hori- 
 zontal diameter ..... 
 
 4. Parallel to a Poles are on 
 
 diameter from (ooi) . . . 
 
 5. Parallel ' to b Poles are on 
 
 primitive circle. 
 
 6. Parallel to c Poles are on 
 
 Horizontal diameter . 
 
 Symbol. 
 
 001 
 
 Joio} 
 
 ioo 
 
 \M\ 
 
 Name of Form. 
 
 Classes to which 
 Form Belongs. 
 
 PlNACOID, Fig. 59. 
 (Basal Pinacoid~) 
 BASAL PLANE. 
 One face of Fig. 59. 
 PINACOID, Fig. 60. 
 (Clino Pinacoid. ) 
 PLANE, One face of Fig. 60. 
 PINACOID, Fig. 6r. 
 (Ortho Pinacoid.) 
 PLANE, One face of Fig. 61. 
 
 PRISM, Fig. 62. 
 (Clino Dome.) 
 DOME, Fig. 65. 
 SPHENOID, Fig. 67. 
 PINACOID, Fig. 63. 
 (Henri Ortho Dome.) 
 PLANE, One face of Fig. 63. 
 PRISM, Fig. 64. 
 (Monoclinic Prism.) 
 DOME, Fig. 66. 
 .SPHENOID, Fig. 68. 
 
 3 i 5- 
 
 > 4, 
 
 , 4, 5- 
 
 3 . 
 3. > 5- 
 
 i 4, . 
 
 > 5- 
 
 4, 
 3. , 
 3. , 5- 
 
 -, 4, 
 
 * Plane of projection the pinacoid (010). 
 
GEOMETRICAL CHARACTERS. 
 
 33 
 
 FIG. 59. 
 
 > -4 
 
 FIG. 60. 
 
 FIG. 61. 
 
 FIG. 62. FIG. 63. FIG. 64. 
 
 LIMIT FORMS OF CLASS 5. 
 
 FIG. 65. FIG. 66. FIG. 67. FIG. 68. 
 
 OTHER LIMIT FORMS WHICH ARE NEW SHAPES. 
 
 PROJECTION AND CALCULATION OF MONOCLINIC FORMS. 
 The plane of projection may be the pinacoid (oio), which is 
 a plane of symmetry in classes 4 and 5, when the poles of all 
 
34 
 
 CHARACTERS OF CRYSTALS. 
 
 planes in the zone of b will lie in the primitive circle as given in 
 preceding table. Or the projection may be made, as in the tri- 
 clinic system, upon a plane normal to the vertical axis, in which 
 case the poles of planes in the zone of b will be projected on the 
 vertical diameter as described in example, p. 24. 
 
 The poles will be found as in the triclinic system. 
 
 When a plane of symmetry appears as a diameter in the pro- 
 jection, all poles and indices will be symmetrical to this diameter. 
 
 DETERMINATION OF ELEMENTS. 
 
 In Fig. 29, p. 13, B and Y will coincide and BC, AB and / 
 become 90. Then AC ft. In the spherical triangle rst, 
 
 OL cos ts 
 
 cos tr = - 
 
 cos s 
 
 costs = 
 
 cos? 
 
 sin r 
 
 sin s ' OK sin ts 
 
 = cot ts 
 
 OL s\n(iSo-AC tr) Tr _ 
 
 gj -. If P is the parametral plane, 
 
 OH: OK: OL = a\b\c, if not, a = OHJi, b = OK.k, c = OL.!. That 
 is three angles suffice for the determination of the elements. 
 
 To DETERMINE THE POSITION OF ANY POLE P. 
 Using notation of Fig. 69, 
 
 r 7 ~-^ 
 
 b 
 .jcot/ 3 
 
 sin CK 
 
 a 
 
 CK 
 
 j sin AK; 
 
 sin AK 
 k sin AD tan OB 
 
 ch % h = sin AD sinCK 
 ~al ' T ~ sin CD ' s'mAK* 
 
 -j ( cot AD cot AC) ; cos PA = sin PB 
 cos AK\ cos PC= sin PB cos CK 
 
 tan PB = ~r 
 
 l s'mAD 
 
 k sinAK 
 
 tan OB. 
 
 Aioo 
 
 FIG. 69. 
 
 Because B is pole of zone circle AA' t CK= PBC, AK = PBA 
 and CK= i8o(PA + CA'), that is the sines and cosines of 
 these angles may be substituted for those of the arcs in any^of the 
 formulae above. 
 
GEOMETRICAL CHARACTERS. 
 
 35 
 
 To FIND THE ARC JOINING TWO POLES, PandP'. 
 Proceed as in triclinic, p. 30. 
 
 ORTHORHOMB1C SYSTEM. 
 All forms in this system must 
 be referable to three nonequiva- c 
 
 lent axes, Fig. 70, at right angles 
 to each other. The three axes 
 are physically of equal impor- 
 tance, any one may be chosen as 
 
 c, the vertical ; the longer of the 
 
 other two will be the macro or b ^^ 
 
 axis ; the shorter axis (from front 
 to back) the brachy or a axis. 
 There are as many series of forms 
 possible as there are irrational 
 
 values for -7 and T 
 b b 
 
 FIG. 70. 
 
 6. CLASS OF THE RHOMBIC BISPHENOID. 
 
 With three axes of binary symmetry at right angles to each 
 other. As shown in the. projection, Fig. 71, four faces satisfy the 
 symmetry of the most general form or RHOMBIC BISPHENOID, Fig. 
 72: EXAMPLE. Epsomite. 
 
 \. ^ X 
 
 a 
 
 FIG. 71. 
 
 FIG. 72. 
 
 7. CLASS OF THE RHOMBIC PYRAMID. 
 
 With two planes of symmetry perpendicular to each other and 
 intersecting in an axis of binary symmetry. As shown in the pro- 
 jection, Fig. 73, four faces satisfy the symmetry of the most gen 
 eral form or RHOMBIC PYRAMID, Fig. 74. EXAMPLES. Calamine, 
 struvite. 
 
 * Story-Maskelyne's Crystallography, p. 436 and Groth's Phys. Kryst., p. 578. 
 
CHARACTERS OF CRYSTALS. 
 
 -ib 
 
 a 
 
 FIG. 73. 
 
 FIG. 74. 
 
 8. CLASS OF THE RHOMBIC BIPYRAMID. 
 
 With three planes of symmetry at right angles to each other 
 which intersect in three axes of binary symmetry. These are 
 shown in Fig. 75, and the planes divide space into eight octants, 
 shown in projection, Fig. 76, as four trirectangular spherical tri- 
 angles. 
 
 Any upper face corresponding to a pole x in the projection 
 would, by rotation of 180 around the vertical binary axis, coincide 
 with an upper face in the alternate octant, these reflected in the 
 vertical symmetry planes coincide with two other upper faces and 
 the four reflected in the horizontal plane coincide with four lower 
 planes the poles of which are marked by circles. Since ft, k and / 
 retain a constant order, there can be sign permutations only cor- 
 responding to one plane in each octant. 
 
 The symmetry of the class is therefore satisfied by eight faces 
 for the most general form or RHOMBIC BIPYRAMID, Fig. 77.* EX- 
 AMPLES. Aragonite, marcasite, barite. 
 
 U-" 
 
 FIG. 75. 
 
 FIG. 76. 
 
 FIG. 77. 
 
 *There may be many different pyramids in a series with rational indices hkl which 
 may be all equal, any two equal or all unequal. 
 
GEOMETRICAL CHARACTERS. 
 
 37 
 
 THE Six LIMIT FORMS. 
 
 In each class there are six limit forms corresponding to special 
 positions of the faces of the general form. These may be tabu- 
 lated as follows : 
 
 Position of Any Face and its Pole. 
 
 Symbol* 
 
 Name of Form. 
 
 Classes to which 
 Form Belongs. 
 
 I. Parallel to a and b 
 Poles are projected at cen- 
 ter. 
 
 i'S 
 
 BASAL PINACOID, Fig. 78. 
 BASAL PLANE. 
 one face of Fig 78. 
 
 6, -, 8. 
 
 , 7, 
 
 2. Parallel to a and c. 
 
 
 PINACOID, Fig. 79. 
 
 6 7, 8 
 
 Poles are at intersections 
 of b axis and primitive 
 circle .... . . 
 
 < OI0 5 
 
 (Brachy Pinacoid). 
 
 
 3. Parallel to b and c . . . . 
 Poles are at intersections of 
 a axis and primitive cir- 
 cle 
 
 '! 
 
 PINACOID, Fig. 80. 
 (Macro Pinacoid). 
 
 6, 7. 8. 
 
 4. Parallel to a. Poles are on 
 the b axis 
 
 \*ki\ 
 
 PRISM, Fig. 81. 
 (Brachy Dome). 
 
 6, -, 8. 
 
 5. Parallel to b. Poles are on 
 the a axis .... 
 
 {*,/] 
 
 DOME, Fig. 84. 
 
 PRISM, Fig. 82. 
 (Macro Dome). 
 
 , 7, 
 6, -, 8. 
 
 6. Parallel to c. Poles are on 
 the primitive circle . . . 
 
 \Uv] 
 
 DOME, Fig. 85. 
 
 PRISM, Fig. 83. 
 (Rhombic Prism). 
 
 7 
 6, 7, 8. 
 
 FIG. 8 1. FIG 82. FIG. 83. 
 
 LIMIT FORMS OF CLASS 8. 
 
 * To obtain type symbols. The order is invariably hkl with reference to a, b t 
 Any may become zero. 
 
CHARACTERS OF CRYSTALS. 
 
 FIG. 84. FIG. 85. 
 
 OTHER LIMIT FORMS WHICH ARE NEW SHAPES. 
 
 PROJECTION AND CALCULATION OF ORTHORHOMBIC FORMS. 
 The pinacoid (ooi) is usually selected as the plane of projection. 
 The poles are as in the table and their exact positions usually re- 
 sult from the intersections of known zones or by Problem 3, p. 22. 
 It is sometimes convenient to calculate the position of a plane 
 (hko] corresponding to the plane (hkl), lay off this on the primitive 
 circle thus determining the zone [/z/o ooi]. 
 
 CALCULATION OF ELEMENTS. 
 
 The interaxial angles are all right angles. 
 
 In the spherical triangle rst, t = 90, r = hkl : oio, s = hkl\ 100 
 Substituting in formulae p. 34. 
 
 cos s cos r 
 
 cos tr = -7 , cos ts = - 
 
 sin r sin s 
 
 OL OL 
 
 m ~cot&, m 
 
 cot 
 
 a = OH.k, b=OK.k, c = OL.l. 
 
 Also, a = tan y 2 (100 : 1 10) and c = tan y 2 (01 1 : 01 1) 
 = a tan y 2 (101 :Toi). 
 
 Vihol 
 
 A 100 
 
 FIG. 86. 
 
 FIG. 87. 
 
GEOMETRICAL CHARACTERS. 
 
 GENERAL EQUATION BETWEEN AXES AND INDICES. 
 
 y cos PA = t cos PB, = 4 cos ^ 
 
 h k I 
 
 To DETERMINE * ANY POLE P. 
 
 cot PA = -^ cos /**= cos 
 
 cot P = ^ cos PBC=~cvs PBA = ^ = 
 
 kc 
 tan PC A = 4/ , tan PBC=~, tan 
 
 For Unit Plane O 
 a:b\c=co$OB cos OC : cos <9<7 cos OA\ cos O4 cos OB. 
 
 To FIND THE ARC JOINING ANY Two POLES Pand P' 
 
 Let 5 denote JPPc* + /&W + / 2 ^ 2 , 5 r a similar quantity from 
 indices h'k'l' of second pole. 
 
 IfP' = A, B, C, H, Kor L of Fig. 87 
 
 cos P(7 = sin PZ = 
 
 If P and P' are faces of the same form f 
 
 a I 
 
 tan P^ = j ^ ; tan J (//^/) : (^/) = - - j cos 
 
 sin J (//>&/) : (//J/) = cos \ (hkl) : (hkT) cos PC A 
 sin J (tiki) : Qikl) = cos J (//>&/) : (^7) sin PC4. 
 
 TANGENT RELATION BETWEEN ^/= P, and h'k'l P 1 which lie in 
 
 a zone with A, B or (7. 
 
 ^ tanPM_^ / _/ / > k'tenP'B _l r _h r m I' tenP'C_h f _k' 
 H ' tan PA~~k~~T' ~k tanP ~7~h ' 7 " tan/^~~/~J 
 
 * Miller's Crystallo'graphy, p. 79, and Story-Maskelyne's Crystallography, p. 443. 
 fGroth, jP^/j. Kryst., p. 574. 
 
CHARACTERS OF CRYSTALS. 
 
 TETRAGONAL SYSTEM. 
 
 FJG. 88. 
 
 All forms of this system must 
 be referable to two equivalent 
 axes, a, at 90 to each other, 
 Fig. 88, and the axis c, conven- 
 tionally vertical, at 90 to both. 
 
 The forms possible on crystals 
 of the same substance can all be 
 
 referred to one value of - 
 a 
 
 9. CLASS OF THE THIRD ORDER BISPHENOID. 
 With composite symmetry to a quaternary axis and a plane at 
 right angles thereto. As shown in the projection, Fig. 89, four 
 faces satisfy the symmetry for the most general form or TETRA- 
 GONAL BISPHENOID OF THIRD ORDER, Fig. 90. No examples are 
 known. 
 
 \ 
 
 O : 
 
 fa 
 
 a 
 
 FJG. 89. 
 
 FIG. 90. 
 
 10. CLASS OF THE TETRAGONAL PYRAMID OF THIRD ORDER. 
 With one axis of quaternary symmetry. As shown in the stere- 
 ographic projection, Fig. 91, four faces satisfy the symmetry for 
 the most general form or TETRAGONAL PYRAMID OF THIRD ORDER, 
 Fig. 92. EXAMPLE. Wulfenite. 
 
 a 
 
 FIG. 91. 
 
 FIG. 92. 
 
GEOMETRICAL CHARACTERS. 41 
 
 II. SCALENOHEDRAL CLASS. 
 
 With two planes of symmetry at right angles to each other and 
 intersecting in an axis of quaternary symmetry. Also two axes 
 of binary symmetry midway between the planes. As shown in 
 the projection, Fig. 93, eight faces satisfy the symmetry for the 
 most general form or SCALENOHEDRON, Fig. 94. EXAMPLE. Chal- 
 copyrite. 
 
 FIG. 93. 
 
 FIG. 94. 
 
 12. TRAPEZOHEDRAL. CLASS. 
 
 Without planes of symmetry, but with the five axes of Class 15. 
 As shown in the projection, Fig. 95, eight faces satisfy the sym- 
 metry for the most general form or TRAPEZOHEDRON, Fig. 96. 
 EXAMPLE. Nickel sulphate, NiSO 4 .6H 2 O. 
 
 / \ 
 
 r 
 
 KO 
 
 FIG. 95. 
 
 FIG. 96. 
 
 13. CLASS OF THE TETRAGONAL BIPYRAMID OF THIRD ORDER. 
 With one horizontal plane of symmetry and one vertical axis of 
 quaternary symmetry. As shown in the projection, Fig. 97, eight 
 faces satisfy the symmetry for the most general form or TETRA- 
 GONAL BIPYRAMID OF THIRD ORDER, Fig. 98. EXAMPLE. Scheelite. 
 
CHARACTERS OF CRYSTALS. 
 
 FIG. 98. 
 
 14. CLASS OF THE DlTETRAGONAL PYRAMID. 
 
 With four vertical planes of symmetry intersecting in an axis of 
 quaternary symmetry. As shown in the projection, Fig. 99, the 
 essential change from Class 15 is the omission of the plane of sym- 
 metry normal to the quaternary axis. The general form is there- 
 fore geometrically like the upper or lower half of Fig. 103. Fig. 
 100 shows the six- faced most general form or DITETRAGONAL 
 PYRAMID. EXAMPLE. Silver fluoride, AgF.H 2 O. 
 
 FIG. ico. 
 
 15. CLASS OF THE DITETRAGONAL BIPYRAMID. 
 
 With four planes of symmetry at 45 to each other, which inter- 
 sect in an axis of quaternary symmetry, and one plane normal to 
 these which intersects the four planes in axes of binary symmetry. 
 
 The planes divide space into sixteen sections, Fig. 101 shown 
 in the projection, Fig. 102, as eight birectangular spherical tri- 
 angles with angles at the center of 45. Any upper face corre- 
 sponding to a pole X, by rotations of 90 coincides successively 
 with three other upper faces. These four reflected in a vertical 
 planes coincide with four others and the eight reflected in a hori- 
 zontal plane coincide with eight lower faces marked with a circle. 
 
GEOMETRICAL CHARACTERS. 
 
 43 
 
 FIG. 10 1. 
 
 FIG. 103. 
 
 Since /remains with the third axis, the letter permutations are 
 only /^/and khl. Each of these is subject to eight permutations in 
 sign + + + ,+ + , --- + ,- + +,+ + -,+ -- , --- ,- + . 
 
 Both symmetry and permutations show that there must be in 
 this class sixteen faces in the most general form or DITETRAGONAL 
 BIPYRAMID Fig. 103. EXAMPLES. Zircon, cassiterite, rutile. 
 
 LIMIT FORMS OF CLASS 15. 
 
 FIG. 104. 
 
 FIG. 105. 
 
 FIG. 106. 
 
 FIG. 107. 
 
 FIG. 108. 
 
 FIG. 109. 
 
44 
 
 CHARACTERS OF CRYSTALS. 
 
 THE Six LIMIT FORMS. 
 
 In each class there are six limit forms corresponding to special 
 positions of the faces of the general form. These may be tabulated 
 as follows: 
 
 Position of any Face and its 
 Pole. 
 
 Symbol* 
 
 Name of Form. 
 
 Classes to which the Form 
 belongs. 
 
 I. Intersects the vertical 
 
 5 OOI i 
 
 * 
 
 BASAL PINACOID, Fig. 
 
 15,, 13,12, II, ,9. 
 
 axis and is parallel to 
 
 
 104. 
 
 
 both basal axes. Poles 
 
 
 
 
 are projected at the 
 
 
 BASAL PLANE, one face 
 
 ,14,,,, io,. 
 
 center. 
 
 
 of Fig. 104. 
 
 
 2. Intersects the vertical 
 
 \hol\ 
 
 TETRAGONAL BIPYRA- 
 
 15, ,13, 12, II,,. 
 
 axis and is parallel to 
 
 
 MID, SECOND ORDER, 
 
 
 one basal axis. Poles 
 
 
 Fig. 105. 
 
 
 tire on cixicil dicimctcrs 
 
 
 TETRAGONAL PYRAMID 
 
 IA ' io 
 
 
 
 SECOND ORDER, Fig. 
 
 HTc'T'D A /TM\T A T T^TCT>TTT7 
 
 Q 
 
 
 
 J. ii.1 KALrUWAL* X51bl rill*- 
 
 NOID, SECOND OR- 
 
 
 
 
 DER, Fig. in. 
 
 
 3. Intersects the vertical 
 
 \hhl\ 
 
 TETRAGONAL BIPYRA- 
 
 15, , 13, 12, , , . 
 
 axis and is equally in- 
 
 i 5 
 
 MID, FIRST ORDER, 
 
 
 clined to both basal 
 
 
 Fig. 106. 
 
 
 axes. Poles are on di- 
 
 
 TETRAGONAL PYRAMID, 
 
 , 14,, , , io,. 
 
 agonal diameters. 
 
 
 FIRST ORDER, Fig. 
 
 
 
 
 114. 
 
 
 
 
 TETRAGONAL BISPHE- 
 
 
 
 
 NOID, FIRST ORDER, 
 
 
 
 
 Fig. no. 
 
 
 4. Parallel to the vertical 
 axis and to one basal 
 axis. Poles are at inter- 
 
 5.00J 
 
 TETRAGONAL PRISM, 
 SECOND ORDER, Fig. 
 
 15,14,13,12, ii, io, 9. 
 
 sections of primitive cir- 
 
 
 107. 
 
 
 cle and axial diameters. 
 
 
 
 
 5. Parallel to the vertical 
 axis and equally inclined 
 to the basal axes. Poles 
 
 {110} 
 
 TETRAGONAL PRISM, 
 FiRST ORDER, Fig. 
 
 15, 14, 13, 12, II, 10, 9. 
 
 are at intersections of 
 
 
 1 
 
 
 primitive circle and di- 
 
 
 
 
 agonal diameters. 
 
 
 
 
 6. Parallel to the vertical 
 
 \ hkv \ 
 
 DlTETRAGONAL PRISM, 
 
 15, 14, ,12, II,, .- 
 
 axis and unequally in- 
 
 < > 
 
 Fig. 109. 
 
 
 clined to the basal axes. 
 
 
 TETRAGONAL PRISM, 
 
 , , 13, , , io, 9. 
 
 Poles are on the primi- 
 
 
 THIRD ORDER, Fig. 
 
 
 tive circle. 
 
 
 112. 
 
 
 * The first and second indices will be h and k or both h if equal, or ho if the face 
 is parallel to one basal axis. The third symbol will be i or o as the face intersects or 
 is parallel to the vertical axis. These must be reduced to simplest form. 
 
GEOMETRICAL CHARACTERS. 
 LIMIT FORMS OF CLASS 15. 
 
 45 
 
 FIG. 112. 
 
 OTHER LIMIT 
 
 FORMS WHICH 
 
 ARE NEW 
 
 SHAPES. 
 
 FIG. 113. FIG. 114. 
 
 PROJECTION AND. CALCULATION OF TETRAGONAL FORMS. 
 The basal pinacoid is usually selected as the plane of projec- 
 tion, the poles lying as stated in the table. On p. 23, Fig. 36, is 
 described the projection of the poles of a crystal of cassiterite. 
 
 DETERMINATION OF ELEMENTS. 
 
 In Fig. 86, cos tr = -. - and = cot tr. c = OL./ 
 sin r OH 
 
 Also in Fig. 115 if hhl = F. 
 
 c = tan FC cos 45 = tan KC 
 sin KC = tan KF cot 45 
 
 tan PC 
 
 To DETERMINE POSITION OF ANY 
 POLE* P. 
 
 cot PA = tan PH = {* cos PAB 
 
 = k , C cos PAC = 
 la 
 
 ch 
 
 Lk&o 
 
 Aioo 
 
 Miller's Crystallography, p. 44, Story-Maskelyne's Crystallography, p. 449. 
 
4 6 
 
 CHARACTERS OF CRYSTALS. 
 
 cot PB = tan PK = f- cos PBA = ^cos PBC 
 
 la 
 
 cot PC = tan PL = cos PCA = =- cos PCB = 
 kc kc 
 
 GENERAL EQUATION BETWEEN AXES AND INDICES. 
 
 -cosPA = ~ 
 n, k 
 
 = cos PC. 
 
 / 
 
 FORMULA FOR ANGLE BETWEEN Two PLANES. 
 cos PP = . 
 
 la 
 
 Formulae for angle between two faces of same form are as on 
 P- 39- 
 
 ANGLE PC WHEN PC KNOWN 
 
 tan PC 
 
 tan PC. 
 
 HEXAGONAL SYSTEM. 
 
 All forms in this system must be referable to three equivalent 
 axes a in one plane at 60 to each other and a fourth axis c nor- 
 mal to these, conventionally placed vertically. Fig. 116. 
 
 The system comprises two grand 
 divisions. The Hexagonal division 
 with five classes in which the fourth 
 axis is an axis of senary symmetry. 
 The Rhombohedral division with 
 seven classes in which this axis is 
 2 an axis of ternary symmetry. 
 
 The vertical and basal axes being 
 non-equivalent, the ratio of the 
 
 parameters - is always an irrational 
 
 number and for each different ratio a different series of forms 
 exists, one series only being capable of occurrence on the crystals 
 of the same substance. 
 
 The basal axes are most conveniently considered in the order of 
 
GEOMETRICAL CHARACTERS. 
 
 47 
 
 the figure, for then* the third index is always the algebraic sum of 
 the first and second. 
 
 THE RHOMBOHEDRAL DIVISION. 
 1 6. CLASS OF THE TRIGONAL PYRAMID OF THIRD ORDER. 
 Without planes of symmetry, but with an axis of ternary sym- 
 metry. As shown in projection, Fig. 117, the symmetry is satis- 
 fied by three faces for the general form, or TRIGONAL PYRAMID 
 OF THIRD ORDER, Fig. 118. EXAMPLES. Sodium periodate 
 NaIO 4 .sH 2 O. 
 
 V: y 
 
 FIG. 117. 
 
 FIG. 1 1 8. 
 
 17. CLASS OF RHOMBOHEDRON OF THE THIRD ORDER. 
 With composite symmetry to a ternary axis and a plane at right 
 angles thereto. As shown in the stereographic projection, Fig. 
 119, the symmetry is satisfied by six faces for the most general 
 form or RHOMBOHEDRON OF THE THIRD ORDER, Fig. 120. EX- 
 AMPLES. Dioptase, phenacite. 
 
 o 
 ia 
 
 FIG. 120. 
 
 FIG. 119. 
 
 1 8. CLASS OF THE TRIGONAL TRAPEZOHEDRON. 
 Without planes of symmetry, but with the four axes of sym- 
 metry of Class 22. As shown in projection, Fig. 1 21, the sym- 
 metry is satisfied by six planes for the general form, or TRIGONAL 
 TRAPEZOHEDRON, Fig. 122. EXAMPLES. Quartz, cinnabar. 
 
 * Simple proof in Bauerman's Systematic Mineralogy, p. 76. 
 
4 8 
 
 CHARACTERS OF CRYSTALS. 
 
 FIG. 1.2 1. FIG. 122. 
 
 19. CLASS OF THE TRIGONAL BIPYRAMID OF THIRD ORDER. 
 With one plane of symmetry at right angles to the axis of ter- 
 nary symmetry. As shown in the projection, Fig. 123, the sym- 
 metry is satisfied by six planes for the most general form, or TRI- 
 GONAL BIPYRAMID OF THIRD ORDER, Fig. 124. No examples are 
 known. 
 
 FIG. 1.23. FIG. 124. 
 
 2O. CLASS OF THE DlTRIGONAL PYRAMID. 
 
 With three planes of symmetry at 60 to each other which in- 
 tersect in the axis of ternary symmetry. These forms are geomet- 
 rically the upper or lower halves of those of Class 22. As shown 
 in projection, Fig. 125, the symmetry is satisfied by six planes 
 for the most general form, or DITRIGONAL PYRAMID, Fig. 126. 
 EXAMPLES. Tourmaline, proustite, pyrargyrite. 
 
 FIG. 125. 
 
 FIG. 126. 
 
 21. CLASS OF THE DITRIGONAL SCALENOHEDRON. 
 With three planes of symmetry at 60 to each other and inter- 
 secting in an axis of ternary symmetry, and three axes of binary 
 
GEOMETRICAL CHARACTERS. 
 
 49 
 
 symmetry bisecting the angles between the planes. As shown in 
 projection, Fig. 127, the symmetry is satisfied by twelve faces for 
 the most general form, or DITRIGONAL SCALENOHEDRON, Fig. 128. 
 EXAMPLES. Calcite, hematite, corundum. 
 
 FIG. 127. 
 
 FIG. 128. 
 
 22. CLASS OF THE DITRIGONAL BIPYRAMID. 
 
 With three planes of symmetry at 60 to each other which in- 
 tersect in an axis of ternary symmetry and one plane normal to 
 these which intersects the others in axes of binary symmetry. 
 The planes, Fig. 129, divide space into twelve equal parts, shown 
 in projection, Fig. 130, as six birectangular spherical triangles 
 with angles at the center 60. 
 
 Anyjapper face, corresponding to a pole X in one of the tri- 
 angles, would by rotations of 120 around the ternary axis coin-, 
 cide with two other upper faces. Each of these by reflection in 
 a vertical plane of symmetry coincides with another face and 
 the sixjby reflection in the horizontal plane of symmetry coincide 
 with six lower faces, the poles of which are indicated by circles. 
 
 The symmetry of the class is satisfied by twelve faces for the 
 most general form or DITRIGONAL BIPYRAMID, Fig. 131. No ex- 
 amples are known. 
 
 FIG. 129. 
 
 FIG. 130. 
 
CHARACTERS OF CRYSTALS. 
 
 THE Six LIMIT FORMS. 
 
 In each class there are six limit forms corresponding to special 
 positions of the faces of the general form. These may be tabu- 
 lated as follows : 
 
 Position of any Face and its 
 Pole. 
 
 Symbol* 
 
 Name of Form. 
 
 Classes to which the forrn 
 belongs. 
 
 I. Parallel to all basal 
 axes. Poles projected at 
 the centre. 
 
 2. Oblique to the vertical 
 axis but parallel to one 
 basal axis. Poles are on 
 
 joooij 
 ShoTiii 
 
 BASAL PINACOID, Fig. 
 153- 
 BASAL PLANE,one plane 
 of Fig. 153. 
 TRIGONAL BIPYRAMID 
 FIRST ORDER, Fig. 
 132. 
 RHOMBOHEDRON FIRST 
 
 22,21, ,19, 18,17, . 
 ,,20,,,, 1 6. 
 22, , , 19, , , . 
 
 the interaxial diameters. 
 
 
 ORDER, Fig. 135. 
 TRIGONAL PYRAMID 
 
 
 3. Oblique to the vertical 
 axis and equally inclined 
 to alternate basal axes. 
 
 Shfahil 
 
 FIRST ORDER, Fig. 
 136. 
 HEXAGONAL BIPYRA- 
 MID SECOND ORDER, 
 
 Fig. 155- 
 HEXAGONAL PYRAMID 
 
 22,21, ,,,,. 
 
 Poles are on the axes. 
 
 
 SECOND ORDER, Fig. 
 1 60. 
 TRIGONAL BIPYRAMID 
 SECOND ORDER, Fig. 
 
 ,,,19,18,,. 
 
 17 -' 
 
 
 
 OND ORDER, Fig. 140. 
 TRIGONAL PYRAMID 
 
 > > > > */ 
 
 ,16. 
 
 4. Parallel to the vertical 
 axis and to one basal 
 axis. Poles are at inter- 
 sections of primitive cir- 
 cle and interaxial di- 
 ameters. 
 5. Parallel to the vertical 
 axis and equally inclined 
 to alternate basal axes. 
 Poles are at intersections 
 of axes and primitive 
 circle. 
 
 6. Parallel to the vertical 
 and unequally inclined to 
 all basal axes. Poles are 
 
 S loloj 
 
 {"*} 
 
 SECOND ORDER, Fig. 
 141. 
 TRIGONAL PRISM FIRST 
 ORDER, Fig. 133. 
 HEXAGONAL PRISM 
 FIRST ORDER, Fig. 
 156. 
 
 HEXAGONAL PRISM 
 SECOND ORDER, Fig. 
 
 "57- 
 
 TRIGONAL PRISM SEC- 
 OND ORDER, Fig. 138. 
 
 DITRIGONAL PRISM, 
 Fig. 134- 
 
 DlHEXAGONAL PRISM, 
 
 22, 21,20, 19, , , 16. 
 ,,,,18,17,. 
 
 22,21,20, -,l8,I7,. 
 
 , , ,19, , ,16. 
 
 22, ,20, ,8 1,,. 
 ,21, , , , , 
 
 
 
 Fig. 158. 
 TRIGONAL PRISM 
 THIRD ORDER, Fig. 
 
 139. 
 HEXAGONAL PRISM 
 THIRD ORDER, Fig. 
 161. 
 
 , , , 19, , , 1 6. 
 
 ,,, ,,17, . 
 
 * See footnote under hexagonal division, p. 54. 
 
GEOMETRICAL CHARACTERS. 
 
 FIG. 132. 
 
 FIG. 133. 
 
 FIG. 134. 
 
 FIG. 135. 
 
 FIG. 136. 
 
 FIG. 137. 
 
 FIG. 138. 
 
 FIG. 139. 
 
 FIG. 140. 
 
 LIMIT FORMS IN RHOMBOHEDRAL DIVISION 
 
 WHICH ARE GEOMETRICALLY SHAPES 
 
 NOT INCLUDED IN HEXAGONAL 
 
 DIVISION. 
 
 FIG. 141. 
 
52 CHARACTERS OF CRYSTALS. 
 
 THE HEXAGONAL DIVISION. 
 23. CLASS OF THE THIRD ORDER HEXAGONAL PYRAMID. 
 Without planes of symmetry but with one axis of senary sym- 
 metry. As shown in the projection, Fig. 142, six planes satisfy 
 the symmetry of the most general form or HEXAGONAL PYRAMID 
 OF THIRD ORDER, Fig. 143. EXAMPLE. Nephelite. 
 
 FIG. 142. FIG. 143. 
 
 24. CLASS OF THE HEXAGONAL TRAPEZOHEDRON. 
 Without planes of symmetry but with the seven axes of Class 
 27. As shown in projection, Fig. 144, twelve faces satisfy the 
 symmetry for the most general form or HEXAGONAL TRAPEZOHE- 
 DRON, Fig. 145. EXAMPLE. (SbO) 2 BaC 4 H 4 O 6 .KNO 3 . 
 
 \ 
 
 v 
 
 FIG. 144. FIG. 145. 
 
 25. CLASS OF THE THIRD ORDER HEXAGONAL BIPYRAMID. 
 With one plane of symmetry normal to the axis of senary sym- 
 metry. As shown in projection, Fig. 146, twelve faces satisfy the 
 symmetry for the most general form or HEXAGONAL BIPYRAMID OF 
 THIRD ORDER, Fig. 147. EXAMPLE. Apatite, pyromorphite. 
 
 FIG. 146. 
 
 FIG. 147. 
 
GEOMETRICAL CHARACTERS. 
 
 53 
 
 26. CLASS OF THE DlHEXAGONAL PYRAMID. 
 
 With six planes of symmetry at 30 to each other and intersec- 
 ting in the axis of senary symmetry. From the projection, Fig. 
 148, it is evident that the general form, or DIHEXAGONAL PYRAMID, 
 Fig. 149, is geometrically the upper (or lower) half of the general 
 form, Fig. 152, of the next class. 
 
 FIG. 148. 
 
 FIG. 149. 
 
 27. CLASS OF THE DIHEXAGONAL BlPYRAMID. 
 
 With seven planes of symmetry, six being normal to the seventh 
 and at 30 to each other as shown in Fig. 150. The common 
 intersection of six planes of symmetry is the axis of senary sym- 
 metry and their intersections with the seventh are axes of binary 
 symmetry. 
 
 FIG. 150. 
 
 FIG. 152. 
 
 In projection, Fig. 151, the planes of symmetry form twelve 
 equal birectangular spherical triangles with angles at the centre, 
 30. Any upper face, with a pole marked x , by rotations of 60 
 around the senary axis coincides successively with five other upper 
 faces represented by crosses in the alternate triangles. Each 
 of the six reflected in a vertical plane of symmetry coincides 
 with another face and the twelve reflected in the horizontal 
 plane coincide with twelve lower faces represented by circles. 
 
54 
 
 CHARACTERS OF CRYSTALS. 
 
 The letter permutations of the general symbol (1ikli\ in which i 
 always must remain in fourth place, are hkli, hlki, khli, klhi, Ihki, 
 
 Ikhi, and each may have four sign permutations, +-j f-, j-+, 
 
 + -j and 1 or twenty-four permutations in all. Pro- 
 jection and permutation both show that the symmetry of the class 
 is satisfied by twenty-four faces for the most general form or Di- 
 
 HEXAGONAL BlPYRAMID, Fig. 152. EXAMPLE. Beryl. 
 
 THE Six LIMIT FORMS. 
 
 In each class there are six limit forms corresponding to special 
 positions of the faces of the general form. These may be tabu- 
 lated a follows : 
 
 Position of any Face and its Pole. 
 
 Symbol* 
 
 Name of Form. 
 
 Classes to which 
 the formbel'ng's 
 
 I. Parallel to all basal axes. 
 Poles projected at the center. 
 
 2. Oblique to the vertical axis 
 but parallel to one basal axis. 
 Poles are on the interaxial 
 diameters. 
 
 3. Oblique to the vertical axis 
 and equally inclined to alter- 
 nate basal axes. Poles are 
 on the axes. 
 
 4. Parallel to the vertical axis 
 and to one basal axis. Poles 
 are at intersections of primi- 
 tive circle and interaxial di- 
 ameters. 
 
 5. Parallel to the vertical axis 
 and equally inclined to alter- 
 nate basal axes. Poles are 
 at the intersections of axes 
 and primitive circle. 
 
 6. Parallel to the vertical but 
 unequally inclined to all j 
 basal axes. Poles are on the 
 primitive circle. 
 
 !OOOI 
 
 \hhzhi\ 
 
 1010 
 
 1 1 20; 
 
 BASAL PINACOID, Fig. 153. 
 BASAL PLANE, one plane of 
 Fig- 153- 
 
 27, ,25,24, . 
 ,26, ,,23. 
 
 HEXAGONAL BIPYRAMID [27, ,25,24, . 
 
 FIRST ORDER, Fig. 154. | 
 HEXAGONAL PYRAMID ,26, , ,23. 
 
 FIRST ORDER, Fig. 159. 
 
 HEXAGONAL BIPYRAMID 27, ,25,24,- 
 
 SECOND ORDER, Fig. 155. 
 HEXAGONAL PYRAMID SEC- ,26, , ,23. 
 
 OND ORDER, Fig. 160. 
 
 HEXAGONAL PRISM FIRST 127,26,25,24,23. 
 ORDER, Fig. 156. 
 
 HEXAGONAL PRISM SEC- 
 OND ORDER, Fig. 157. 
 
 27,26,25,24,23- 
 
 DlHEXAGONAL PRISM, Fig. 27,26, ,24, . 
 
 158. 
 
 HEXAGONAL PRISM THIRD , ,25, ,23. 
 ORDER, Fig. 161. 
 
 * In the type symbols the first and second indices relate to the relative inclination of 
 the faces to the alternate basal axes. If unequally inclined hk, if equally inclined hh, 
 if parallel to one ho. The third index is /, the algebraic sum of the first and second 
 with the opposite sign. 
 
GEOMETRICAL CHARACTERS. 
 
 55 
 
 FIG. 153. 
 
 FIG. 154. 
 
 FIG. 155. 
 
 FIG. 156. 
 
 FIG. 157. 
 LIMIT FORMS OF CLASS 27. 
 
 FIG. 158. 
 
 FIG. 159. FIG. 1 60. 
 
 LIMIT FORMS GEOMETRICALLY NEW IN THE 
 OTHER CLASSES. 
 
 FIG. 161. 
 
 PROJECTION AND CALCULATION OF HEXAGONAL AND RHOMBOHE- 
 
 DRAL FORMS. 
 
 The basal pinacoid is the plane of projection. The position of 
 the poles are as in the tables, pp. 50 and 54. 
 
 The indices of a face truncating an edge of any form are the 
 sum of the indices of the faces forming the edge, for example, the 
 edge between ion and oui is truncated by 1122. 
 THE ZONAL RELATIONS 
 
 Are calculated from three indices of which one must relate to 
 the vertical axis. For instance, by method of cross multiplication 
 
CHARACTERS OF CRYSTALS. 
 
 the zone indices of the planes hkliji'k'l'i' w\\\ be determined by hki 
 and h'k'i' : u = ki' ik' , v = ih' hi', w = hk' kh' . 
 
 DETERMINATION OF ELEMENTS. 
 
 Two equal basal axes at 120 to each other and at right angles 
 to the vertical are sufficient. In Fig. -f u the spherical triangle rst 
 has /== 120. 
 
 cos s -f- V 2 cos r OL 
 
 cos tr = - : -^= ; cot tr = () rr] c = OL.i 
 
 or 
 
 // c 
 tan (oooi) :(II22) = ^ = - = c 
 
 Cooo/JfT 
 
 ono 
 
 FIG. 162. FIG. 163. 
 
 To DETERMINE THE POSITION OF ANY POLE P. 
 
 Using notation of Fig. 163 and denoting by M the quantity 
 
 31 
 
 c(2/i -f k) 
 cos PA = - --~, cos 
 
 m 
 To FIND THE ARC JOINING TWO POLES P AND P. 
 
 cos PP' = - 
 TANGENT PRINCIPLE. 
 
 c(h + 2k) c(k - Ji) 
 
 - -^ , cos /^ = A ^- 
 
 /v/3 
 
 MM' 
 
 tan PC _ k _ h 
 tarTF^ ~" V ~ Ji' 
 
GEOMETRICAL CHARACTERS. 
 
 57 
 
 THE ISOMETRIC SYSTEM. 
 
 All forms in this system must be referrable to three equivalent 
 axes at right angles to each other. The system includes five 
 classes. 
 
 28. CLASS OF THE TETARTOID. 
 
 The forms have seven axes of symmetry but no planes of 
 symmetry. Three of the axes are binary normal to the faces of 
 the hexahedron and four are ternary through diagonally opposite 
 solid angles of the hexahedron. 
 
 As shown in the projection Fig. 164, twelve faces in each alter- 
 nate octant satisfy the symmetry for the most general form or 
 TETARTOID, Fig. 165. EXAMPLES Sodic chlorate, baric nitrate. 
 
 .-";;*. -.. 
 
 a 
 FIG. 164. 
 
 FIG. 165. 
 
 29. CLASS OF THE GYROID. 
 
 Without planes of symmetry but with all the axes of symmetry 
 of class 32. As shown in the projection Fig. 166, twenty-four 
 faces, three in each octant, satisfy the symmetry for the most gen- 
 eral form* or GYROID, Figs. 167 and 168. EXAMPLES, Halite, 
 sylvite, cuprite. 
 
 FIG. 1 66. 
 
 FIG. 167. 
 
 FIG. i 68. 
 
 *The right form, Fig. 168, and the left form, Fig. 167, are enantiomorphic, that is, 
 their elements are equal and the faces of the one are parallel but oppositely placed 
 with respect to those of the other. 
 
CHARACTERS OF CRYSTALS. 
 
 30. CLASS OF THE DIPLOID. 
 
 The forms have three cubic planes of symmetry, the intersec- 
 tion of these are three axes of binary symmetry and there are four 
 diagonal axes of ternary symmetry. 
 
 As shown by the projection, Fig. 169, the symmetry of the 
 class is satisfied by twenty-four faces for the most general form or 
 DIPLOID, Figs. 170 and 171. EXAMPLES Pyrite, smaltite, cobal- 
 tite. 
 
 FIG. 169. 
 
 FIG. 170. 
 
 FIG. 171. 
 
 31. CLASS OF THE HEXTETRAHEDRON. 
 
 The forms * have the six dodecahedral planes of symmetry of 
 Class 32 and the seven axes of symmetry formed by their intersec- 
 tion. Of the latter, four are ternary and three binary. 
 
 As shown in the projection, Fig. 172, the symmetry of this 
 class is satisfied by twenty-four faces for the most general form or 
 HEXTETRAHEDRON, Figs. 173 and 174. EXAMPLES. Sphalerite, 
 tetrahedrite, diamond. 
 
 FIG. 172. 
 
 FIG. 173. 
 
 FIG. 174. 
 
 32. CLASS OF THE HEXOCTAHEDRON. 
 
 /\ 
 The forms of this class have three planes of symmetry parallel 
 
 to the faces of the cube and six planes of symmetry through diag- 
 
 * The forms of this class exist in pairs which are said to be congruent, that is either 
 by a revolution of 90 about an axis may brought into the position of the other. They 
 are distinguished as right and left forms. 
 
GEOMETRICAL CHARACTERS. 
 
 59 
 
 onally opposite edges of the cube, that is parallel to the faces of 
 the rhombic dodecahedron Fig. 181. 
 
 FIG. 175. 
 
 FIG. 176. 
 
 FIG. i 7 || 
 
 The intersections of these planes are axes of symmetry; three 
 are quaternary the intersections of the planes parallel the hexahe- 
 dron ; four are ternary, each the intersections of three diagonal 
 planes ; and six are binary, each the intersection of a plane of each 
 kind. 
 
 These planes and axes of symmetry are shown in stereographic 
 projection, Fig. 176. 
 
 The number of faces in the most general form may be deter- 
 mined as follows : 
 
 In any quadrant the pole marked I of a face in the upper half 
 of the crystal by rotations of 120 around the ternary axis must 
 coincide successively with faces the poles of which are 2 and 3. 
 These three by reflection in the planes of symmetry must coincide 
 I with 4, 2 with 5, 3 with 6. Since there is a vertical axis of 
 quaternary symmetry the entire quadrant must coincide succes- 
 sively pole for pole with the second, third and fourth quadrants 
 and finally the resulting 24 faces reflected in the horizontal 
 plane of symmetry must coincide with 24 lower faces. That 
 is, the form must consist of 48 faces with a pole in each of the 
 right spherical triangles made by the planes of symmetry. 
 
 This result could also be reached by considering the possible 
 permutations of letter and sign in the symbol of the general form 
 \hkl \ . The letter permutations are : 
 
 hkl, hlk, klh, khl, Ikh, Ihk. 
 
 Each of these can undergo eight permutations in sign according 
 to the octant in which the plane occurs. 
 
 Upper octants + + +,_ + +,- . +, + +, 
 Lower octants + -\ , ( , , H , 
 
6o 
 
 CHARACTERS OF CRYSTALS. 
 
 or 48 permutations in all. That is to satisfy the symmetry the 
 most general form or HEXOCTAHEDRON, Fig. 177, must consist of 
 48 faces six in each octant. EXAMPLES. Garnet, fluorite, spinel. 
 
 THE Six LIMIT FORMS. 
 
 In each class there are six limit forms corresponding to special 
 positions of the faces of the general form. These may be tabu- 
 lated as follows : 
 
 Position of any Face. 
 
 Symbol* 
 
 Name of Form. 
 
 Classes to which 
 Form belongs. 
 
 I. Equally inclined to all three 
 axes. Poles on ternary axes. 
 
 J'"J 
 
 OCTAHEDRON, Fig. 178. 
 TETRAHEDRON, Fig. 184. 
 
 3 2, ,30,29, . 
 ,31, ,,28. 
 
 2. Equally inclined to two 
 axes, more nearly parallel 
 the third. Poles are on the 
 short s'des of triangles. 
 
 {Ml] 
 
 TRIGONAL TRISOCT'AHEDRON, 
 Fig. 179- 
 TETRAGONAL TRISTETRAHE- 
 DRON, Fig. 185. 
 
 32, ,30, 29, . 
 
 ,3' .28- 
 
 3. Equally inclined to two 
 axes less nearly parallel the 
 third. Poles are on hypothe- 
 nuse. 
 
 {Mi} 
 
 TETRAGONAL TRISOCTAHE- 
 
 DRON, Fig. 1 80. 
 
 TRIGONAL TRISTETRAHE- 
 DRON, Fig. 1 86. 
 
 32,, 30,29,, 
 .31, ,,28. 
 
 4. Equally inclined to two 
 axes,parallel thethird. Poles 
 at vertices cf right angles. 
 
 S"i 
 
 RHOMBIC DODECAHEDRON, 
 Fig. 181. 
 
 32,31,30,29,28. 
 
 5. Unequally inclined to two 
 axes parallel thethird. Poles 
 are on the long sides. 
 
 w 
 
 TETRAHEXAHEDRON, Fig. 182. 
 
 PENTAGONAL DODECAHE[ 
 DRON, Fig. 187. 
 
 32,31, ,29, . 
 
 , , 30 28. 
 
 6. Parallel to two axes. Poles 
 are on the quaternary axes. 
 
 l,oo] 
 
 HEXAHEDRON, 183. 
 
 32,31,30,29,28. 
 
 FIG. 178. 
 
 FIG. 179. 
 
 FIG. 1 80. 
 
 * The type symbols of the forms are easily obtained without direct reference to the 
 intercepts by noting parallelism and relative nearness to parallelism to the axes. Zero 
 being the symbol of parallelism and conventionally h > k > /. 
 
GEOMETRICAL CHARACTERS. 
 
 61 
 
 FIG. 181. 
 
 FIG. 182. 
 LIMIT FORMS OF CLASS 32. 
 
 FIG. 183. 
 
 FIG. 184. 
 
 FIG. 185. 
 
 FIG. 1 86. 
 
 LIMIT FORMS GEOMETRICALLY NEW 
 THE OTHER CLASSES. 
 
 IN 
 
 FIG. 187. 
 
 PROJECTION AND CALCULATION* OF ISOMETRIC FORMS. 
 The forms are usually projected on a cubic face and the poles 
 are as stated in the table, the positions being usually found by zone 
 intersections. The planes of symmetry divide space into forty- 
 eight similar parts each shown in stereographic projection as a 
 right triangle in which : the hypothenuse (54 44'), is the angle 
 between a binary and a ternary axis ; the longer side (45) is the 
 half angle between two binary axes; the shorter side (35 16') is 
 the half angle between two ternary axes. 
 
 CALCULATION OF ELEMENTS. 
 
 The parameters are all equal and the angles between the axes 
 are right angles. 
 
 * Millers' Crystallography, pp. 25; Story-Maskelyne's Crystallography, p. 453. 
 
62 
 
 CHARACTERS OF CRYSTALS. 
 
 GENERAL EXPRESSION OF RELATION BETWEEN ANGLES AND INDICES* 
 
 cos PA _ cos PB _ cos PC 
 
 ~k~ ~T~ ~T~ 
 
 To DETERMINE POSITION OF ANY POLE. P=(hkl). 
 Adopting notation of Fig. 188 equations become 
 
 tan BAP= \ tan = 
 
 cos PA = sin PH= -^=; 
 
 k 
 
 Vihol 
 
 Lk&Q 
 
 cos PC = smPL = -W n which 
 
 A ioo 
 
 FIG. 188. 
 
 To FIND THE ARC JOINING P= hkl and P* = h'k'l' 
 
 cos/y-^ + ^tf 
 
CHAPTER V. 
 
 MEASUREMENT OF CRYSTAL ANGLES. 
 
 The instruments used in measuring the angles between the faces 
 of crystals are called goniometers, the simplest form being the appli- 
 cation goniometer invented for 
 Delisle by Carangeot and con- 
 sisting of a semicircular grad- 
 uated arc and two arms moving 
 upon a pivot, the position of 
 which may be changed accord- 
 ing to the size and position of 
 the crystal. In the older instru- 
 ments the arms are fastened to 
 FlG l8 9- the arc, but in the later types, as 
 
 in Fig. 189, they are detached from the arc during measurement 
 and replaced for the reading. In use the arms are each in close 
 contact with a face and at right angles to the edge between the 
 , faces. They cannot be relied upon closer than one degree, and 
 are, therefore, practically limited to the identification of previously 
 measured angles. 
 
 All measurements of ac- 
 curacy are obtained by ro- 
 tating the crystal around 
 the edge between the faces 
 and determining, by the aid 
 of a ray of light fixed in 
 direction, the angle between 
 the position in which the 
 first face gives a reflection 
 and that in which the sec- 
 ond face does the same. 
 
 In Fig. 190 the crystal 
 
 is so adjusted that an edge 1( , 90> 
 
 coincides with the axis of 
 
6 4 
 
 CHARACTERS OF CRYSTALS. 
 
 rotation O. The fixed direction of the ray of light is CO 7 
 that 'is, the incident ray is CO and the eye or telescope is at 
 T. Then OA will be the direction of the first face when it acts as 
 a reflecting surface, and OB will be the direction of the second 
 face at that time, and only when rotated to the respective direc- 
 tions, OA' and OB f t will a reflection be obtained from the second 
 face, that is, after a rotation measured by the equal arcs A A' , BB 1 
 or NN' t the normal angle between the faces. 
 
 GONIOMETERS WITH HORIZONTAL AXES. 
 
 The original reflection goniometer is that of Wollaston.* In 
 this a base and column support in a collar a hollow axle to which 
 there is attached at one end a vertical disc with a graduated rim 
 and at the other end a handle. Through this axle passes a second 
 axle, with at one end the crystal holder and at the other a handle. 
 These two axles may be clamped to turn together, or, by the inner 
 axle, the crystal may be rotated without the graduated circle. 
 
 In the simplest forms of this goniometer the crystal is fastened 
 
 FIG. 191. 
 
 with wax to a thin, flexible, brass plate, and this is fixed in a 
 holder which has several simple motions by means of which the 
 
 *,W. H. Wollaston, Description of a reflective goniometer, Phil. Trans., 1809, p. 
 253-25.9. 
 
GEOMETRICAL CHARACTERS. 65 
 
 faces of the zone to be measured can be made to project clear of 
 the apparatus and the edge to coincide with the axis of rotation. 
 The signal may be a horizontal window bar twenty to thirty feet 
 away, or better, a horizontal slit in a screen before an artificial light 
 and it must be parallel to the axis of rotation. The eye is brought 
 almost into contact with the crystal and there watches for the re- 
 flection of the signal as the faces successively move into position. 
 
 To secure a constant line of sight a reference mark may be made 
 below the crystal parallel to the signal, or better, a second image 
 of the signal may be obtained in a small mirror, the plane of which 
 is parallel to the axis of rotation, or still better, a telescope with 
 cross hairs may be used. In each case the rotation is continued 
 until the image of the signal is bisected by or coincides with the 
 reference mark. 
 
 Mallard's* modification of the Wollaston goniometer, shown 
 in Fig. 191 on the left, differs from the earlier types in the sub- 
 stitution of a better crystal holder ; the crystal is supported in the 
 manner suggested by Groth, that is, it is fixed with wax on a small 
 circular disc, d, and, by turning the screws v, v, and v' f v' , can re- 
 ceive two movements of rotation on two arcs of circles perpendic- 
 ular to each other which have their common center near the middle 
 of the crystal, A, so that the changes of orientation of the crystal 
 do not too much displace its center of gravity. 
 / For the proper centering, the entire system which holds the 
 crystal is attached to two sliding planes, g and g* , which, by means 
 of the screws, u and ', impart to the crystal two movements of 
 translation in a plane perpendicular to the axis of rotation. There 
 is also a tangent screw for fine rotation and the mirror, M, has four 
 motions of adjustment parallel to the axis of rotation. Any signal 
 may be used, but preferably the collimator and artificial light, as 
 shown in the figure. 
 
 The collimator is a cylinder, C, with at one end a large lens, L, 
 and at the other, which is the exact focal plane of the lens, an ad- 
 justable plate, Fig. 192, pierced with signals, /,/',/", of various 
 forms beneath each of which is a fine reference slit. The light 
 from an Argand gas lamp, R, or a Welsbach burner passes through 
 the central signals, emerges as parallel rays from the lens and is 
 reflected to the eye at the same time from the crystal face and the 
 mirror. 
 
 * Er. Mallard, Annales des Mines, Nov.-Dec. 1887. 
 
66 
 
 CHARACTERS OF CRYSTALS. 
 
 A greater degree of accuracy would be secured by the addition 
 of an observation telescope, as in the Mitscherlich* modification 
 of the Wollaston, but this is accompanied by loss of light and 
 usually makes a dark room necessary, whereas the Mallard-Wol- 
 laston gives good results in ordinary light. 
 
 H h 
 
 ffl. 
 
 FIG. 192. 
 
 Professor Groth describes-)- a simple, inexpensive Wollaston with 
 a telescope fixed parallel to the plane of the graduated circle and 
 centered on the goniometer axis. The crystal holder is like that 
 of the Mallard instrument, but the inner axle, which ordinarily 
 rotates the crystal independently, is replaced by a screw which 
 moves the crystal horizontally in the direction of the axis. The 
 signal used by Professor Groth is a very small incandescent light at 
 a distance of about thirty feet where it appears like a luminous 
 point. 
 
 The principal objection to the Wollaston type of goniometer is 
 that the weight of the crystal tends to throw it out of adjustment. 
 
 GONIOMETERS WITH VERTICAL AXES. 
 
 Babinet, v. Lang, Miller, Websky and others have gradually de- 
 veloped this more perfect and more generally applicable variety of 
 goniometer, and there is little doubt that for simplicity of adjust- 
 ments and perfection of construction the instrument known as the 
 Fuess Model II. at present excels all others. It is shown in Fig. 
 193 and consists of: 
 
 a. THE STAND. A tripod with a conically bored head-piece sup- 
 porting three concentric axles the outer axle carrying the vernier 
 circle and telescope, and turned by the latter ; the fine adjustment 
 is by the tangent screw, F, and the clamp screw, , the middle axle 
 carrying the graduated circle and turned by the disc, g> or more 
 
 *Ueber ein Goniometer Abhandl. Berlin Akad , 1843, 189-197. 
 \Physikalische Krystallographie, III. Ed., p. 613. 
 
GEOMETRICAL CHARACTERS. 
 
 67 
 
 FIG. 193. 
 / 
 
 conveniently by ';* the fine adjustment is by the tangent screw, 
 G, and the clamp screw, /?, the inner axle guiding the rod of the 
 crystal carrier and which may be turned separately or clamped to 
 the others. 
 
 The circle is graduated to half degrees. The vernier circle is 
 protected by a ring, A", with glass windows over the verniers, while 
 between the tripod and the vernier circle is a ring with two arms 
 each with a magnifier and a mirror, s, by means of which the vernier 
 can be read to half minutes and estimated to quarter minutes. 
 
 a. THE CRYSTAL CARRIER is like that described in the Mallard- 
 Wollaston, the crystal being attached to a platef by wax, and this to 
 
 * The part g' is omitted in many instruments and the clamping of the inner axle is 
 by a vertical screw. 
 
 f Two or three sizes are furnished. For very small crystals platinum wire may be 
 used with a cement of gelatine and acetic acid. By bending the wire the different 
 zones may be adjusted. 
 
68 CHARACTERS OF CRYSTALS. 
 
 two cylinder arcs, which are moved by tangent screws around inter- 
 secting axes at right angles to each other and to the goniometer axis. 
 The common point of intersection is within the crystal, so that 
 their motions tip the crystal without moving it much out of center. 
 The arcs rest upon sliding parts, which are moved by micrometer 
 screws, in the direction of the axes of the arcs. 
 
 c. THE TELESCOPE L. The observation telescope turns with the 
 vernier circle, to which it is attached by a pillar, and its axis 
 always intersects and is perpendicular to the axis of the gonio- 
 meter. The vertical cross hair is parallel to the axis. Before the 
 infinite distance objective swings an extra lens of a focal length 
 equal to the distance to the goniometer axis. When in use this 
 converts the telescope into a weak microscope. There are four 
 eye pieces magnifying respectively six diameters, three diameters 
 (generally used), two diameters (for distorted reflections) and a 
 diminishing combination which is used with very small faces to re- 
 duce the signal to one third of its natural size. 
 
 d. The COLLIMATOR. The infinite distance collimator is fixed 
 firmly to the tripod by the pillar C and its axis and the telescope 
 axis are in the same plane perpendicular to the goniometer- axis. 
 The signals are either separate tubes or all four may be on a revolv- 
 ing target in the focal plane of the collimator lens. 
 
 The Schrauf signal is a light diagonal cross on a dark back 
 ground, with, at the center, very minute cross hairs which are used 
 in the adjustments and with rare very perfect faces. 
 
 The Websky Signal is an orifice bounded by two circular discs 
 of equal diameter, the distance apart of which is regulated by a 
 screw. This admits more light than a narrow cleft though still 
 very narrow in the center and if dilated by a narrow face (see p. 
 23) and diminished in intensity the shape may still be divided 
 symmetrically with some accuracy. 
 
 The Slit Signal is an adjustable vertical cleft especially adapted 
 to broad faces or for measurement of indices of refraction. 
 
 The Pinhole Signalis a round J^mm. opening used in adjusting 
 and in recognizing slight deviations from the zone or, by multiple 
 reflection, cracked and facetted faces. 
 
 After the instrument is in adjustment* once for all the process 
 of measurement is as follows : 
 
 *The complete adjustments are : 
 I. Adjtistmeiit of the vertical hair. 
 
GEOMETRICAL CHARACTERS. 69 
 
 Several of the smallest and most brilliant crystals are selected, 
 carefully cleaned with chamois skin and thereafter handled only 
 with pincers or a tapering pencil of wax. Sketches are made 
 showing striations, flaws and other peculiarities as revealed by the 
 magnifier and later by the micro-telescope. Letters are assigned 
 to all faces and thereafter represent them. 
 
 The most prominent zone is first measured and is designated by 
 the letters of its two most prominent faces, say [A C~\. The crystal 
 is fastened with a conical pencil of wax to the plate so that the 
 zone axis is approximately vertical and one face is parallel to one 
 of the motions of translation. 
 
 The crystal is brought into view by the extra lens and raised or 
 lowered by the mother screw k. With one arc at right angles to the 
 telescope axis one of the edges of the zone is made coincident with 
 
 A needle is adjusted in the axis of the goniometer, so that during rotation it does not 
 appear to move. 
 
 An eyepiece is adjusted so that its cross hairs are distinct and is then placed in 
 the telescope tube, focussed on some distant object, the extra lens dropped into position 
 and the eyepiece turned until one hair is parallel to the needle, then the cross hairs are 
 moved horizontally until this hair coincides with the needle. 
 
 Making Telesccope axis normal to Goniometer axis. 
 
 A plate of parallel glass is fastened to the holder approximately parallel to the 
 goniometer axis and perpendicular to the telescope axis and is made exactly so by re- 
 flecting light upon the cross hairs from another plate fastened in front of the eyepiece 
 at about 45 to the telescope axis. The reflected light throws an image of the cross 
 hairs upon the parallel glass, which is reflected back into the telescope. 
 
 By aid of the arcs the images reflected from opposite sides of the parallel glass are 
 made to take the same position in the field, that is, the plate is made exactly parallel to 
 the axis of the goniometer, and by raising or lowering the cross wires their center is 
 brought into coincidence with its image as seen after reflection. The telescope axis 
 is then normal to the plate and^therefore to the axis of the goniometer. 
 Adjustment of Horizontal hair. 
 
 During rotation of the outer axle the image of the intersection of the cross wires 
 should move across the field in coincidence with the horizontal wire, if not, raise, lower, 
 or turn the wire as needed. 
 
 The Collitnator Adjustment. 
 
 The pin hole signal is first inserted, focussed, and its center made to coincide with 
 the horizontal wire of the telescope by the screw which fastens the pillar C to the tripod. 
 The vertical slit and Websky's signal are focussed and made symmetrical to the tele- 
 scope cross hairs and finally the cross hairs of the Schrauf signal are brought into ap- 
 parent coincidence by their traversing screws. 
 the Other Eyepieces, 
 
 Are adjusted by means of Schrauf 's signal. All eyepieces and signals have sliding 
 collars with a projecting tooth which fits into a notch in the tube of the telescope or col- 
 limator. The collar is tightened when the tube is in final adjustment and thereafter is 
 fixed. (M. Websky, Zeit. /. Krystallographie, IV., 545.) 
 
70 CHARACTERS OF CRYSTALS. 
 
 the vertical hair by means of the corresponding micrometer screws ; 
 then with the other arc at right angles to the telescope axis the same 
 adjustment is made with the other screws, and this is repeated until 
 during a rotation the edge and vertical hair appear to be one, and on 
 raising the extra lens the image of the signal from any face in the 
 zone is symmetrically bisected by the cross hairs. 
 
 Except in the case of very narrow faces* it is not necessary to 
 recenter on each edge, but is sufficient by a slight motion with the 
 centering screws to make the vertical hair coincident with an imag- 
 inary central axis within the zone. 
 
 With the telescope at some convenient angle to the collimator 
 (100 to 1 20 degrees) and with only /? undamped, the graduated 
 circle and -crystal are turned together by the disc g until the re- 
 flected signal is seen through the telescope, then ft is tightened, the 
 signal moved by the tangent screw G until it is bisected by the 
 vertical cross hair and the vernier is read and recorded. 
 
 The screw /9 is again loosened and the rotation continued until 
 the signal is received from a second face, and this is centered by 
 G and /? and recorded as before and the difference between the two 
 readings is the normal angle. The position of a third face is de- 
 termined in the same way and so on around the zone. 
 
 At least three measurements of any angle should be made and 
 preferably with different portions of the circle. For the second 
 measurement when the signal from the second face is centered the 
 clamp / is loosened and the crystal alone is turned until the signal 
 from the first face is centered, then / is tightened and /5 loosened and 
 the crystal and circle turned together as before. This is repeated 
 before a third measurement. 
 
 It is important to assign a quality mark to each image of the 
 signal. For instance, a distinct image may be 2, a poor image I, 
 a band or shimmer o, and the proportionate value of any measure- 
 ment may then be taken as the sum of the quality marks of the 
 readings. These may be recorded in tabular shape, as in the fol- 
 lowing example : 
 
 * See page 23. 
 
GEOMETRICAL CHARACTERS. 
 
 CRYSTAL i. Zone [m m'"] 
 
 | ! 
 
 
 
 Face. 
 
 
 READINGS. 
 
 Mean 
 
 Times to be 
 
 
 
 First. 
 
 Second. 
 
 Third. 
 
 Normal Angle. 
 
 counted. 
 
 
 
 
 
 , 
 
 
 
 / 
 
 
 
 / 
 
 / 
 
 
 m 
 
 m' 
 
 \ 
 
 137 
 
 S4 
 
 53 
 
 54 
 332 
 
 IS 
 
 332 
 249 
 
 21 
 
 mm' 8231*- 
 m'a' 48 45! 
 
 1 + 2 = 3 
 
 2+2^4 
 
 a' 
 
 2 
 
 6 
 
 6 
 
 
 36 
 
 201 
 
 5 
 
 a'm" 48 44^ 
 
 
 m" 
 m'" 
 
 I 
 
 2 
 
 317 
 334 
 
 S6* 
 
 234 
 *5 2 
 
 5i 
 
 22 
 
 "8 
 
 20| 
 
 m"m'i' 82 28f 
 '" 48 44i 
 
 1 + 2 = 3 
 0+2=2 
 
 a 
 
 O 
 
 867 
 
 1 1 
 
 103 
 
 3i 
 
 21 
 
 6 
 
 am 4845! 
 
 0+1 = 1 
 
 m 
 
 I 
 
 112 
 
 24 
 
 54 
 
 534 
 
 332 
 
 21 
 
 
 
 The angles indicate that m, m' , 
 ;//', /"' are faces of one form and 
 that a, a! are faces of another form 
 truncating the first as in Fig. 194. 
 The most probable angles will be 
 determined by averaging. For in- 
 stance, the probable angle m-m" r 
 may be the average of: 
 
 FIG. 1 94. 
 
 Minutes. 
 
 2 m a 
 2m'" a 
 2 m' a' 
 
 =97 3 
 =97 2 
 =97 3 
 
 2 m'" a' =97 
 
 180 mm' = 97 
 
 o m"m'" = 97 
 
 m m'" = 97 
 
 m'.m" =9J 
 
 Times to be counted. 
 
 i or i 
 i 
 
 2 
 
 n 
 
 3 
 3 
 3 
 3 
 
 = 97 2 9F 
 
 ERRORS DUE TO IMPERFECT CENTERING. 
 
 Let E denote the error. 
 
 Let a denote the angle between the crystal edge and the goni- 
 ometer axis. 
 
 Let f denote the angle obtained. 
 
 * The errors in the first four are already doubled, hence they can be allotted only 
 half the sum of their quality marks. 
 
 2 
 
 57g~ 
 
 4 
 
 "5* 
 
 3 
 
 85 
 
 6 
 
 171 
 
 6 
 
 187 
 
 6 
 
 177 
 
 6 
 
 179 
 
 34 
 
 1012 
 
72 CHARACTERS OF CRYSTALS. 
 
 a 2 
 Then m\\E=*--y= [cos 45 cos (2 r + 45)]. 
 
 This value is a maximum when cos (2^ -f- 45) is I, that is, 
 when Y = 67^2 for which E= -f^ a 1 (approximately) and a mini- 
 mum when cos (2 ? + 45) = + i or Y = 1 57^ for which = T L a 2 . 
 
 This error is guarded against by the coincidence of signals from 
 different faces of the zone. 
 
 In addition. 
 
 Let / denote angle of incidence of the light. 
 
 Let d and d' denote the distances of the faces from the axis. 
 
 Let r denote the distance to the light source. 
 
 Then will E=2=*smi. 
 r 
 
 This diminishes as r increases or i decreases and is zero if r is 
 infinite or d= d' ; that is, the second error is eliminated by an in- 
 finite distance collimator, and even without it is only about 9" if 
 d d' = j4 mm. and r = I o m. 
 
 SPECIAL CASES IN MEASUREMENT. 
 
 Narrow faces tend to broaden the image by a phenomenon 
 analogous to diffraction. The diminishing eyepiece may obviate 
 this, but as the error is proportionatell to the eccentricity of the 
 face, in such a case each edge must be separately centered. 
 
 Transparent Crystals give colored images due to total reflections 
 within the crystal, if the crystal and telescope are turned steadily 
 in one direction these images at a point move backwards (angle of 
 least deviation) ; when such an image coincides with the true one 
 it is only necessary to alter the angle of incidence. 
 
 Finely Striated Faces yield a bright colorless image with colored 
 images on each side. The bright image is due to the plane tan- 
 gent at the edges of the striae and only when an image is obtained 
 after the plane has been turned into some other zone is it proved 
 that there is a definite crystal face underneath the striae. 
 
 Bent or Cracked Faces yield distorted or manifold images. An 
 risblende eyepiece will limit the reflection to a selected best por- 
 tion, or an approximate measurement may be obtained from the 
 outer member of each group, or by measuring in several zones 
 
 f Story Maskelyne's Crystallography, p. 414 
 \ Ibid, 402. 
 
 For other reasons i is commonly 50 to 60. 
 I! Mull. Soc. Min. de France, I., 35. 
 
GEOMETRICAL CHARACTERS. 73 
 
 to the opposite face the properly oriented portion may be' found. 
 The best way is to use another crystal. 
 
 7 win Crystals and Composite Crystals in approximately parallel 
 position will give good images in part of a zone and in the rest the 
 images will be a little out of center. 
 
 Dull Faces* may be coated with thin varnish or cover glasses 
 may be glued to them or the extra lens may be used, the col- 
 limator slit narrowed to the smallest width which will give an illu- 
 mination and the position of BRIGHTEST ILLUMINATION of both faces 
 recorded several times. Traube's attachment to the collimator of 
 the Fuess goniometer^ shown in Fig. 195 may be used. It is 
 
 FIG. 195. 
 
 practically a little extra dark room ; the light passing through the 
 collimator and the conical tube d to the crystal. The cylinder e 
 shuts out much of the extraneous light giving a darker field in 
 which the luminous signal is by contrast brighter. Still more 
 light can be shut out by laying a card over the top of e or better by 
 the cap g t which may be turned to diminish the window /to a 
 mere slit. 
 
 Small and 'nearly parallel faces which yield a combined image may 
 be distinguished by changing the angle of incidence. 
 
 Crystals which alter in the air may be protected during measure- 
 ment by replacing the crystal plate by a hollow cylinder a t 
 Fig 196, supporting a hemisphere z and plate /, the whole cov- 
 ered by a little glass bottle with bottom pressed in and ground to fit 
 air-tight on the oiled hemisphere. In the channeled bottom is placed 
 sulphuric acid or chloride of lime for deliquescent crystals, and 
 
 *Groth describes an elaborate application goniometer " das Fuhlhebel goniometer " 
 for measuring dull crystals Physikalische Krystal., III. Ed., pp. 604-608. 
 \Neus Jahrb. f. Mineralogie, 1894, Bd. II. 
 \Jahrb. d. geol. Reichsanstalt, 1884, 329. 
 
74 CHARACTERS OF CRYSTALS. 
 
 FIG. 196. 
 
 water, etc., for efflorecent crystals. The bottle is steadied by the 
 spring clamp A, and the light enters and emerges normal to the 
 plane windows of the bottle. 
 
 Imbedded crystals may be measured to within one degree by means 
 of impressions in sealing wax. 
 
 THEODOLITE OR TWO-CIRCLE GONIOMETERS. 
 
 In the reflection goniometers described all faces of the zone which 
 is placed perpendicular to the graduated circle reflect the signal and 
 if the circle is considered to be the plane of projection their poles 
 are in the circumference at angles apart equal to those meas- 
 uring the rotation. 
 
 If a second direction of rotation at right angles to the other be 
 added, any face in any zone may be referred to this first zone by the 
 two motions necessary to bring it into the field, the one giving the 
 point where its meridian cuts the primitive circle, the other giving 
 the number of degrees on this meridian from the primitive circle. 
 Czapski, Goldschmidt* and v. Federovv have described such instru- 
 ments; that of the latter, shown in Fig. 197, consists of a telescope 
 B y which is also the collimator. The signals are on the revolving 
 disc b y and the light entering at the focal plane of the objective 
 passes through a small total reflecting prism and emerges as parallel 
 rays. There is an extra lens / to bring the crystal into focus or, by 
 
 *Zeit.f. Kryst. XXI., 210-232. 
 
GEOMETRICAL CHARACTERS. 
 
 75 
 
 FIG. 197. 
 
 means of the spring clamp h and the rack and pinion H and 
 K, one of the weaker objectives of a microscope may^be focussed 
 'on the crystal. The eye-piece has an adjustable irisblende. 
 
 The stand consists of a tripod with a bored conical head in which 
 rests the graduated circle C turned by d and read by the fixed ver- 
 nier N, using clamp e and tangent screw/. The stand also sup- 
 ports D, the carrier of the vertical circle, which can be clamped to 
 the stative by g, or to the horizontal circle by e. 
 
 The vertical circle is a complete Wollaston goniometer, but 
 rotates with the horizontal circle when e is clamped. By trans- 
 ferring the centering apparatus to the horizontal circle the instru- 
 ment may be used as a goniometer with vertical axis. 
 
CHAPTER VI. 
 
 CRYSTAL PROJECTION OR DRAWING. 
 
 For the purposes of calculation and for a comprehensive 
 view of the relations between the faces, the latter are more con- 
 veniently represented either by points, Fig. 198, as described pp. 
 20-24, under stereographic projections, or by lines as in the so- 
 called Linear projection. For description and illustration some 
 projection which actually figures the shape of the crystal is usually 
 preferred. 
 
 LINEAR PROJECTIONS. 
 
 In Quenstedt's linear projection* each face of the crystal is as- 
 sumed to be moved parallel to itself until it cuts the vertical axis 
 at a unit's distance above the the plane of the basal axes. The 
 line in which the face then intersects the plane of the basal axes 
 represents the plane. 
 
 For example, the crystal of topaz, Fig. 201, is orthorhombic in 
 crystallization with a : b : ^=.529 : I : .477. The basal axes are drawn 
 at right angles, and the proportionate lengths .529 : I laid off upon 
 them. As the crystal is symmetrical to the axial planes it is only 
 necessary to determine the projections of the faces of one octant, 
 say the upper right hand. The indices of these, therefore, are 
 written, then their reciprocals, which are the axial intercepts (p. 
 12), and finally, these divided by the third term, that is, the inter- 
 cepts of the faces when they cut the vertical axis at unity 
 
 Indices. Reciprocals. Unit c intercepts. 
 
 in ill in 
 
 110 III 001 
 
 120 i^o ooi^: 
 
 223 I** . If i 
 
 041 i\i i\l 
 
 * Beitrdgen zur rechnenden Krystallographie, Tubingen, 1848. 
 
 f In Goldschmidt's Euthygraphic Projection the faces cut a unit's distance below 
 the basal axes. Ueber Projection u. graph, Krystal., p. 25. 
 
 \ Prisms are all projected as lines through the center parallel to the projections of 
 pyramids of the same zone. In this case such a pyramid as 121. 
 
GEOMETRICAL CHARACTERS. 77 
 
 The points on a and b corresponding to the first and second 
 term of each unit c intercept, are then connected by a straight line 
 which is the linear projection of the face. Fig. 200. 
 
 The projection of the edge between any two faces" is evidently 
 the line from the center of the projection to the intersection of the 
 two lines representing the faces, and the direction of that edge is 
 the line from the intersection to the unit point on the vertical axis. 
 Hence all faces projected in lines with a common intersection in- 
 tersect in parallel edges or lie in one zone. Evidently also a face 
 lying in two zones will be projected in the line through the two 
 common intersections or zonal points. 
 
 The zonal equations may be here used as described, pp. 17-20, 
 and calculations may be made, though less conveniently than with 
 the stereographic projection. 
 
 PROJECTIONS IN PARALLEL PERSPECTIVE. 
 
 If the eye be conceived to be at an infinite distance the visual 
 rays become parallel, and all parallel lines remain so * when pro- 
 jected, -and the proportionate lengths into which any line is divided 
 are .not changed, although the line may be foreshortened. 
 
 ORTHOGRAPHIC PARALLEL PERSPECTIVE. 
 
 The plane of projection is at right angles to the visual rays. If 
 the plane is the base the basal axes are full length, but if it is at 
 right angles to the vertical axis the axis a in monoclinic crystals 
 will be foreshortened to a sin p and in triclinic crystals a will be- 
 come a sin ft and b will become b sin a. 
 
 The faces of the zone normal to the plane of projection will be 
 projected as lines bounding the figure at their true angles. The 
 edges between other faces will be parallel both to the edge projec- 
 tions (see above) of the linear projection on the same plane or to 
 the tangent to the primitive circle at the intersection with the zone 
 circle of the two faces. For instance, in Fig. 199, the direction of 
 the orthographic projection of any edge, say [041, 223], is both 
 that of the line Oc, Fig. 201, and of the tangent at O, Fig. 198. 
 
 * Lines parallel in projection are not necessarily so in the crystal, for all lines in a 
 plane through two rays are projected in the same line. 
 
CHARACTERS OF CRYSTALS 
 
 FIG. 200. 
 
 FIG. 201. 
 
 FIG. 202. 
 PROJECTIONS OF A TOPAZ CRYSTAL. 
 
 198. Stereographic. 199- Orthographic Parallel. 
 
 200 . Linear. 201. Clinographic Parallel. 
 
 202. Linear upon axial cross. 
 
GEOMETRICAL CHARACTERS. 
 
 79 
 
 CLINOGRAPHIC PARALLEL PERSPECTIVE. 
 
 The plane of projection is oblique to the visual rays. This method 
 gives an appearance of solidity and usually is made upon a vertical 
 plane, the point of sight being to the right and above the crystal. 
 
 The Axial Cross, or perspective view of the axes, is first prepared. 
 
 If the angle to the right', or horizontal angle, is denoted by d and 
 the angle of elevation by e, then the following relations* exist be- 
 tween these angles and the projected lengths and angles of the 
 isometric axes in which the notation is as in Fig. 203. 
 
 f^V.i^ cos 2 ^ cos 2 e ; OB = ^/ i sin 2 d cos 2 e ; OC= cos e 
 cot AOC= d sin e ; cot BOC= tan <5 sin e ; cot 8 = I 
 
 cot AOC 
 
 cot BOC 
 
 COS SiUL = 
 
 OB* - OC*) (OB 1 + OC* - OA Z ) 
 ~ ' 
 
 cos BOC = 
 
 \/(OA 2 + OC* OB*) (OA* 4- OB? OC*) . 
 2 OB.OC 
 
 The following table gives a series of projections, of which num 
 bers 3, 4, 5, 6 and 10 are usually preferred. 
 
 Number." 
 
 Angles of 
 Revolution. 
 
 Angles 
 between projected 
 Axes. 
 
 Foreshortened 
 lengths of axes for 
 true length = i. 
 
 Approximate 
 Proportionate 
 lengths. 
 
 
 6 
 
 e 
 
 ^<?C 
 
 BOC 
 
 OA 
 
 OB 
 
 OC 
 
 OA 
 
 OB 
 
 OC 
 
 i 
 
 n 35' 
 
 O o 49 , 
 
 93 58' 
 
 90 10' 
 
 .200 
 
 .980 
 
 999 
 
 10 
 
 49 
 
 5 
 
 2 
 
 3 
 
 14 35' 
 180 26' 
 
 lO I 7 / 
 60 20' 
 
 94 55' 
 1080 19' 
 
 900 20' 
 920 06' 
 
 .250 
 239 
 
 .968 
 .828 
 
 999 
 994 
 
 8 
 29 
 
 IOO 
 
 32 
 
 120 
 
 4 
 
 1 80 26' 
 
 70 1 1/ 
 
 i ioO 34/ 
 
 920 23' 
 
 338 
 
 950 
 
 .992 
 
 36 
 
 IOO 
 
 i5 
 
 c 
 
 1 80 26' 
 
 90 28' 
 
 ii60 i 7 / 
 
 93 8' 
 
 353 
 
 95 
 
 .986 
 
 37 
 
 IOO 
 
 104 
 
 6 190 16' 
 
 20 I 4 / 
 
 9 60 23' 
 
 900 47 / 
 
 333 
 
 944 
 
 .998 
 
 6 
 
 17 
 
 18 
 
 8 
 
 27 59' 
 
 95 0/ 
 35 1 6' 
 
 1070 49' 
 
 1200 
 
 95 ii' 
 
 1200 
 
 493 
 .817 
 
 .887 
 .817 
 
 985 
 .817 
 
 5 
 
 i 
 
 9 
 
 i 
 
 10 
 
 i 
 
 9 
 
 200 42' 
 
 190 28' 
 
 1310 25' 
 
 970 1 1/ 1 .471 
 
 943 
 
 943 
 
 i 
 
 2 
 
 2 
 
 10 
 
 13 38' 
 
 13 15' 
 
 133 24' 
 
 930 1 1/ 1 .324 
 
 973 
 
 
 i 
 
 3 
 
 3 
 
 ii 
 
 ioO 10' 
 
 IOO 
 
 1340 6' 
 
 9i 47' 
 
 .246 
 
 985 
 
 .985 
 
 i 
 
 4 
 
 4 
 
 * Story-Maskelyne's Crystallography, p. 480. Jos. Barrett, Lchigh Quarterly. 
 
\ 
 
 \ 
 \ 
 
 8o CHARACTERS OF CRYSTALS. 
 
 In drawing the cross, OC is vertical and the directions of OA 
 and OB result from the angles AOC and BOC. The proportionate 
 lengths may be laid off at any scale desired. 
 
 When cot d and cot s are simple numbers as in Nos. 3 and 5 in 
 which cot =3 and cot e = g and 6 respectively, the projection 
 of the axial cross may be obtained as follows : 
 
 A horizontal line ss, Fig. 203, is bisected by a perpendicular, and 
 perpendiculars are drawn as indicated at the ends and at points 
 / and t f and distances laid off so that 
 
 O= Of = sg=* 5 and s'e = Os - 
 
 COt d COt 
 
 (in the figure cot d = 3, cot e = 6). 
 
 The point * determines the line eO 
 and AA'\ the projection of the front 
 horizontal axis is the portion of this 
 line which is included between the 
 perpendiculars at t and f. The point 
 g determines the radius Og, which is ^., 
 the length of half the vertical axis CO. II 
 For the projection of the third axis 
 draw Af parallel to ss', draw/<9, and 
 from the intersection v draw vB par- 
 allel to ss'. BB' completes the de- FlG 203 
 sired projection or axial cross. 
 
 The axial cross for crystals of other systems is derived from the 
 isometric cross by assuming the latter to be the projection of lines 
 each equal in length to that axis of the crystal which extends con- 
 ventionally from left to right (ft). 
 
 In the TETRAGONAL and ORTHORHOMBIC crosses the proportion- 
 ate values of the axes c and a are laid off on CO and AA' re- 
 spectively. In Zircon, for instance, the vertical axis is made ^ T of 
 CO. 
 
 In the MONOCLINIC crosses, Fig. 204, the vertical axis is a propor- 
 tionate length on CC, but A A is the projection of a line at right 
 angles to CO and the direction of the clino axis is first found by 
 laying off Or=OC cos /? and OnOA sin and completing the par- 
 allelogram OAtr. The proportionate value of a is then laid off 
 upon the diagonal Ot. 
 
 1 8 
 
GEOMETRICAL CHARACTERS. 
 
 81 
 
 This and other projections are quickly obtained by using a metal 
 quadrant.* Fig. 204 the center of which is at B. The edge AB 
 and the arc AC are tapered to a thin edge for greater exactness in 
 marking. With AB ten centimeters long the results will be cor- 
 rect within the limits of a drawing. 
 
 FIG. 204. 
 
 A scale line OS is drawn at will from the center and axial 
 lengths are transferred directly from AB to the scale line approxi- 
 mately to the third decimal. Sines and cosines are transferred as 
 follows. If the edge BC and the scale line are made coincident 
 and then, by use of a triangle, the quadrant is slid along in a direc- 
 tion at right angles thereto (BC remaining parallel to the scale line) 
 until the scale line cuts the arc at the given degree and the approxi- 
 mate minute, the intercept on the scale line will be the sine. Simi- 
 larly with the edge AB coincident with the scale line and a motion 
 at right angles thereto, the intercept on the scale line will be the 
 cosine. 
 
 All measurements are transferred to other lines, by lines parallel 
 to the direction between the ends of .the scale line and the unit 
 line. For instance, in a monoclinic crystal in which a : b : c= 
 1.092 : I : 0.589 and ,3=74 10'. Ok is sine 74 10' when radius is 
 OS and transferred to OA is On\ Odis cosine 74 10' when radius 
 is OS and transferred to OC is Or\ by completing the parallelo- 
 gram t and Ot result. For the axial lengths, Oi is 1.092 times OS 
 and Oe is 0.5 9 times OS and transferred are respectively Oa= 
 
 * Described A. J. Moses, Am. Journ. Science ', I., June, 1896. 
 
82 
 
 CHARACTERS OF CRYSTALS. 
 
 In the triclinic crosses, Fig. 205, the same methodjis carried fur- 
 ther, for instance: The constants for axinite are a : b : c 0.492" 
 I :o.479, a f\ <r=/2=9i 52', b A ^=^=82 54' (100) /, (oio)= 
 131 39'. 
 
 Vertical Axis. Make Cfc= (97x 479- 
 
 Macro Axis. Make Oe=OB sin 131 
 39', and <9^= <9^4 cos 1 3 1 39'; com- 
 plete the parallelogram <^(9^^. Make 
 Ot = 0n sin 82 54' and (9^r=(7Ccos 
 82 54' ; complete the parallelogram 
 rOxb. Then is the projection of 
 one-half the desired axis. 
 
 Brachy Axis. Make 01= OC cos 
 91 52', and Op=OA sin 91 52'. 
 Complete the parallelogram ; pOIt 
 make6>#= 0.492 = <9/; then is Oa 
 the projection of one-half the desired axis. These results are more 
 quickly made with the quadrant previously described. 
 
 In the HEXAGONAL CROSSES, Fig. 206, the proportionate value of 
 c is laid off on CO and the basal axes are derived as follows : 
 
 Make Op= 6^4^.1.732 ; draw pB 
 and pB' ; bisect Op by a line parallel 
 to BB'\ then are OB, Oa and Oa^ 
 the projections of desired semi-axes. 
 
 DETERMINATION 
 OF EDGES. 
 
 OF THE DIRECTION 
 
 FIG. 206. 
 
 The unit form always results from 
 joining the extremities of the axial 
 cross by straight lines and other sim- 
 ple forms are easily drawn by meth- 
 ods which suggest themselves, for in- 
 stance, the unit prism bylines through 
 the terminations of the basal axes 
 It is always possible, also, to obtain 
 two points of any edge by actually constructing the two planes 
 and rinding the intersection of their traces in two axial planes. 
 The method, however, is cumbersome. 
 
 In all systems the projected intersection may be most easily 
 found by the following method : A linear projection of the faces 
 
 parallel to the vertical axis. 
 
GEOMETRICAL CHARACTERS. 83 
 
 is made (Fig. 202) upon the basal axes of the axial cross precisely 
 as described for the ordinary linear projection. One point of the 
 edge between any two planes is the unit point on the vertical axis ; 
 another is the intersection of the linear projections of the two 
 planes and the line connecting these is the edge. For instance, 
 Oc, Fig. 202, is the direction of the edge [041, 223] and is so drawn 
 in Fig. 201. 
 
 CONSTRUCTION OF THE FIGURE. 
 
 The edges thus formed must be united in ideal symmetry, yet so 
 as to show, as far as possible, the relative development of the forms. 
 
 A second axial cross is drawn parallel to that used in determin- 
 ing the edge directions and these are transferred by triangles, care 
 being taken that all corresponding dimensions are in their proper 
 proportions and in accord with the planes of symmetry. Gen- 
 erally it will be best to pencil in and verify the principal forms 
 and later work in the minor modifying planes. 
 
 The back (or dotted) half of many crystals can be obtained by 
 marking the angles of the front half on tracing paper, turning the 
 paper in its own plane 1 80 and pricking through. This is also a 
 test of accuracy, for the outer edge angle for angle should coin- 
 cide. 
 
 TWIN CRYSTALS. 
 
 These have two set of axes, the second so related to the first that 
 it corresponds to a revolution of 1 80 about the twin axis or line 
 normal to the twining plane. 
 
 The two individuals may be in apposition, that is, the twin plane 
 coinciding with the combination face. In this case the twin axis 
 which passes through both centers normal to the twin plane will 
 be bisected by the latter. 
 
 In interpenetrating twins the two centers may coincide or be 
 near together. The orientation, however, will be as in the former 
 case. 
 
 Given the axial cross of one crystal to find that of a second crystal 
 in apposition thereto.* 
 
 Let OA OB OC, Fig. 207, be the axial cross of the first crystal 
 and HKL the twinning plane. To find first the point Zat which 
 the normal from cuts HKL, draw H'L and HL f parallel AC 
 and draw KL" and K' L parallel BC. Complete the parallelograms 
 
 *Groth, Physikalisoche Krystallographie, p 599, III Ed. 
 
8 4 
 
 CHARACTERS OF CRYSTALS. 
 
 FIG. 207. 
 
 OH' ML' and OK'NL" and draw their diagonals OM and ON and 
 from the intersections of these with HL and LK draw RK and SH 
 respectively ; their intersection is the desired point Z. 
 
 Because the crystals are in apposition prolong OZ till ZO' = OZ, 
 then is the center of axes for the second crystal and as the face 
 HKL is common 0'H t 0' K and O'L are in direction and length 
 the coordinates of this face on the new axes. 
 
 The unit lengths will be found by drawing AA' t BE' and CO 
 parallel to O'Z. 
 
 All constructions on the axes of the second crystal follow the 
 rules previously given. 
 
OPTICAL CHARACTERS. 87 
 
 of time. The ray front in the first medium at the expiration of 
 this time would be TE, E being a point of the front just impinging 
 on AB, therefore also a point of the new ray front, that is of the 
 tangent plane ES. The refracted ray is therefore * OS to the 
 point of tangency. OYis the corresponding reflected ray. 
 In the triangles OETand OES, Fig. 208, 
 
 >, os 
 
 OE = - ^r~ = -r^ and OE = 
 
 x-\ j * rt-\ - C*Vt \S ** ' x-~\ 7"" {~* "' '"' * 
 
 sin OET sin 2 sm OES sin /> 
 
 hence, 
 
 7', sin i 
 v^ sin p 
 
 Usually the ratio recorded is that of the crystal with respect to 
 air. Denoting the velocity in air by v and the indices of the outer 
 medium and crystal with respect to air by n L and n, we have 
 
 v - v 
 
 n. = and n = . 
 v \ v z 
 
 and substituting these values we have 
 
 sin i 
 
 n = 
 
 1 sin p 
 There is no refraction for normal incidence, for sin z o, hence 
 
 o 
 
 sin p 
 which is only possible if p o. 
 
 With a plane-parallel plate the ray emerging is parallel to the 
 entering ray, for at entrance, Fig. 210, 
 
 sin i sin z, 
 
 and at emergence n^ = n 
 
 sin p sin^ 
 
 #j sin p sm ^ 
 
 but / t = p hence i= p^ 
 
 An approximate determination of the 
 index of refraction may be made by meas- 
 FIG. 210. uring the displacement of the focal distance 
 
 * This is called the Huyghens construction, each point of the border surface becom- 
 ing a new center of propagation of light. A still simpler construction is that of Snel- 
 lius, Fig. 209, ^ is a sphere with radius the index of refraction of outer medium, R.^ a 
 sphere with index of crystal. Prolong IO to T. Draw TS parallel the normal ON, 
 then is OS the refracted ray, and OK the corresponding reflected ray. 
 
88 CHARACTERS OF CRYSTALS. 
 
 of a microscope caused by the interposition of a known thickness of 
 crystal as described, p. 120, but usually one of the following meth- 
 ods will be employed. 
 
 Determination of Index of Refraction by Prism Method. 
 
 Let AOC, Fig. 211, be the section of the prism at right angles 
 to the refracting edge O. About O describe the circles R^ and R 2 with 
 radii proportionate to the indices of refraction of the outer medium 
 and prism respectively. Let 10 be the incident ray, then, by the 
 construction of Snellius (foot note p. 89), is OS the direction of the 
 ray in the prism. From Sdraw SP normal to the surface CO, then 
 is OP the direction of the ray on emergence. Denoting the prism 
 angle by / NON 1 the total deviation by d = TOP, the incident 
 angle by z = TON, and the angle of refraction at the second surface 
 by Pl =PON l . 
 
 <5 + 7 = TOP + NOW = 2 TOP + PON + TON 1 , 
 i + ^ = TON+ PON 1 = 2 TOP + PON + TON 1 
 hence, 
 
 3 + X = * + Pl or d = i + Pl - x . 
 
 d -4- y 
 
 This value of d is least * when i= /> lt then <5 = 2 z 7 z = - . 
 
 When z = />! the ray RS, Fig. 212, within the prism must be nor- 
 mal to BD, the bisectrix of the refracting angle, therefore will NRS 
 = ABD or p = y 2 /. Hence substituting in the formula for index 
 
 r f ,.- Sm * u 
 
 of refraction ;/ = n,~ , we have, 
 sm/> 
 
 PRACTICAL MANIPULATION. Two perfect faces of a clear trans- 
 parent crystal are required, making such an angle (40 to 70) with 
 each other that at the second surface the ray is incident at less than 
 the angle of total reflection, p. 90, or with a larger angle, the prism 
 may be immersed in a strongly refracting liquid in a parallel walled 
 
 *The arc TP or rf, Fig. 211, cut by SP and ST, is least when they make equal 
 angles with OS that is when the first deviation i p equals the second p l i l for 
 with any other position of IO the point 7' moves a certain number of degrees and one 
 of the arms TS or PS approaches OS, becoming less oblique and cutting off a part of 
 TP, the other recedes becoming more oblique and adding a larger arc to TR, hence 
 the combined change yielding a larger value for J. 
 
OPTICAL CHARACTERS. 
 
 89 
 
 glass vessel. If necessary, faces may be ground at the proper angles. 
 The other faces of the crystal should be coated with lamp black. 
 
 FIG. 211. FIG. 212. 
 
 The most satisfactory instrument is a goniometer with vertical 
 axis seep. 66-74. The angle / is centred and measured as de- 
 scribed, p. 69. The telescope is then clamped at 7] Fig. 2 12, directly 
 opposite the collimator K and a reading made, the crystal is moved 
 by the centring screws so that the edge B is a little beyond the 
 centre, and, the telescope and graduated circle remaining clamped, 
 is revolved into, for example, the position shown in the figure. 
 The telescope is then undamped and turned towards the left until 
 the image is in the field. The crystal is turned towards the right 
 and if the image moves it is recentred by a movement of the tele- 
 scope towards the right, and this double motion is continued until 
 the image appears for a moment to be stationary and then starts 
 to move in the opposite direction. 
 
 The position of rest T r is the position of least deviation and 
 
 Monochromatic light* is essential. 
 
 * Certain solids vaporized in the flame of a Bunsen burner emit light essentially 
 monochromatic. Three very commonly used are RED /, = 0.000670 Lithium sul- 
 phate, YELLOW ? = 0.000589 Sodium sulphate, Green ?= 0.000535 Thallium sul- 
 phate. Purer light may be obtained by using portion of a spectrum. A. E. Tutton de- 
 scribes an instrument for thus producing light of any desired wave length of greater 
 brilliancy than that yielded by a colored flame. Proc. Royal Soc. 1894, v. 55, p. ill. 
 
 The production of monochromatic light by absorption of the other colors is not pos- 
 sible. Blue cobalt glass permits to pass only blue and extreme red. Certain solutions 
 absorb certain rays; aniline blue absorbs yellow; permanganate of potash, green ; sul- 
 phate of copper, red ; chromate of potash, blue. The light passed through a series of 
 these may become essentially monochromatic, e, g. t Sulphate of copper, aniline blue 
 and chromate of potash, leave green. 
 
CHARACTERS OF CRYSTALS. 
 
 If the instrument used does not permit of independent rotation 
 of the crystal the position of minimum deviation and the reading 
 T' are obtained by alternate movements of the telescope and di- 
 vided circle and the reading of the collimator or T is obtained last. 
 
 Determination of Index of Refraction by Total Reflection. 
 
 When the index of refraction n of the outer medium is greater 
 than n of the crystal there is a so-called "critical" angle of inci- 
 dence for which the angle of refraction is 90 ; that is, the refracted 
 ray travels along the border surface. 
 
 If p =3 90, sin p = I ; hence, n n^ sin i or sin 2 = . 
 
 n \ 
 
 For any angle of incidence 
 greater than this the light is to- 
 tally reflected. 
 
 The following is the construc- 
 tion of Snellius: If AB, Fig. 21 3, 
 is the border surface, R and R z 
 circles with radii proportionate to 
 the indices of refraction of the 
 first and second substances, then 
 for the critical angle the direction 
 of the refracted ray must be OP 
 and from the tangent at P results the direction lOTot the limit 
 incident ray. 
 
 According to the relative position of the observation telescope 
 and the incident light two different results are obtained. 
 
 i. TOTAL REFLECTION PROPER. If diffused light is admitted in 
 the quadrant AN, Fig. 213, and the telescope axis is in the direction 
 OW, all rays incident at less than the critical angle are in part re- 
 flected, e. g. t Y along OM and in part penetrate the crystal ; while 
 all rays incident at more than the critical angle are 
 entirely reflected ; that is, the telescope field re- 
 ceives on one half totally reflected rays, on the 
 other partially reflected rays ; between these is a 
 sharp line, Fig. 214, which is the intersection of 
 the focal plane of the telescope with a limit sur- 
 face or cone the vertex of which is at O and the 
 elements of which make the critical angle with 
 the normal to the reflecting surface. This limit line is a curve, but 
 
 FIG. 214. 
 
OPTICAL CHARACTERS. 
 
 FIG. 215. 
 
 within the limits of the field of the telescope is essentially straight. 
 2. GRAZING INCIDENCE, or Observation of Transmitted Light.* If 
 the light is shifted to the quadrant AL, or, which is equivalent, the 
 light remains in y^Vand the telescope axis is made to coincide 
 with OT and the transmitted rays are viewed, then, assuming the 
 faces at entrance and emergence to be parallel, 
 the incident and emerging rays are parallel. 
 All rays incident at less than the critical angle 
 are partially transmitted, but all of greater angle 
 are totally reflected at the first surface ; the tele- 
 scope field is on that side dark, Fig. 215, but 
 on the other side is illuminated by the rays in- 
 cident at less than the critical angle. 
 
 In the Kohlrausch apparatus the substance is supported in a 
 liquid of higher index of refraction than the crystal, in such a way 
 that the reflecting surface is vertical and the crystals can be 
 rotated about a vertical line in the reflecting surface. The crystal 
 holder may be simply a metal plate with a window-like opening 
 bisected by a platinum wire and adjusted once for all so that the 
 back surface is in the desired position of the reflecting face and 
 the wire coincides with the vertical cross-hair of the telescope. It 
 is only necessary with this to fasten the crystal face over the win- 
 dow. A more elaborate 
 holder permits rotation of 
 the crystal in its own plane 
 and other adjustments. 
 
 The rotation is observed 
 by a horizontal telescope 
 placed in the direction 719, 
 Fig. 216, normal to a plane 
 front of the vessel holding 
 the liquid, and the rotation 
 is recorded upon a gradu- 
 ated circle. The best posi- 
 tion for the light is found 
 by trial. When the sharp 
 limit linebetween thetotally 
 
 * Apparatus for observation of the transmitted ray with liquids were constructed by 
 Christiansen, Pogg. Ann., 1871, 143, p. 250 and others; and for solids by Quinke 
 Zeit. f. Jfryst., 1879, 4,540. 
 
 FIG. 216. 
 
92 CHARACTERS OF CRYSTALS. 
 
 and partially reflected rays has been made to coincide with the 
 vertical hair of the telescope the light and screen are moved to the 
 opposite side and the plate is rotated until the limit line is again 
 obtained and centered. 
 
 Since the angle between the telescope axis and the normal is the 
 critical angle the rotation NON 1 = 22 whence by ;/ = ;/ A sin i the 
 index of the solid results. 
 
 A simpler apparatus is made as an attachment to the No. 2 Fuess 
 Goniometer* and is shown in Figs. 217, 218. The pin a fits in 
 place of the pin of the usual crystal plate of the instrument ; pro- 
 vision for approximate adjustment is made but the accurate adjust- 
 ments of the Fuess instrument are the principal reliance. 
 
 To secure approximately constant temperature the holder is 
 
 covered by a box of asbestos with 
 proper openings. The mineral is at- 
 tached to the little plate/, and the con- 
 trol mineral at g. The rough adjust- 
 ments are made by the eye, so that 
 the necessary rotation can be secured ; 
 then the finer adjustment is made with 
 the Fuess centring screws, and the 
 vessel filled with the refracting liquid. 
 FIG. 217. FIG. 218. The cover box is then put on, the light 
 
 adjusted, and the boundary found 
 
 and centred. After standing, say V 2 hour, with light burning, the 
 boundary is recentred and this repeated till no change takes 
 place. The usual readings are then made, after which, without 
 disturbing vessel or cover, the control mineral f is raised into the 
 field by the vertical screw and readings obtained from it From 
 these the index results by the formula 
 
 , sin t 
 n = n'. 
 
 sin P 
 
 in which 
 
 i = half the angle of rotation of the mineral tested. 
 F= " " " " " " " control mineral. 
 n' = index of refraction of the control mineral. 
 
 *A. J. Moses and E. Weinschenk, Zeit f. Krystallog., XXVI., 150 and S cf M. 
 Quarterly, XV ill., p 12. 
 
 f For a control mineral fluorite is very suitable, as its refraction has been carefully 
 determined, and further, it is isotropic, has a low index of refraction and is easily 
 obtained. 
 
OPTICAL CHARACTERS. 
 
 93 
 
 These instruments require either monochromatic light, or sun- 
 light may be used if the eye-piece is replaced by a spectroscope, 
 in which case at the proper angle the light of all colors will be 
 totally reflected and the field will show on one side a bright 
 spectrum, on the other a relatively dark spec- 
 trum, separated by a line oblique to the verti- 
 cal hair, Fig. 219. The critical angles cor- 
 responding to the Frauenhofer lines can be 
 successively determined by bringing the points 
 of intersection of these lines with the limit 
 line into contact with the vertical hair. 
 
 The Liebisch apparatus, Fig. 220, employs 
 a glass prism P of high index of refraction, 
 firmly mounted* with the refracting edge 
 vertical and one face normal to the axis of the holder. Diffused 
 monochromatic light is admitted at the side AB, Fig. 221, reflected 
 at the second side JSCand emerges at the third side AC. 
 
 The edges of the prism are first made vertical; the crystal is 
 then glued to the cap z and adjusted by the screws q so that a 
 collimator signal from the crystal remains fixed in the telescope 
 during a complete rotation by T. By the centring screws y y of the 
 goniometer a central line of the crystal is made to coincide with 
 the goniometer axis. 
 
 FIG. 219. 
 
 FIG. 220. 
 
 *As made by Fuess two prisms with indices 1.6497, '-7849 are furnished. 
 
94 CHARACTERS OF CRYSTALS. 
 
 The prism is then moved into contact with the crystal, close 
 contact being secured by a drop of strongly refracting liquid. 
 The determination of the index of the plate requires the meas- 
 urement of j the index of the prism, of = ACB, the refracting 
 v angle of the prism, and of d, the de- 
 
 \ viation of the emerging ray from the 
 
 \ normal to the face of emergence. 
 
 To determine o three readings are 
 ' ^ needed : First t the telescope and col- 
 
 limator are placed opposite each 
 other, say at T and V respectively. 
 Second, the telescope is arbitrarily 
 \ moved to some position L and the 
 
 carrier turned until the limit line is 
 centred. Third, the carrier is turned 
 T still further until AC gives a signal, 
 FlG - 221 - that is, until RN bisects ZFat OH, 
 
 This gives TL 9 NH t and HL= y 2 (180 - TL). 
 Then the deviation d = LN= NH HL. 
 
 To find an expression for the index of refraction of the substance : 
 
 Since i is the critical angle of the prism, n = n^ sin i (p. 90). 
 
 From the figure, a = ACB = AN,R = N^OR + N^RO = i + ft, 
 
 whence z = a /9, or in general i= a =fc p t since as n approaches 
 
 u ft diminishes and may become negative ; that is, the ray RL may 
 
 be on the other side of RN. Substituting, n = ^ sin (a =t /9) = 
 
 ;/! (sin a cos /5 cos a sin /?.) The reflected ray OR is refracted at R 
 
 sin /9 i 
 
 where -~- = - , whence 
 sin d n 
 
 sin 8 I 
 
 sin /? = - and cos /? = - \/// 2 sin 2 
 
 Substituting, 
 
 sin a \/n 2 sin 2 <5 cos a sin 
 
 / ///^ Pulfrich apparatus the glass prism is replaced by a vertical 
 glass cylinder, the substance resting upon the upper base and il- 
 luminated from below by diffused light, no light being permitted 
 to enter at the top. The angle d of deviation from the normal 
 on emergence is measured by a right-angled telescope revolving 
 
OPTICAL CHARACTERS. 95 
 
 on a horizontal axis. For this 
 
 n =-v 2 
 
 In the Abbe apparatus the glass prism is replaced by a hemi- 
 sphere of glass with the substance resting upon the horizontal 
 base. The critical angle is measured directly by a telescope 
 centred upon the centre of the sphere. For this n ;/ x sin i. 
 
 The refracting liquid used may be : Thoulet solution, Rohrbach 
 solution, mono-Bromnapthalin, Methylene iodide alone or satu- 
 rated with iodoform or sulphur or any other sufficiently stable and 
 transparent liquid, the index of refraction of which is higher than 
 that of the mineral to be tested. 
 
 Density. Temp. Li or B. Na = D. Tl or E. Decrease for i increase, 
 Glycerine. 
 
 C 3 H 8 O 3 1.2594 20 i.47 2 93 
 
 Thoulet Solution. 
 
 Potassium. 3.122 18 B 1.6960 1.7167 E 1.7391 
 
 Mercury, 2.493 '8 B 1.5855 1.6001 E 1.6160 
 
 Iodide. 2.091 18 B 1.5129 1.5235 E 1.5347 
 
 Rohrbach Solution. 
 
 Barium. 3 564 23 I-793 1 E 1.8265 
 
 Mercury. 
 
 Iodide. 
 Bromnapthalin a. 
 
 C 10 H 7 Br. 8 1.66264 .00045 Na 
 
 1.4914 20 1.65820 
 
 Methylene Iodide* 
 
 CH 2 T 2 8 Li 1.746 1.7466 Tl 1.7584 .00067 Li. 0007 1 Na 
 
 19 1.7421 .00073X1 
 
 II. OPTICALLY ISOTROPIC CRYSTALS WHICH ARE CIRCU- 
 LARLY POLARIZING. 
 
 Certain isometric crystals possess the power of rotating the 
 plane of polarization (see p. 100) of the incident light ivhatever 
 the direction of transmission ; they are therefore still isotropic, but 
 doubly refracting, this rotation having been experimentally proved 
 to be due to two rays transmitted with different velocities and 
 circularly polarized in opposite directions. 
 RAY SURFACE. 
 
 Either circularly polarized ray is transmitted with a constant 
 velocity in any direction, but with respect to each other the 
 velocities have a constant difference, hence the ray surface must 
 
 * Saturation with iodoform raises the index about .02 and with sulphur as much as 
 .04. Li line is at 32; B at 28. Tl line is at 68; E at 71. 
 
96 CHARACTERS OF CRYSTALS. 
 
 be two concentric spheres. Since on reversing any section the di- 
 rection of observed rotation is not changed it follows that at the 
 extremities of any diameter of the ray surface the rotations in the 
 same shell must be opposite in direction. There can, therefore, 
 be no planes of general symmetry, though every diameter is an 
 axis of isotropy. 
 
 The division is, therefore, necessarily limited to classes 28 and 
 29 which have no planes of symmetry. Examples in class 28 are 
 barium nitrate, sodium chlorate and sodium bromate. The 
 phenomenon has not yet been observed in class 29 and it is evi- 
 dent that symmetry is not the only determining cause. 
 
 The phenomena and testing of circularly polarizing crystals will 
 be described more fully under uniaxial division IV. 
 
 ABSORPTION IN ISOTROPIC CRYSTALS. 
 
 In optically isotropic crystals monochromatic light diminishes 
 steadily in intensity as the distance traversed increases, but is in- 
 dependent of the direction of transmission. 
 
 With white light the different component colors are absorbed at 
 different rates. 
 
 A section of any given thickness therefore of an isometric 
 crystal* will transmit the same color tint whatever the direction 
 in which the crystal may be cut. 
 
 By decomposing this color with a prism the absorption spec- 
 trum is obtained, which usually shows a gradual change in absorp- 
 tion in adjoining portions, perhaps increasing from one end 
 towards the other, perhaps increasing in both directions from the 
 centre. With even a moderate thickness certain colors may be 
 absorbed completely so that the spectrum shows dark bands. 
 
 *The color tints due to the combination of the partially and unequally absorbed rays 
 may vary greatly in specimens of the same substance, which may be properly colorless or 
 faintly colored in ordinary thicknesses and yet frequently occur of brilliant colors, which, 
 nevertheless, conform perfectly in absorption to the crystal symmetry. It is prob- 
 able that this is due to minute amounts of oxides of rarer metals titanium, zirconium, 
 cerium, etc., dissolved like coloring matter in solution. See Weinschenk, Zcit.f. Anorg. 
 Chemie, XII., 372. 
 
OPTICAL CHARACTERS. 
 
 97 
 
 CHAPTER VIII. 
 
 THE OPTICALLY UNIAXIAL CRYSTALS. 
 
 In every crystal of the hexagonal or tetragonal system the direc- 
 tions equally inclined to the crystallographic axis c are optically 
 equivalent, so that c is an axis of isotropy and being a fixed 
 crystallographic direction may be called an Optic Axis.* , All 
 diameters normal to c are axes of binary symmetry. 
 
 III. OPTICALLY UNIAXIAL CRYSTALS IN WHICH THE OPTIC 
 AXIS IS A DIRECTION OF SINGLE REFRACTION. 
 
 DOUBLE REFRACTION. 
 
 In a moderately thick calcitef cleavage, Fig. 222, mounted with a 
 rhombic face vertical and so that it can be revolved about a hor- 
 
 FIG. 223. 
 
 izontal axis normal to a vertical face, any light ray, IT, nor- 
 mally incident, Fig. 223, at the vertical face, is transmitted in the 
 rhomb as two rays of essentially equal brightness;}: (giving twa 
 images of any signal), and as the rhomb is turned about the axis 
 
 * It will be seen later that while the optic axis in uniaxial crystals is fixed, the so- 
 called optic axes in biaxial crystals change with the light or by heat or pressure. 
 
 f Calcite is chosen because of the marked divergence of the two rays. The discus- 
 sions, however, are general. 
 
 \ Absorption is more marked in the case of one image than the other. 
 
9 8 
 
 CHARACTERS OF CRYSTALS, 
 
 one of these remains fixed in position, the other moves around the 
 first and always so that both remain in a plane parallel to a b c d 
 (the so-called principal section) and at a constant distance apart. 
 
 With crystals optically isotropic and normal incidence the fixed 
 image only would have been seen, hence this is called the ordinary 
 and the other by contrast the extraordinary. 
 
 If a second calcite rhomb similarly mounted is placed in front of 
 the first and revolved, the other remaining stationary, each of the 
 two rays from the first is again split into two rays, an ordinary 
 and an extraordinary, lying in the principal section of the sec- 
 ond calcite, and these are no longer of equal brightness, but wax 
 and wane in turn, the sum of their intensities remaining constant. 
 PLANE OF VIBRATION. 
 
 The changes in intensity (brightness) corresponding to different 
 
 values of , the angle between the 
 principal sections, correspond exactly 
 to the assumption that the varying 
 elliptical vibrations of common light 
 are converted by > the first calcite into 
 two sets of straight-lined vibrations, 
 one parallel to the principal section, 
 one at right angles thereto. Since 
 the rays are of equal intensity, with 
 equal vibration amplitudes, if we de- 
 note the ordinary and extraordi- 
 nary rays from the first calcite by O 
 
 and E and their ordinary and extraordinary components in the 
 second calcite by and E E e , it will be seen from Fig. 224. 
 
 If the principal section is the plane 
 of vibration of the ORDI- 
 NARY RAY. 
 
 Ray. 
 
 Direction. 
 
 Amplitude. 
 
 
 E 
 
 ab 
 
 Ok = cos a 
 Og = sin a 
 Oc=l 
 
 
 
 
 a i^i 
 
 O&r=s\n a 
 
 
 e 
 
 <A 
 
 Of= cos a 
 
 
 O cd 
 
 Oc=i 
 
 If the principal section is the plane 
 of vibration of the EXTRA- 
 ORDINARY RAY. 
 
 1 
 
 c \ d \ 
 *& 
 
 O/= cos a 
 Ok = sin a 
 
 Og sin a 
 Oh = cos a 
 
OPTICAL CHARACTERS. 
 
 99 
 
 Both assumptions give, therefore, the same amplitudes for the 
 four rays, viz : O = cos a, O e = sin , Q = sin a, E t = cos a, 
 therefore both correspond to the same relative intensites (propor- 
 tionate to squares of amplitudes), moreover O e and O e = E Q 
 for all values of , and also -f O e = O and E Q -f E e = E since 
 sin 2 -f cos 2 a. = i. 
 
 We shall hereafter assume* //&?/ the plane of vibration of the extra- 
 ordinary ray is parallel to the principal section, and that of the ordinary 
 is at right angles to the principal section. 
 
 Oo 
 
 t>. 
 
 
 
 E \< 
 
 ) 
 
 : <S 
 
 e \ 
 
 &'' 
 
 F. C *O O % 
 
 o ( 
 
 s 
 
 I * 
 
 
 
 '. : 
 
 ; 
 
 FIG. 225. 
 
 The results* corresponding to different values for a are illus- 
 trated in Fig. 225, a b representing the principal section of the 
 first calcite a l b l that of the second calcite and the diameter in each 
 circle being the assumed vibration direction. The intensities cor- 
 responding are proportionate to the squares of the vibration ampli- 
 tudes. 
 
 Intensity proportionate to cos 2 a 
 
 
 
 
 E e 
 
 Oe 
 
 E 
 
 
 
 i 
 
 i 
 
 
 
 
 
 45 
 
 X 
 
 X 
 
 X 
 
 X 
 
 900 
 
 
 
 
 
 I 
 
 I 
 
 igoO 
 
 i 
 
 i 
 
 
 
 
 
 2250 
 
 
 X 
 
 X 
 
 X 
 
 2700 
 
 
 
 
 
 I 
 
 I 
 
 600 
 
 X 
 
 X 
 
 
 
 X. 
 
 Intensity proportionate to sin 2 a 
 
 That is at zero two rays are extinguished and the same two at 1 80. 
 From these points they gradually increase at the expense of the 
 
 * This supposition is purely for convenience as best connecting this work with the 
 common usage in "Optical Mineralogy and Petrography." The question is one for 
 the physicists and by them seems to be more generally decided in the other way. See 
 Jas. MacCullagh, Trans. Royal Irish Soc., XVIII., XXI., W. H. C. Bartlett, Amer. 
 Jour. Science, Nov., 1890. F. Neumann, Vorlesungen fiber der festen Korper una 
 des Lichtdthers. 
 
ioo CHARACTERS OF CRYSTALS. 
 
 other pair and are at a maximum at 90 and 270, the others being 
 then totally extinguished. At all diagonal positions there are visible 
 four rays of equal intensity, and for all other positions as at 60 
 the intensity of two rays are greater than those of the other 
 two. 
 
 PLANE OF POLARIZATION. 
 
 Common light reflected at a particular angle of incidence char- 
 acteristic of the reflecting substance acquires the same peculiar 
 characters as the rays produced by double refraction. 
 
 If normally incident at a rhombic face of a calcite rhomb an or- 
 dinary (unrefracted) image is obtained when the plane of reflection 
 (through incident and reflected ray), is parallel to the principal sec- 
 tion of the calcite, an extraordinary when these are at right angles 
 to each other and, for all other angles, both ordinary and extra- 
 ordinary of varying intensity, just as with calcite. 
 
 Malus* described the reflected ray as POLARIZED with reference 
 to the plane of reflection and called the latter the PLANE OF POLAR- 
 IZATION of the ray. 
 
 In the same sense the two rays produced from common light 
 by double refraction in calcite are said to be POLARIZED. 
 
 The principal section of the analyzing calcite becomes the plane 
 of reference. Whatever it is parallel to wlien a ray undergoes ordi- 
 nary refraction is the plane of polarization of that ray\ that is, the 
 plane of polarization of the ordinary ray is the principal section and 
 the plane of polarization of the extraordinary ray is at right angles 
 to the principal section. 
 
 RAY SURFACE. 
 
 By measurement of the indices of refraction for different direc- 
 tions of transmission it is found : 
 
 1. That the velocity of the ordinary ray is constant. 
 
 2. That the velocity of the extraordinary ray in any section 
 through c varies for different directions of transmission. For the 
 direction parallel to c it is equal to that of the ordinary and dif- 
 fers most for the direction of right angles to c andf or any other 
 direction, as discovered by Huyghens for calcite, the velocity is 
 given by the corresponding radius-vector of an ellipse the axes of 
 which are the least and greatest velocities. 
 
 The ray surface for light of any wave-length is therefore a double 
 surface, the extraordinary shell being an ellipsoid formed by the 
 
 *Mem. de la Soc. de Strasbourg, 1811, I., 284. 
 
OPTICAL CHARACTERS. 
 
 101 
 
 revolution of the ellipse, the axes of which are the least and greatest 
 velocities, about one of its axes ; and the ordinary shell a sphere 
 with a diameter equal to the axis of revolution of the extraor- 
 dinary. This surface is symmetrical to all planes through the optic 
 axis and to the diametral plane at right angles thereto. 
 
 Denoting the indices of refraction of the fastest and slowest 
 rays, that is of the two rays transmitted normal to the optic 
 axis, by and y and their vibration directions by a and c,* 
 there will be two cases arbitrarily distinguished as POSITIVE in 
 which c is parallel to the crystallographic axis c and NEGATIVE in 
 which a is parallel to c. In the former case, therefore, the extra- 
 ordinary ray with vibration direction parallel c will be the slower 
 ray and the extraordinary shell will be wholly within the 
 ordinary as in Fig. 226, while in the negative surface the 
 extraordinary ray is the faster and the extraordinary shell will 
 enclose the ordinary as in Fig. 227. 
 THE OPTICAL INDICATRIX. 
 
 All the relations between the optical characters of a crystal can 
 be expressed by the geometrical characters of the extraordinary 
 shell or its equivalent/)* the ellipsoid of revolution, the axis of revo- 
 lution of which is the index of refraction of the extraordinary ray 
 transmitted normal to c and the equatorial diameter the index of 
 the ordinary ray. 
 
 FIG. 226, -f , c c. FIG. 227, , c= a. 
 
 In Fig. 226 let the section of the extraordinary shell represent 
 the corresponding section of the indicatrix, then let IO be any di- 
 ection of transmission. Draw ID tangent at /, OE parallel to 
 
 * Often called axes of elasticity. 
 
 f In indicatrix axis in direction a a, in shell = In indicatrix axis in direction 
 
 7 
 
 c = 7, in shell = But a : y = : 
 
102 CHARACTERS OF CRYSTALS. 
 
 ID, ED tangent at E and EN normal to JO, then are 01 and OE 
 conjugate radii and the parallelgram OEDI is of constant area 
 Oc x Oa, hence, Oc x Oa = 01 X 7V. Let OA denote the nor- 
 mal at equal Oa> 
 
 Describe the circumscribing circle or section of the ordinary 
 shell, then are Of and OH the velocities of extraordinary and or- 
 dinary ray v e and V Q . 
 
 From Oc X Oa = Of x EN, have 01= v^ = -,., 
 
 JH.V 
 
 Ocx Oa 
 
 From <9# = 6>c and OA = Oa, 0/f = z> = 
 
 OA 
 that is v* : v n = 
 
 Moreover, jSTV in the principal section and OA normal to it are 
 respectively the directions of vibration of rays to which they cor- 
 respond. 
 
 That is : For any diameter of the ellipsoid considered as a direc- 
 tion of transmission there are two points of the surface, the nor- 
 mals from which are also normal to that diameter. These normals are 
 at once the directions of vibration and the reciprocals of the velocities 
 of the rays transmitted in the direction of the diameter. 
 
 DERIVATION OF POSITIVE RAY SURFACE. Upon a and c, Fig. 226, 
 the directions of vibration of the fastest and slowest rays, make 
 Oa = a Oc = Y- Oc is the axis of rotation and the ellipsoid result- 
 ing is the indicatrix. 
 
 Section ac of Ray Surface. For the direction Oc the two normals 
 are Oa and Oa, for the direction Oa the two normals are Oa and Oc, 
 for any other direction, Of, the two normals are OA = Oa and 
 EN, in which EN varies between Oa and Oc according to the di- 
 rection of transmission. 
 
 According to the rule then this section of one shell is a circle 
 
 with constant radius -^ =- and of the other shell is an ellipse 
 Oa a 
 
 with axis in direction a = 7^- = - and in direction c = 7^- = - 
 
 Oc r Oa a 
 
 corresponding exactly to Fig. 226. 
 
 SECTION cm- For every direction of transmission two normals 
 exist, Oc and Oa ; that is, this section of the double surface is two 
 
 I i . i I 
 
 concentric circles with radii -=- = and -=-=. 
 
 Oc r Oa a 
 
OPTICAL CHARACTERS. 
 
 103 
 
 Determination of Optical Characters. 
 
 The determination of the optical characters of a uniaxial crys- 
 tal consists essentially in the determination of the axes of the Indi- 
 cate ix ; that is, of the principal indices of refraction (indices of the 
 two rays transmitted normal to the optic axis c\ The determina- 
 tion may be direct or indirect. 
 
 Direct Determination of Principal Indices of Refraction. 
 
 (a) Prisms with refracting edge B t Fig. 212, parallel to the optic 
 axis give for minimum deviation, p. 88, a direction of transmis- 
 sion RS normal to the optic axis. 
 
 (b) In prisms with refracting edge B, Fig. 212, perpendicular to 
 the optic axis and faces AB and AC equally inclined thereto the 
 optic axis is BD and the direction of transmission RS for minimum 
 deviation is normal to it. 
 
 Because the horizontal section aa of the ray surface is two con- 
 centric circles, the formula p. 89 holds good for both rays 
 
 sin y 2 ( + x] sin V 2 ($' -f x\ 
 
 a n, . v . r n \ . , 
 
 sin y 2 x sin y 2 x 
 
 (c] In a prism ABC, Fig. 228, with one face 
 AB parallel to the optic axis, rays normally in- 
 A cident* at that face experience no refraction, 
 because in sections normal to the optic axis 
 both shells are circles, but on emergence from 
 the second face the two rays are differently re- 
 fracted. 
 
 Denoting the prism angle by x, the deviation 
 of the faster ray by d and that of the slower ray 
 by S r and measuring these only we have : 
 
 a = 
 
 sin PSN 
 
 w 
 
 M sin (x 4- d) 
 
 'Sin 737V- 
 
 sin OSN 
 Sin 737V ~ 
 
 t sin (x 4 8') 
 
 1 sin;r 
 
 * With this face vertical and with collimator and telescope at any convenient angle 
 obtain a signal from the face, then turn the crystal through one-half this angle until 
 normal to the collimator. 
 
104 CHARACTERS OF CRYSTALS. 
 
 (d) The indices and f may be calculated from the indices ob- 
 tained with other prisms.* 
 
 (e) In any crystal face or section one of the extinction direc- 
 tions, p. 117, is in a plane through the optic axis and the other is 
 at right angles thereto and, therefore, is itself at right angles to the 
 optic axis. 
 
 If this direction is made horizontal in a total 
 reflectometer the transmission will be at right 
 angles to the optic axis and the methods and 
 formulae of pp. 90-95 will be available. There 
 will be two distinct limit lines which may both 
 be in the field at once as in Fig. 229 or may 
 not, and are successively brought into coinci- FIG. 229. 
 
 dence.with the vertical hair of the telescope. 
 
 The measurements determine the relative values of and ?-. By 
 means of a nicols prism, p. 105, which transmits only light vibra- 
 ting in a plane through its shorter diagonal, the ordinary and extra- 
 ordinary rays may be distinguished, the former being transmitted 
 when the shorter diagonal is at right angles to the optic axis, the 
 latter when the shorter diagonal is parallel, as previously explained. 
 
 In positive crystals the ordinary is the faster, that is, corresponds 
 to . 
 
 In negative crystals the ordinary is the slower, that is, corresponds 
 to Y- 
 
 Indirect Determination with Plane Polarized Light. 
 
 Parallel faced (plane-parallel) sections of known orientation are 
 prepared. Cleavages are used when obtainable, or if the section 
 is to be parallel to a crystal face this face is cemented to a glass and 
 an opposite artificial face ground on with emery and polished with 
 rouge. When the desired section is not parallel to any known 
 face it is fastened to glass by slowly hardening cement, adjusted at 
 the proper angle and ground, the new face being verified gonio- 
 metrically with reference to other faces. 
 
 Crystals soluble in water are ground in some other liquid, as 
 mono-bromnapthalin or benzine, and if fragile are ground only on 
 a glass plate. After grinding the sections are cleaned and trans, 
 ferred to another plate. 
 
 A very perfect apparatus in which true planes may be rapidly 
 
 *Th. Liebisch, Phys. Kryst., 1891, 384-390. 
 
OPTICAL CHARACTERS. 
 
 105 
 
 ground within 10' of any desired direction has been described by 
 A.E. Tutton.* 
 
 PLANE POLARIZED LIGHT may be produced from common light, 
 
 (a] By reflection at a particular angle of incidence (tan /= ), 
 
 the vibrations being at right angles to the plane of reflection 
 
 (plane through incident and reflected ray) in accordance with the 
 
 assumption of p. 99. 
 
 (&} By refraction through a series of parallel glass plates, each 
 plate increasing the proportion of polarized light. In this case the 
 vibrations are in the plane of reflection. 
 
 In (a] and (b] common light is always present. 
 (c] By double refraction and total reflection of one of the rays. 
 The best known device for securing this effect is the so-called 
 Nicol's prism,t made from a cleavage of calcite with a length about 
 twice its thickness, Fig. 230. The two small 
 rhombic faces at 71 to the edge are ground 
 away and replaced by faces at 68 to the edge. 
 The prism is then cut through by a plane at 
 right angles both to the new terminal faces and 
 to the principal section. The parts are carefully 
 polished and cemented by Canada balsam, the 
 index of refraction of which is 1.54 or about that 
 of the extraordinary ray bd^ which, therefore, 
 passes through the balsam with but little change 
 in direction; the ordinary ray be, however, with 
 an index of refraction of 1.658, being incident at 
 an angle greater than its critical angle, is totally 
 reflected. The vibration direction of the emerging 
 light is, therefore, parallel to the short diagonal 
 of the face of the nicol, as shown by the arrow. 
 (d) By double refraction and absortion. Cer- 
 tain substances absorb one ray much more 
 FIG 2 o rapidly than the other, hence thicknesses can be 
 
 * Proc. Royal Soc. 1894. Vol. 55, p. 108. 
 
 t Described Jamesons New Journal, V. 6 1828. Various modifications of this 
 prism have been made to decrease the cost and increase the field. See Ze.it. f. Kryst., 
 XL, 179, 410, for instance. 
 
 The Foucault's prism uses a layer of air instead of Canada balsam and is cut at a 
 different angle requiring a shorter prism but giving a smaller field. 
 
 In the Hartnack prism the terminal planes are at right angles to the axis, the prism 
 is shorter and the field reaches 420. It is much used. 
 
 The Bertrand prism is of flint glass with the high index of refraction of 1.658. It 
 is bisected by a plane at 76 43' to the base and between the two halves is a thin calcite 
 
io6 CHARACTERS OF CRYSTALS. 
 
 chosen for which one is totally absorbed, the other is partially 
 transmitted as light, the vibrations of which are in one plane. In 
 tourmaline the ordinary ray is the more rapidly absorbed. 
 
 INTERFERENCE. A ray of polarized monochromatic light AB 
 Fig. 231, incident at the lower surface of a plane-parallel doubly 
 refracting plate at any angle, is broken into two rays BC and BD Y 
 vibrating in planes at right angles to each 
 III other and following different paths in the 
 
 1 C J >] plate. On emergence they follow parallel but 
 
 not coincident paths and do not produce inter- 
 ference. 
 
 But among the other incident rays from the 
 same source and parallel to AB there are 
 
 rays EG and FH, such that from all points 
 
 231. anc j o ^ ^ U pp er sur f ace there will emerge 
 
 the ordinary component of one ray and the extraordinary of 
 another following the same path. These rays will have travelled 
 over slightly different paths in the plate with different velocities. 
 
 If a second polarizer is placed in the path of these rays each ray 
 will be by it resolved into components the vibrations of which are 
 in and at right angles to the plane of vibration of the polarizer 
 and only the former will be transmitted. That is, there will 
 emerge two rays advancing in the same line and with parallel vi- 
 brations. If these vibrations are alike in phase the intensity of the 
 resultant ray will be proportionate to the square of the SUM of their 
 amplitudes, but if unlike in phase the intensity will be propor- 
 tionate to the square of their DIFFERENCE. 
 
 POLARISCOPES. The instruments used for producing and study- 
 ing the interference phenomena are called POLAHISCOPES. In these 
 parallel rays of plane polarized light, or converging bundles of par- 
 allel rays, are incident at one surface of the plate at a known 
 angle, traverse the plate undergoing single or double refraction ac- 
 cording to its nature; and, if doubly refracting, the rays following; 
 the same path are reduced by the analyzer to one plane of vibra- 
 tion producing interference phenomena. 
 
 The essentials of a polariscope for parallel light are shown in 
 
 cleavage properly oriented. The light enters the prism, and reaching the calcite is 
 doubly refracted ; the ordinary ray, with a refractive index about that of the glass, con- 
 tinues its course; the extraordinary with a much lower index is totally reflected. The 
 field is about 45. Compte Rendu, Acad. Sci., Sept. 29, 1884. 
 
OPTICAL CHARACTERS. 
 
 107 
 
 Fig. 232. The mirror M sends parallel rays through the lower 
 lens L, which concentrates them at the centre of the polarizer P\ 
 this point is also the focus of the equivalent upper lens L. On 
 emergence from L the rays are again parallel, undergo refraction 
 in the plate 5 of the substance and reach the analyzer A, which 
 transmits only those components of the resultant rays the vibra- 
 tions of which are in its own plane. 
 
 Convergent light is obtained by the 
 
 S addition of a lens or system of lenses 
 
 ,, of short focal length just above the 
 plate 5 and a corresponding system 
 just below the plate, Fig. 233. Any 
 point/ of the focal plane of the lower 
 lens system is illuminated by a cone 
 of rays the base of which is the lens. 
 This cone is made a cylinder of par- 
 allel rays by the lens. The rays of 
 each cylinder which traverse the plate 
 are again concentrated by the upper 
 lens system at points />', etc., are 
 sorted by the analyzer and finally 
 exhibit a picture or image the shape 
 brightness and tints of which depend 
 upon the structure of the plate for all 
 the directions traversed by the cylin- 
 ders of parallel rays. 
 
 The polariscope of to-day is usually a polarizing microscope. 
 In the simpler types, such as the Seibert * n A, the polarizer below 
 the stage can be raised, lowered and turned ; the analyzer above 
 the objective can be pushed in and out and convergent light 
 images can be obtained with high power objective by placing a 
 small convergent lens on top of the polarizer, raising the latter till 
 it touches the section and removing the eye-piece. An orifice 
 above the objective is always provided for the insertion of test plates, 
 p. 146, and with increasing complexity there are added special 
 micrometer eye-pieces, Bertrand lens for magnification of con- 
 vergent light image, sliding motions of the stage and so on. 
 
 _ 
 P\ 
 
 232. 
 
 FIG. 233. 
 
 * For description of this instrument and its manipulation, see L. Mel. Luquer, 
 S. of M. Quarterly, 1896, p. 442-445. 
 
io8 
 
 CHARACTERS OF CRYSTALS. 
 
 FIG. 234. 
 
 The Fuess microscope * model VI., Fig. 234,13 at present prob- 
 ably the finest instrument made for this work. The stage reads to 
 minutes and has quick rotation by hand, slow rotation by ratchet 
 and sliding motions in two directions. There is an independent 
 
 * Aeues Jahrbuch f. Mineralogie Beilage, Bd., X., 180, by C. Leiss. 
 
OPTICAL CHARACTERS. 
 
 109 
 
 FIG. 236. 
 
 focussing screw for the interference figure and a special device of 
 cog wheels r Z, r l Z l , by which there may be a simultaneous 
 rotation of polarizer and a special cap analyzer, the object remain- 
 ing at rest, but the same relative change taking place as if the stage 
 were revolved and the nicols at rest. 
 
 Two forms of the Norremberg apparatus as constructed by 
 Fuess are here shown. Fig. 235 shows the so-called Universal 
 
no 
 
 CHARACTERS OF CRYSTALS. 
 
 Apparatus, ee' are collecting lenses on each side of the polarizer ; 
 above e' are four plano-convex lenses, n, forming the condenser 
 and just over these the stage. 
 
 In a separate tube system above are the objective, composed of 
 four similar plano-convex lenses o, and at their focal plane the 
 glass plate r, on which a cross and a scale are marked ; the image 
 there formed is magnified by t and viewed through the analyzer q. 
 
 By removal of n, o, r and / the apparatus yields parallel rays. 
 
 Fig. 236 shows a later less expensive type, in which the lower 
 nicol is replaced by a pair of mirrors. The high cost of iceland 
 spar is the principal reason for the change and the results are very 
 satisfactory. 
 
 WITH PARALLEL MONOCHROMATIC LIGHT, AND CROSSED NICOLS. 
 
 With crossed nicols none of the light from the polarizer can 
 pass through the analyzer and the field must be dark. 
 
 FIG. 237. 
 
 FIG. 238. 
 
 In sections normal to the optic axis the field remains dark through- 
 out the entire rotation of the stage, and no interference phenomena 
 are possible, because the light from the polarizer traverses the 
 section in the direction of the optic axis, therefore, without change. 
 
 In all other sections there is double refraction and interference. 
 The field is dark at intervals of 90 ; that is, whenever the planes 
 of vibration of the rays produced in the section coincide with 
 the planes of vibration of the nicols. For all other positions the 
 field is illuminated by the components of the rays which pene- 
 trate the analyzer and this brightening is most intense in the diag- 
 onal positions. 
 
OPTICAL CHARACTERS. in 
 
 The rays pursuing the same path are by the analyzer brought 
 into one plane of vibration and there interfere, the kind of inter- 
 ference being determined by J, the difference in the retardations 
 which the two rays have undergone, the formula being* 
 
 Or for normal incidence, 
 
 A = /(, - n) 
 
 In which J is the retardation (difference in retardation) in ////. 
 millionths of a millimeter. 
 
 / is the thickness of the plate in ////. 
 
 A is the wave-length in /J./J.. 
 
 i is the angle of incidence. 
 
 i? l is the index of refraction of the slower ray. 
 
 ;/ is the index of refraction of the faster ray. 
 
 When J = A, 2A, 3 A, etc., the field ts dark during an entire revolu- 
 tion, tor Fig. 237, the components of PP,on emergence from the 
 plate with vibration directions RR and DD, must be of the same 
 phase that is the simultaneously displacing forces' acting upon any 
 ether particle are Or and Os, which when reduced to the plane 
 of the analyzer are Oa and Oa lt in opposite directions and equal. 
 
 When J = 1A, fA, fA, etc., the light will 
 be at its brightest because the components 
 of PP must then on emergence from the 
 plate be of opposite phase, Fig. 238, and 
 the simultaneous displacing forces acting 
 on any ether particle O are Or and Os, 
 
 which reduced by the analyzer to its plane are Oa and Oa l in the 
 same direction and equal. 
 EXPERIMENT. 
 
 If a wedge of double refracting crystal, Fig. 239, cut so that its 
 planes of vibration are parallel to the length and breadth, is placed 
 between crossed nicols and illuminated by perpendicularly incident 
 monochromatic light and there revolved. 
 
 It will be perfectly dark when in the normal positions and in all 
 others will show a series of dark and light parallel stripes which 
 are most marked in the diagonal position. If the nicols are made 
 parallel the portions formerly light become dark. With light of a dif- 
 ferent wave-length, the distance between the dark bands is changed. 
 
 * Reduced from formula, p. 364, Glazebrook's Physical Optics. 
 
1 1 2 CHARACTERS OF CR YSTALS. 
 
 *These relations may be deduced from the formula for intensity 
 of emerging light 
 
 /= a 2 sin 2 2 (. sin 2 
 
 in which 
 
 a = amplitude of incident ray, /^wave-length, J = retardation, 
 <P = angle between vibration plane of lower nicol and slower ray. 
 / will be a minimum : 
 
 (a) when sin 2 2 <p = o or 2 ? = o, 1 80, 360, 540, etc., or <p = 
 0, 90, 1 80, 270; that is, four times in a revolution of the plate 
 or whenever the planes of vibration of the plate coincide with 
 those of polarizer or analyzer. 
 
 (b) when sin 2 ( I = o which will be whenever - = I, 2, 3, etc., 
 
 \ A / / 
 
 that is whenever the phase difference J is a multiple of A, for then 
 sin 2 I I becomes sin 2 180, or a multiple thereof. 
 
 This is independent of <p. hence with this condition the plate- will 
 remain dark throughout an entire revolution. 
 / will be a maximum: 
 
 When sin 2 2^=1 or 2^ = 90, 270, etc., and ^ = 45, 135* 
 225, etc. 
 
 When f =i, 2", f, for then sin 2 (~ J \ = sin' 2 90, sin 2 270, etc. 
 
 = I. This will be independent of c? and the illumination will 
 exist throughout an entire revolution. 
 
 WITH PARALLEL WHITE LIGHT AND CROSSED NICOLS. 
 
 In sections normal to the Optic Axis the field is dark throughout 
 the rotation, the optic axis being the same for all colors. 
 
 In all other sections there is extinction every 90 and greatest 
 brightness in the diagonal positions, but, since J may be at the 
 same time approximately an even multiple of ^/ and an odd mul- 
 tiple of j/', light of one wave-length may be greatly weakened 
 while that of another wave-length is practically undimmed; that is, 
 there will result a tint due to unequal changes in all the colors. 
 
 As J increases in value it passes alternately through */, I, 3/2, 
 2, 5/2 times each wave-length as shown in Fig. 240, in which the 
 
 *Th. Liebisch, Grundriss der Phys. Kryst.> 1896, p. 271-275. 
 
OPTICAL CHARACTERS. 
 
 Blaci 
 
 / ron -gray 
 
 Lavender- gray 
 
 WhiU 
 
 YelUo 
 
 Oranye-ytttow . 430 
 Orange 4 5o 
 
 Jnd'igo 
 
 Blue 
 
 Green 
 
 . yellow yio 
 
 Orange 94 J 
 
 F 
 
 ft eddn/i -orange <oo 
 
 Biuuf, eioUt 
 
 n 5 
 D H 
 
 .As/ / / / 
 
 YY 
 
 
 / Y NX 
 
 
 
 x/ \ 
 
 ^)O 
 
 \ 
 
 () 
 
 zigzag lines show the changes 
 in brightness for six of the 
 colors.* The tint correspond- 
 ing to any value of A maybe 
 judged by noting which colors 
 are near maximum and mini- 
 mum. 
 FIRST ORDER COLORS. 
 
 A = o. All light is shut out; 
 as A increases *4/ of violet is 
 first reached and J^A of blue 
 and green next. 
 
 J = 100 /jt, /jt. The weakly 
 coloring violet and still fainter 
 blue and green give a total 
 impression of lavender gray, 
 which gradually brightens as 
 the other colors show through. 
 
 j = 259 //, /Jt. The color is 
 pure white i after which the 
 violet end begins to diminish 
 in intensity and the red end 
 to increase. 
 
 j = 300 /A, //.. Bright yel- 
 low is at a maximum and violet 
 nearly extinguished. Green 
 and red are weakened and to- 
 gether produce white, hence 
 the predominating color is 
 yellow. 
 
 A = 450 /Jt, //. The yellow 
 is still high, but the red rays 
 are at or near their maximum 
 and the other rays relatively- 
 weak; the color is therefore 
 orange. 
 
 A = 5 30 nn Violet and red 
 
 FIG. 240. 
 are about equally near a maximum, green is extinguished and 
 
 * The values of A for these in ///z (millionths of a millimeter) are /^(violet), 393.3 ; 
 /^(blue), 486.0; ^(green), 526.9; /^(yellow), 589.5; C(ved), 656.2; ^(red) 
 760.4. 
 
ii 4 CHARACTERS OF CRYSTALS. 
 
 blue and yellow weak, hence the strongly coloring red predomi- 
 nates. 
 SECOND ORDER COLORS. 
 
 J = 575 fj.fjL. This is 3/2 A for the brightest violet, A for bright- 
 est yellow and red and blue much weakened, hence the total im- 
 pression is violet, often called sensitive violet and used in testing be- 
 cause with very slight change in A it becomes either red or blue. 
 
 J = 589 yields indigo blue. 
 
 A = 664 up. This is 3/2 A for the brightest blue and is A far 
 orange red and near it for yellow, hence predominating tint is blue. 
 
 A = 800 fjLfi. This is 3/2 for bright green and near A for outer- 
 most red and 2 A for violet, hence the color is a mixture of green, 
 blue and yellow ; that is, green. 
 
 A =? 900 AV*. This is 3/2 A for yellow and 2 A for blue. Some 
 red and violet emerge with the yellow; that is, the prevailing color 
 is orange. 
 
 A io60fj. t a. This 2 A for green, 3/2 A for some red, 5/2 A for 
 indigo, and as red is the stronger color red predominates. 
 COLORS OF HIGHER ORDERS. 
 
 A = 1 1 30 pp yields sensitive violet No. 2. 
 
 With increasing values for A the latter becomes an approxi- 
 mately perfect multiple of 1/2 A or A for an increasing number of 
 wave-lengths and the colors resulting are less pure and brilliant, 
 for example : 
 
 A = 1 590 /JL/J.. This is 5/2 A for orange red, 7/2 A for indigo, 3 A for 
 green, 2 A for red, 4 A for violet. The resultant total effect is a red. 
 
 With the still higher values this is further noticeable and beyond 
 the fourth order the tints resulting are not to be distinguished 
 from white. Hence thick crystals show no interference colors. 
 
 Experiment. The quartz wedge with white light will show colors 
 in the order named, or the Federow mica wedge may be used, or 
 assuming the value of (n n) for special minerals and measur- 
 ing /, the colors in plates of different thicknesses may be com- 
 pared. 
 
 The colors of corresponding thicknesses of the mica wedge, in 
 which n\ n = .042 ///./. are much higher than for quartz with 
 ;/ x n = .oog/j.fj., one fifth the thickness producing the same retar- 
 dation in mica. 
 
 The thicknesses corresponding to an interference color of red of 
 first order,, a// = .009, for several common minerals are in millimeters 
 approximately. 
 
UNIVERSITY 
 
 OPTICAL CHARACTERS. 
 
 Calcite 0.003 
 Muscovite 0.013 
 Chrysolite 0.015 
 Barite 0.048 
 
 Quartz 0.060 
 Gypsum 0.06 1 
 Orthoclase 0.079 
 Apatite 0.124 
 
 WITH CONVERGENT LIGHT AND CROSSED NICOLS. 
 
 The bundles of parallel rays, p. 107, each produce interference 
 phenomena similar to those described, but since for oblique inci- 
 dence 
 
 A 
 
 n n 
 
 the values of J corresponding to odd or even multiples of y 2 A will 
 depend not only on the value (n 11) and the thickness, but upon 
 the angle of incidence. 
 
 In sections normal to the Optic Axis there will be a dark cross, the 
 arms of which intersect in the optic axis (centre of field) and re- 
 main parallel to the vibration planes of the nicols during rotation 
 of the stage. 
 
 With monochromatic light, if the section is not too thin,* the 
 
 optic axis will be surrounded by con- 
 centric circles alternately dark and 
 light, Fig. 241 . If the greatest value 
 of A is less than A no rings will show. 
 The distance apart of the dark 
 rings decreases as the thickness of 
 the section increases and as the dis- 
 tance from the centre increases, for 
 both causes merging sooner or later 
 into a uniform brightness. 
 
 With white light the rings be- 
 come color rings strictly in the order 
 of Newton's colors if the space per- 
 mits, but often overlapping and finally merging into essentially uni- 
 form tints. 
 
 * In the accidental orientation of rock sections there is rarely found a perfect basal 
 section and the thickness is frequently insufficient to produce rings. The uniaxial figure 
 is nevertheless to be recognized in not too oblique sections by one or both arms of the 
 straight armed cross which remain straight and parallel to the original position on rota- 
 tion of stage, whereas in biaxial not only do the arms curve into hyperbola, but revolve 
 in opposite directions to the rotation of stage. 
 
n6 CHARACTERS OF CRYSTALS. 
 
 The dark rings correspond to A = A, 2/, 3/1, etc., and since (p. 100) 
 the indices of the extraordinary ray are alike for all directions, 
 equally inclined to the optic axis, n ;z, must be constant for any 
 one value of i\ that is, these rings must be circles. 
 
 The cross results from the planes of vibration of the different 
 bundles being parallel and normal to different principal sections 
 through bundle and optic axis. For any one position of the 
 stage the vibration planes of certain bundles will be in the diagonal 
 positions and those of others will coincide with the planes of the 
 nicols ; that is, the light circles will grade from greatest brightness 
 in the diagonal positions to total darkness parallel to the nicols and 
 as the stage is rotated successive rays will come into these posi- 
 tions, maintaining the same effect. 
 
 In Sections oblique to the Optic Axis the curves must be 
 symmetrical to a principal section through the plate normal and 
 the optic axis. If the optic axis shows in the field it will change 
 its position with rotation of the stage, but the arms of the dark 
 cross will always remain parallel to the planes of the nicols, Fig. 
 242. One arm only may show. 
 
 FIG. 242. FIG. 243. 
 
 hi sections parallel to the optic axis the curves are symmetrical to 
 the principal section through the plate normal and optic axis and to 
 a plane at right angles to the optic axis and are conjugate hyper- 
 bolae, Fig. 243. They are usually only visible with monochro- 
 matic light. 
 WITH PARALLEL NICOLS. 
 
 The change from crossed to parallel nicols reverses all the 
 phenomena of interference, p. in. When J = /., 2A, 3/, etc., the 
 
OPTICAL CHARACTERS. 117 
 
 light is at its brightest and when J = /, ^A, |A, etc., the light is 
 extinguished. That is with the quartz wedge the thicknesses 
 which were dark bands in monochromatic light with crossed nicols 
 will be light and the light bands dark and with white light the 
 colors produced will be complementary to those obtained with 
 crossed nicols. 
 
 With convergent light all bright portions of the field are ex- 
 tinguished and all dark portions become bright. With white 
 light the complementary colors result. 
 
 If an interference color is passed through a prism the spectrum 
 will show dark bands in the positions of the extinguished colors 
 
 Tests with Parallel Light and Crossed Nicols. 
 
 The polarizing microscope, p. 107, is nearly always used in de- 
 termination, the cross hairs being made parallel to the previously 
 determined vibration planes of the nicols. 
 
 DETERMINATION OF PLANES OF VIBRATION OR " EXTINCTION." 
 
 In any section of a uniaxial crystal the plane of vibration of the 
 extraordinary ray passes through the optic axis and the corre- 
 sponding extinction direction is the projection of that axis on the 
 section ; the plane of vibration of the ordinary ray is at right angles 
 to this. 
 
 The positions of maximum darkness or extinction directions 
 are always either parallel or symmetrical to cleavage cracks and 
 crystal outlines. 
 
 A color contrast is more easily judged and may be obtained either 
 by inserting a test plate of quartz or gypsum in the slot always 
 provided between the nicols or by use of a special eye-piece such 
 as the Bertrand.* The section is placed to cover only part of the 
 field and for the extinction positions is of the same color as the 
 rest of the field, but for any other position there is a difference of 
 tint. 
 
 * Four quadrants which are equally thick basal sections of alternately right and 
 left handed quartz; the lines of contact take the place of the cross hairs. Before the 
 introduction of the crystal section the four quadrants are of the same color. 
 
 In the extinction positions the crystal section is colored like the rest of the field, 
 but the slightest divergence from this raises the color of the section in two diagonally 
 opposite quadrants and lowers the color in the other two. A special cap nicol must 
 be used instead of ordinary analyzer. 
 
1 1 8 CHARACTERS OF CR YSTALS. 
 
 VIBRATION DIRECTIONS OF FASTER AND SLOWER RAYS. 
 
 With the extinction (vibration) directions in diagonal position, a 
 test plate of some mineral, in which the vibration directions have 
 been distinguished and marked, is inserted between the nicols (in 
 a slot always provided) with these directions also diagonal. If the 
 interference color is thereby made higher, the vibration direc- 
 tions of the corresponding rays are parallel ; if the color is lowered, 
 the corresponding directions of vibration are crossed. 
 
 MICA TEST PLATE OR QUARTER UNDULATION MICA PLATE. A thin sheet of mica 
 on which is marked c, the vibration direction of the slower ray, which in mica is the 
 line joining the optic axes. The thickness chosen is usually that corresponding to a 
 blue gray interference color or say 140^ which is /l for a medium yellow. 
 
 GYPSUM TEST PLATE OR GYPSUM R,ED OF FIRST ORDER. A thin cleavage of 
 gypsum on which is usually marked a, the vibration direction of the faster ray. The 
 thickness chosen corresponds to an interference color of red of first order or say $6ou/Li, 
 which is essentially /I for a medium yellow. 
 
 QUARTZ WEDGE. A thin wedge of quartz, cut so that one face is exactly parallel 
 to the optic axis. The length of the wedge is parallel to the optic axis, and as quartz 
 is positive this direction is t, the vibration direction of the slower ray. 
 
 THE V-FEDEROW MICA WEDGE. Fifteen quarter undulation mica plates superposed 
 in equivalent position, but each about 2 mm. shorter than the one beneath it. 
 
 The mica plate raises or lowers the value of J by one quarter 
 wave-length, the gypsum, by about one wave-length and the two 
 wedges by amounts increasing with the distance inserted. 
 
 When sections showing only low colors of first order are tested 
 by the gypsum plate, there is practically considered the effect of 
 the plate on the red of the gypsum. 
 
 DETERMINATION OF THE RETARDATION* A. 
 
 The v-Federow mica wedge inserted with corresponding vibra- 
 tions directions crossed will for each interposed plate reduce A 
 
 A 
 
 by 140 fjL/j.. To render the field dark will require - - = ;/ plates. 
 
 1 40 
 
 Conversely n. 140 = /J, in which n is determined by count. 
 
 The quartz wedge similarly used will give an approximate value 
 by counting the number of times the original color reappears, if ;/ 
 times, then is the color a red, blue, green, etc., of n -f I order, for 
 which the value may be looked up in a chart. 
 
 The Babinet Compensator consists of two equal quartz wedges 
 
 * Difference in Retardation. 
 
OPTICAL CHARACTERS. 119 
 
 A and B, Fig. 244, so cut that the optic axis c of say A is parallel 
 
 to the length y and that of B is 
 parallel to the breadth z. Any 
 normally incident ray / will be 
 divided in B into a faster (ordi- 
 i ;, nary) ray vibrating parallel toj/ 
 
 and slower (extraordinary) vi- 
 brating parallel to z. But reach- 
 ing A the faster ray becomes the slower, and vice versa. 
 
 For the central position these will exactly balance, and with 
 either monochromatic or white light there will be here a dark band 
 with which a cross hair is made to coincide, and on each side, with 
 monochromatic light, there will be at equal distances from this other 
 parallel dark bands corresponding to A = A, 2A, 3/, etc. 
 
 Let A be made moveable by a micrometer screw and B be fixed 
 and denote the movement necessary to bring the second dark band 
 into coincidence with the cross-hair by d, then d corresponds to A of 
 the light used, and a movement of nd corresponds to a difference 
 of retardation of A. , 
 
 With the compensator in the zero position and diagonal to the 
 planes of the nicols, introduce a mineral section also in diagonal 
 position and determine the motion D necessary to bring back the 
 central band under the cross-hair, this change being due to the 
 
 mineral, measures the value of A in the section, that is, A = - in 
 
 
 
 wave-lengths, or A in millionths of millimeter // /*. 
 d 
 
 If on the scale used, 8 is unity, then D = A in wave-lengths. 
 
 DETERMINATION OF THE STRENGTH OF THE DOUBLE REFRACTION. If 
 the thickness of the section is known (n^ ;/) results from 
 A = t(n^ n). 
 DETERMINATION OF THICKNESS OF SECTION. 
 
 From formula A = /(;/ t n), t may be calculated if the value of 
 n^ n is known either for the crystal or for fragments of other 
 known minerals ground with the section. 
 
 Moderately thick sections (0.5 mm. and upwards) may be meas- 
 ured by fastening on a thin cover glass larger than the section with 
 thin balsam, cleaning away the balsam at the edges by alcohol 
 and focusing successively on dust upon the lower surface of the 
 cover and the upper surface of the glass. The difference in ele- 
 vation is measured by the micrometer focussing screw and is t. 
 
120 
 
 CHARACTERS OF CRYSTALS. 
 
 Measurements with a micrometer eye-piece, the section being set 
 on edge are sometimes possible. 
 APPROXIMATE DETERMINATION OF PRINCIPAL INDICES. 
 
 The method of the Due de Chaulnes* as improved by Sorbyf 
 depends upon the fact that the focal distance of a microscope is 
 altered when a plane-parallel plate is inserted between the objec- 
 tive and the focus. . Sorby focussed upon fine lines ruled on glass 
 and placed some distance below the objective. Using the lower nicol 
 only the indices corresponding to rays of definite vibration direction 
 were determined either by measuring the displacement due to the 
 unmounted section or successively those due to the glass alone 
 and glass plus section. The thickness of the section and the dis- 
 placement due to one revolution of the micrometer focussing screw 
 must be known. Then denoting the thickness by t and the dis- 
 placement by d 
 
 11, n - i 
 
 t d 
 in which if the outer medium is air n 1 = I. 
 
 Becke determines the relative indices of two substances in con- 
 tact in a section by focussing upon the dark boundary line and 
 raising the telescope tube, upon which the dark boundary appears 
 to move towards the substance with the higher index of refraction. 
 
 TESTS WITH CONVERGENT LIGHT. 
 DETERMINATION OF CHARACTER J OF RAY SURFACE. 
 
 The MICA TEST PLATE inserted diagonally above a section normal 
 
 FIG. 245. FIG. 246. 
 
 to the optic axis will destroy the black cross and break the rings 
 
 *Mem. de I Acad. Paris, 1767-68. zz^ \Mineral Mag. I., 193, II. I. 
 
 JWhen the direction of the optic axis is known the character may be determined 
 in parallel light by the extinction direction which is the projection of the optic axis. 
 If this correspond to the slower ray, p. 1 18, the crystal is positive, if to the faster ray the 
 crystal is negative. 
 
OPTICAL CHARACTERS. 121 
 
 into four quadrants, the relative effects in positive and negative 
 crystals being shown in Figs. 245, 246. The corresponding signs 
 _j_ an d are suggested by the relative positions of the dark flecks 
 and the arrow showing the direction t of the test plate. 
 
 After insertion of the test plate a vertical plane through t will 
 contain the vibration directions of the slower ray in the mica, of 
 the extraordinary rays in the quadrants passed through (first and 
 third) and of the ordinary rays in the other quadrants (second and 
 fourth). 
 
 In positive crystals the ordinary ray being the faster there will be 
 an increase of 1/4 / in the first and third quadrants and a decrease 
 of 1/4 A in the second and fourth. 
 
 At the centre J will now be 1/4 A and no longer dark. In the 
 first and third quadrants the distances between the rings will be 
 decreased about one-fourth by the increase of J by 1/4 L In the 
 second and fourth quadrants by the lessening of A the spaces be- 
 tween the rings will be increased and near the centre the portion 
 formerly bright with J=i/4 / will become J=o; that is, two new 
 dark flecks will be developed as in Fig. 245. 
 
 In negative crystals the extraordinary ray being the faster the 
 phenomena are exactly reversed. The centre is light, the rings 
 are narrowed in the second and fourth quadrants and widened in 
 the first and third and in these two dark flecks are developed near 
 the centre as in Fig. 246. 
 
 THE GYPSUM RED OF FIRST ORDER inserted with a parallel to the 
 arrow is particularly advantageous for very thin or feebly re- 
 fracting sections in which the rings are almost out of the field. 
 The centre will be red. In positive crystals the first and third 
 quadrants will lower the red to say yellow and in the second and 
 fourth near the centre will raise the red to blue. In negative 
 crystals the reverse will take place ; that is, the " blue quadrants " 
 correspond in position to the black flecks. This determination 
 must be made in white light. 
 
 BY SUPERPOSITION OF A BASAL SECTION OF A MINERAL OF KNOWN 
 SIGN. If the two sections are alike in character this will simply 
 act like a thickening of the plate. If unlike they will partially 
 neutralize each other. This test is little used. 
 
CHAPTER IX. 
 
 THE OPTICALLY UNIAXIAL CRYSTALS (Continued), 
 
 IV. OPTICALLY UNIAXIAL CRYSTALS IN WHICH THE OPTIC 
 AXIS IS A DIRECTION OF CIRCULAR POLARIZATION. 
 
 In certain hexagonal and tetragonal crystals monochromatic 
 plane polarized light transmitted in the direction of the crystal 
 axis c (optic axis) is not extinguished by crossed nicols, but is ex- 
 tinguished after a definite rotation of the analyzer. 
 
 PLANE, CIRCULAR AND ELLIPTICAL POLARIZATION. 
 
 If light with vibrations in one plane is decomposed into light 
 with vibrations in two planes at right angles, any ether particle in 
 the path of two interfering rays, p. 106, will be acted upon by two 
 vibrations which will be at right angles, but, generally speaking, 
 these will be neither of the same phase nor of the same intensity. 
 The effects are analogous to those produced upon a swinging 
 pendulum by a second impulse at right angles to the first. 
 
 i. In general the resultant motion is elliptical and the light is ellip- 
 tically polarized. For instance, the light emerging from a doubly 
 refracting plate in a polariscope is in general elliptically polarized, 
 
 2. If the phase difference is a half 
 vibration (J/ 2 A), or an entire vibration, 
 the resulting motion is in a straight 
 line and the light is plane polarized. 
 For example, Fig. 247, if AU is the 
 original direction, OD and OC, the 
 components at right angles y 2 A apart, 
 the resultant motion will be El equal 
 AU, but not in the same direction. 
 
 3. If the vibrations are of equal 
 amplitude, but possess a phase differ- 
 ence of y A, that is, if one vibration is 
 at the center of its swing when the other reverses its direction, the 
 resultant motion is circular and the light is circularly polarized. 
 For example, yellow rays emerge from a quarter undulation mica 
 plate in the polariscope circularly polarized. 
 
 FIG. 247. 
 
OPTICAL CHARACTERS. 
 
 123 
 
 In the Fresnel rhomb, Fig. 248, the index of refraction of the 
 glass and the angles of the rhomb are so related that a plane 
 polarized ray S, normally incident, experiences a total reflection 
 at E and another at U at such angles that at each reflection 
 there is developed a phase difference of y& A |between the com- 
 ponent vibrating in the plane of reflection and that vibrating in a 
 plane at right angles thereto. That is, the vibration of any 
 particle in the emerging ray T is the result of two component vi- 
 brations at right angles with a phase 
 difference of ^ A, but not in general of 
 equal intensity. The ray is therefore 
 elliptically polarized and if examined by 
 a nicol will show variations in inten- 
 sity, but never complete extinction. 
 
 If the plane of vibration of the in- 
 cident light is at 45 to the plane of 
 reflection of the rhomb, the component 
 vibrations will be equal in intensity, 
 the emerging light will be circularly 
 polarized, and the analyzing nicol will 
 show no variation in intensity. 
 
 If two rhombs are used, the four successive total reflections de- 
 velope a phase difference of y 2 A, that is, the impulses are in the 
 'Same straight line, the emerging light is plane polarized and is 
 extinguished whenever its plane of vibration coincides with that 
 of the long diagonal of the nicol. 
 
 ROTATION OF THE PLANE OF POLARIZATION. 
 
 Two^cfrcular vibrations in opposite directions in the same plane 
 are equivalent to a straight lined vibration of double amplitude 
 and vice versa. A particle at A, Fig. 249, simultaneously sub- 
 jected to two motions tending to carry it in any time /, at a uni- 
 form angular velocity through the equal arcs AR and AL would 
 evidently in this time travel along the path AU = 2AS, for the 
 components SL and SR at right angles to AS are equal and oppo- 
 site and of no effect. For the arcs ART and ALT, the distance 
 travelled would be AN ' = 2AT. Conversely, the vibration AN is 
 equivalent to the two circular motions ART and ALT. 
 
 If a plane polarized ray be decomposed by any substance into 
 two rays circularly polarized in opposite directions, these rays, in 
 
 * Due to difference in velocity and length of path. 
 
124 
 
 CHARACTERS OF CRYSTALS. 
 
 traversing a plate of given thickness, will develope a phase differ- 
 ence, * and on emergence will combine to a plane polarized ray. 
 
 The planes of vibration (or of polarization, see p. 100) of the in- 
 cident and emerging rays will not be parallel. 
 
 Let AN, Fig. 250, represent the vibration of the incident ray, 
 R and L the advance points of the emerging rays at the moment 
 
 of emergence, then will DF be the vibration of the emerging ray, 
 D being midway between R and L, for only for this direction will 
 the components SL and SR neutralize each other. 
 
 The rotation for a thickness of I mm. is usually denoted by a and 
 for any other thickness /, by / . The value of in terms of the wave 
 length and the indices of refraction of the two circularly polarized 
 
 ray s* 
 
 s a= 
 
 THE OPTIC Axis A DIRECTION OF DOUBLE REFRACTION. 
 
 In this division of uniaxial crystals, any incident plane-polarized 
 ray transmitted in the direction of the optic axis, is converted into two 
 diverging rays circularly polarized in the opposite sense (direction). 
 
 From the formula above given, the divergence n"n r may be 
 calculated, or, even when minute as in quartz, it may be experi- 
 mentally shown by the method of Fresnel with three prisms of 
 right and left quartz cut as in Fig. 251. 
 
 In A P D the right-handed ray is the faster, reaching PA Cit 
 is the slower, hence is bent towards the normal while the left-handed 
 
 *Th. Liebisch. Grundriss dcr Fhys. Kryst. 1896, p. 124. 
 
OPTICAL CHARACTERS. 
 
 125 
 
 ray is bent away from the normal. Reaching ACE the right is 
 again the faster and is bent from the normal while the left is bent 
 towards it, and finally on emerging both are bent from the normal : 
 all changes acting to increase the divergence until with PA C= 
 152 a divergence of 4 is reached which can be measured by a 
 goniometer. 
 
 C P 
 
 It can be shown by a nicol that these emerging rays are circu- 
 larly polarized and in the opposite sense. 
 
 RAY SURFACE. 
 
 The ray surface is closely that of the preceding division, but with 
 the ellipsoid and sphere not quite in contact at the optic axis, one 
 of them a little flattened, the other a little drawn out.* 
 
 It is probable that there is a grading from plane polarized light 
 transmitted normal to the optic axis, through elliptically polarized 
 light, the eccentricity of the ellipse decreasing with the angle be- 
 tween the transmission direction and the optic axis, to circularly 
 polarized light in the direction of the optic axis. 
 
 The optic axis remains an axis of isotropy, every diameter perpen- 
 dicular to it is an axis of binary symmetry, but there are no planes of 
 symmetry because the direction of rotation in corresponding shells 
 at the extremities of the optic axis are reversed. t 
 
 The division is therefore limited to crystals in which the crystal 
 faces are not symmetrical either to any plane or to the centre. A 
 further condition seems to be that the right and left forms shall be 
 " enantiomorphs " (the one can not by rotation about any axis be 
 brought into coincidence with the other). For example, Group 17 
 has no planes of symmetry, but the forms are not enantiomorphs 
 and circular polarization does not occur. 
 
 * Lang with quartz shows the velocity of the ordinary rays near the axis is not quite 
 constant and that the extraordinary does not follow exactly the Huyghens law of com- 
 mon uniaxial crystals. 
 
 f The sense of rotation is not reversed on inverting the plate. 
 
126 CHARACTERS OF CRYSTALS. 
 
 The classes in which circular polarization is possible are : 
 TETRAGONAL SYSTEM. Class 9, to which no crystals have yet 
 been referred. Class 10, circular polarization is not proved in any 
 crystal of this class. Class 12, examples: strychnine sulphate,, 
 guanadin carbonate, diacetylphenol-thalein. HEXAGONAL : (R.HOM- 
 BOHEDRAL DIVISION). Class 1 6, example: sodium periodate, Na 
 IO 4 . 3H 2 O. Class 1 8, examples: quartz, cinnabar, matico cam- 
 phor C 12 H 20 O, tartrates of rubidium and caesium, subsulphates of 
 lead, potassium, calcium and strontium ; (HEXAGONAL DIVISION). 
 Class 23, example: potassium-lithium sulphate, K Li SO 4 . Class 
 24, circular polarization is not proved in any crystal of this class. 
 
 The direction of rotation corresponds to the right and left of 
 the enantiomorphs.* . With one or two exceptions,^. ., sulphate of 
 strychnine, circular polarization is destroyed by dissolving, fusing 
 or vaporizing the crystal. Certain substances, however, showing 
 in crystalline condition no power to rotate the plane of polariza- 
 tion show it in solution or in vapor, and are called optically active. 
 These substances when in crystals are enantiomorphic right or left 
 as the solution is right-handed or left-handed. If acids are optic- 
 ally active their salts usually are so.f 
 
 WITH PARALLEL MONOCHROMATIC LIGHT AND CROSSED NICOLS. 
 
 In sections normal to the Optic Axis the field will be illuminated,, 
 though only feebly in thin sections, and will only become dark when 
 the analyzer has been turned an amount equal to the rotation 
 
 That is, an amount increasing with the thickness and double re- 
 fraction in the direction of the optic axis and decreasing with the 
 wave-length. Approximately 
 
 A B 
 
 * Herschel in Trans. Cambridge Soc. I, 43, showed that circular polarization in 
 quartz was apparently produced by the same cause which determined the position of 
 the so-called plagihedral faces, for a plate of quartz cut perpendicular to the vertical 
 axis of a crystal in which the left plagihedral faces occur will turn the plane of polar- 
 ization of the incident ray to the left and a similar plate from a crystal showing the 
 right plagihedral faces will turn this plane to the right. 
 
 f See Pasteur's lectures before Societe chimique de Paris 1860, published as No. 28, 
 Ostwald's Klassiker der Exacten Wissenschaften. Ueber die Asymmetric. Leipzig,. 
 1891. 
 
OPTICAL CHARACTERS. 
 
 127 
 
 In all other sections there is extinction every 90 and brightest 
 illumination in the diagonal positions. Extinction also takes place 
 
 for = i, 2, 3, etc., as described p. in. 
 
 WITH PARALLEL WHITE LIGHT AND CROSSED NICOLS. 
 
 In sections normal to the optic axis, since the rotation increases as 
 the wave-length decreases, the different colors will be dispersed 
 and emerge vibrating in different planes. As the analyzer is 
 turned its plane of vibration will be successively at right angles to the 
 planes of vibration of the different colors. That is, at any time 
 only one color can be shut out completely and the rest in varying 
 degree. The color with vibrations most nearly parallel to the 
 vibration planes of the analyzer will be least shut out and will de- 
 termine the predominating tint in the color seen. 
 
 In all other sections there will be extinction every 90 and great- 
 est brightness in diagonal positions and the color tint due to inter- 
 ference as described p. 113. 
 
 WITH CONVERGENT LIGHT AND CROSSED NICOLS. 
 
 In sections normal to the optic axis, with monochromatic light, thin 
 sections will show the dark cross 
 the arms of which will always be 
 parallel to the vibration planes of 
 the nicols and somewhat indis- 
 tinct near the centre. In thicker 
 sections the bars will not reach 
 the centre, Fig. 252, and no dark 
 rings will form there, the first ring 
 being some distance out, no mat- 
 ter how thick the section may be, 
 beyond this there will appear the 
 alternate dark and light rings 
 with radii nearly* as in the last 
 division. 
 
 In general the inner circle is bright except when the thickness for 
 the light used gives la. = 180 x , in which n = I, 2, 3, etc., when 
 it is black. With white light the central circle has the color tint 
 
 * Exact measurements show that the radii of the circles near the centre do not obey 
 the law of common uniaxial crystals. J. C. McConnell, Phil. Trans. Roy. Soc, 1886, 
 177, 299. 1887. 
 
 FIG. 252. 
 
128 
 
 CHARACTERS OF CRYSTALS. 
 
 which is obtained over the entire field when parallel light is used. 
 On turning the analyzer each ring is decomposed into four arcs, 
 which widen when the direction is that proper to the section and 
 contract when it is the opposite. 
 
 In sections oblique to the axis essentially as described p. 1 16. 
 
 In sections parallel to the optic axis essentially the hyperbolic 
 curves of Fig. 243. 
 
 Determination of Optical Characters. 
 
 Most of the determinations are exactly as described under the 
 preceding division. References may therefore be made as follows; 
 Direct determination of principal indices, p. 103 ; Extinction direc- 
 tions, p. 117; Faster and slower ray, p. 118; Retardation, p. 118 
 Strength of double refraction, p. 119. 
 
 DETERMINATION OF DIRECTION OF ROTATION OF PLANE OF VIBRA- 
 TION. 
 In a section normal to the optic axis and with crossed nicols : 
 
 (a) By the widening of the arcs when the analyzer is turned in 
 the direction of rotation. 
 
 (b) A quarter undulation mica plate inserted diagonally above 
 
 FIG. 253. 
 
 FIG. 254. 
 
 the plate converts the rings and cross into two intervvound spirals, 
 which start near the centre and wind in the direction of rotation,* 
 Fig. 253, Right; Fig. 254, Left. If the mica is placed below the 
 plate the spirals are reversed. 
 
 * A plate of left and another of right-handed substance give a fourfold spiral in 
 which the direction of winding conforms to the lower plate. 
 
OPTICAL CHARACTERS. 
 
 129 
 
 (c) The direction in which it is necessary to turn the analyzer to 
 change the tint of passage (see below) into red is the direction 
 of rotation. 
 
 DETERMINATION OF ANGLE OF ROTATION OF PLANE OF VIBRATION. 
 
 In a section normal to the optic axis and with crossed nicols : 
 
 (a) With monochromatic light the rotation of the analyzer neces- 
 sary to produce darkness is la. 
 
 (b) With white light and Tint of Passage or Sensitive Tint. When 
 the analyzer is at right angles to the brilliantly coloring yellow 
 green, for which A = 550 /AM in air, the remaining colors unite to an 
 
 easily recognized grayish violet, 
 called the tint of passage or sensi- 
 tive tint, because the slightest fur- 
 ther rotation in the one direction 
 uer "' ed gives red and in the other blue. 
 g Starting with the polarizer and an- 
 yell alyzer parallel, the angle turned by 
 the analyzer to produce the tint is 
 la for A=55o/Jt/Jt (middle yellow). 
 Fig. 255 shows the rotations pro- 
 duced by a quartz plate of 3.75 mm. 
 thickness. With parallel nicols the 
 vibration plane of yellow is then at 
 
 right angles to PS, the vibration plane of the analyzer, that is yellow 
 is extinguished and the tint of passage results. With crossed nicols 
 the tint of passage results when the vibration plane for yellow is 
 PS, that is, for a rotation of 180 or a thickness of 7.50 mm. 
 
 (c) With white light and spectrum. 
 
 The light polarized in P, Fig. 256, passes through the substance 
 vS, the collimator C, the analyzer A and through a slit and is de- 
 composed by the prism R and viewed by the telescope T. As the 
 analyzer is turned 'a dark line corresponding to the extinguished 
 ray will move along the spectrum and the angles corresponding to 
 the rotations (from crossed position) which bring this line into co- 
 incidence with the Frauenhofer lines will correspond to la. for the 
 corresponding values of A. 
 
 If a quartz plate is placed at 0, the light passes through the 
 quartz, analyzer, slit and decomposing prism and yields a spectrum 
 
 FJG. 255. 
 
130 CHARACTERS OF CRYSTALS. 
 
 with a dark line. The analyzer is turned until the dark line coin- 
 
 FIG. 256. 
 
 tides with a Frauenhbfer line; then the substance is introduced and 
 the rotation necessary to bring the two lines again into coinci- 
 dence is the value of / for that color. 
 
 If a biquartz,* Fig. 257, of 7.50 mm. thick- 
 ness be used, with crossed nicols, or one of 3.75 
 mm. with parallel nicols, a black line will appear 
 for yellow (A = 550 P.;J.\ which for slight rotations 
 of the analyzer will move into the blue or the red. 
 With the introduction of the substance at 5 the 
 black line is disturbed and the rotation necessary 
 F G - 2 57- to re-establish it is the value of la for yellow. 
 
 ABSORPTION AND PLEOCHROISM. 
 
 In uniaxial crystals with monochromatic light the two rays cor- 
 responding to any direction of transmission are in general absorbed 
 at a different rate. The faster ray may be either more absorbed or 
 less absorbed than the slower, the absorption of the ordinary ray 
 being independent of the direction of transmission, while that of 
 the extraordinary varies with the inclination to the optic axis, but 
 is constant for the same angle and differs most from that of the 
 ordinary for transmission normal to the optic axis. 
 
 With white light the color seen in any direction will be due to 
 a combination of the ordinary and extraordinary rays, and this 
 combination will give different tints in different directions. 
 
 Usually the two rays are viewed separately by means of a polar- 
 izing microscope with the analyzer out. The color varies as the 
 
 *Two semicircular basal plates of right and left quartz, of equal thickness 
 (Biot's 3.75 mm., Soleil's 7.50 mm.) and with their twinning plane either parallel or at 
 right angles to the vibration plane of the polarizer. 
 
OPTICAL CHARACTERS. 131 
 
 stage is turned and the maximum color differences are obtained 
 when the extinction directions coincide with the vibration plane 
 of the polarizer (shorter diagonal). 
 
 In sections normal to the optic axis the color is constant for all 
 positions of the stage. 
 
 In sections parallel to the optic axis. If the extinction direction 
 parallel to the optic axis is placed first parallel then at right 
 angles to the short diagonal there will be seen first the extraordi- 
 nary, second the ordinary. 
 
 In all other sections the extraordinary will be found to approach 
 the constant tint of the ordinary as the sections become more 
 nearly perpendicular to the optic axis. 
 
 The colors of ordinary and extraordinary may be contrasted 
 side by side by means of a " dichroscope" consisting, Fig. 258, of 
 
 f 
 
 ' 
 
 e b d 
 
 FIG. 258. 
 
 a rhomb of calcite in which ab and cd are the short diagonals of 
 opposite faces. To these faces glass wedges aeb, dfc are cemented 
 and the whole encased. The section is placed at P and the light 
 from the substance passes through a rectangular orifice, a double 
 image of which is seen by the eye at E. 
 
 When the planes of vibration of the section and the calcite coin- 
 cide the two rays originated in the section will have the maximum 
 color difference of the section. 
 
 If the plate is turned around the axis of the dichroscope each 
 ray from the crystal is decomposed in the calcite and contributes 
 a portion of its intensity to each of the two images and at 45 each 
 will contribute half of its brightness to each image ; that is, the 
 the two images will then be alike in color and brightness. 
 
 Basal sections of dark tourmaline or of magnesium platinocyan- 
 ide show phenomena analogous to the absorption tufts described 
 under biaxial absorption. 
 
CHAPTER X. 
 
 THE OPTICALLY BIAXIAL CRYSTALS. 
 
 In any orthorhombic, monoclinic or triclinic crystal the ray sur- 
 face for light of any definite wave-length is symmetrical to three 
 planes at right angles to each other which intersect in the vibration 
 directions of the fastest and slowest rays, the third intersection 
 being the vibration direction of a ray of some intermediate velocity. 
 
 The planes of symmetry are called Optical principal sections and 
 their intersections are called Principal vibration directions.* No 
 diameter is an axis of isotropy, that is a direction about which the 
 crystals are optically equivalent, but in two directions there is sin- 
 gle refraction and these directions from analogy with the uniaxial 
 are called the " Optic axes ''for light of that wave-length. 
 
 As in the uniaxial crystals the relative values of the principal in- 
 dices of refraction, that is, the indices of the rays with the principal 
 vibration directions, change with the wave-length. But as a 
 consequence the directions of the optic axes change also, 
 the amount of change or so called " dispersion " varying from a 
 few minutes to over forty degrees. That is, the optic axes in this 
 group of crystals are not fixed in position as in the uniaxial. 
 
 With decreasing geometric symmetry not even the optical 
 principal sections or principal vibration directions remain fixed in 
 direction, but alter with the change of light. Upon this so-called 
 " Dispersion of the Bisectrices " are based optical distinctions be- 
 tween the systems. 
 
 In Orthorhombic Crystals the geometric axes are always princi- 
 pal vibration directions and the axial planes are optical principal 
 sections for all colors. 
 
 In Monoclinic Crystals the ortho axis b is always a principal vi- 
 bration direction and the clinopinacoid oio is an optical principal 
 section for all colors, the other two principal vibration directions 
 lie in oio, but vary in position with the wave-length. 
 
 *Often called axes of elasticity. 
 
OPTICAL CHARACTERS. 
 
 133 
 
 In Triclinic Crystals there are no principal vibration directions or 
 principal sections which are constant for all colors. 
 
 THE OPTICAL INDICATRIX. 
 
 The inductive method by which the present conception of a biax- 
 ial ray surface has been reached may be briefly outlined as follows : 
 
 It was shown by Huyghens that the ray surface of calcite was a 
 double surface composed of a sphere and an ellipsoid of revolu- 
 tion, and the ray surfaces of other tetragonal and hexagonal sub- 
 stances were found to be similar and symmetrical always to the 
 crystallographic planes of symmetry. It was later shown that 
 from an ellipsoid of revolution all the relations between direction 
 of transmission, velocity and vibration direction could be deduced. 
 
 By analogy it was conceived that for any less symmetrical crys- 
 tal some correspondingly less symmetrical figure or Optical Indi- 
 catrix existed from which these relations and the shape of the ray 
 surface could be geometrically determined. Experiment has 
 proved this to be true, and that for orthorhombic, monoclinic and 
 triclinic crystals the optical indicatrix is an ellipsoid the three rec- 
 tangular axes of which are in length proportionate to the so-called 
 principal indices of refraction. 
 
 Let a < /? < Y be these principal indices of refraction and a, b and 
 c the directions of vibration of the corresponding rays, that is, of 
 the rays with greatest, intermediate and 
 least velocity of transmission. Upon the 
 directions a, b, c lay ofif <9a = , Ob = ft, 
 and <9c = y and upon these as axes con- 
 struct an ellipsoid that is the Optical In- 
 dicatrix, Fig. 259. Then, by analogy 
 with the uniaxial, p. 102, for any diam- 
 eter considered as a direction of trans- 
 mission there are two points of the sur- 
 face the normals from which are also 
 normal to that diameter. These normals 
 are at once the directions of vibration and 
 the reciprocals of the velocities of the rays 
 transmitted in the direction of the diameter-. 
 
 RAY SURFACE. 
 
 The shape of the ray surface may be judged from the shape of 
 the optical principal sections ab, cb and ac. 
 
134 
 
 CHARACTERS OF CRYSTALS. 
 
 Optical Principal Section ab. For the direction aO t Fig. 259, the two 
 
 normals are b(9 and c(9 corresponding to velocities- and - ; for the 
 
 A r 
 
 direction bO the normals are aO 
 and c<9, corresponding to velocities 
 
 and -, for any other direction 
 a r 
 
 one normal is c<9, the other a line 
 in ab, found as described on p. 102 
 and varying in length between aO 
 and W, according to the direction 
 of the ray, like the radius vectors 
 of an ellipse. Hence for any di- 
 rection of transmission one ray vi- 
 brates in some direction in ab, but 
 is transmitted with a velocity vary- 
 ing like the radius vectors of an 
 
 ellipse between - in the direction b to - in the direction a, Fig. 
 
 260, and corresponds to an extraordinary ray. 
 
 The other ray for any direction vibrates in the direction of c, nor- 
 mal to ab and is transmitted with constant velocity-. It corresponds, 
 therefore, to an ordinary ray. 
 Hence for ab the sections of the 
 two shells are as in Fig. 260. 
 
 Optical Principal Section, cb. 
 In precisely the same manner it 
 is shown that for any direction 
 of transmission in the plane cb 
 one ray is extraordinary in that 
 its vibration direction is always 
 b and its velocity varies between 
 
 , in the direction b, to I in the 
 
 r 
 
 direction c ; whereas the other 
 
 ray is ordinary with a constant 
 
 vibration direction a normal to 
 
 FIG. 260. 
 
 I 
 a 
 
 FIG. 261. 
 
 cb and a constant transmission velocity . That is for cb the sec- 
 
 a 
 
 tions of the two shells are as in Fig. 261. 
 
OPTICAL CHARACTERS. 
 
 135 
 
 Optical Principal Section ac or Plane of tJie Optic Axes. In this 
 case for any direction of transmission one ray is extraordinary 
 with its vibration direction in ac and velocity varied between 
 
 - in the direction a and - in the direction c, while for the ordinary 
 ray the constant vibration direction is & and the constant velocity 
 -* That is, in this section of the two shells, there must be four 
 
 symmetrically placed points of in- 
 tersection , as shown in Fig. 262. 
 Each principal section then con- 
 sists of an ellipse and a circle. The 
 entire surface may be generated as 
 follows. 
 
 Revolve the section ac about one 
 of its axes a, the curves of the sec- 
 tion touching always those in the 
 section ab and altering according 
 to a definite law until after 90 
 rotation they coincide with the 
 curves cf the section . 
 OPTIC AXES.* 
 
 In Section ac of the indicatrix, 
 Fig- 2 59 there must be a radius 
 
 vector OL equal /?, for these vary between a. and Y Similarly 
 OLJ (not shown) symmetrically opposite OL will equal 'ft. The 
 sections of the indicatrix through OL Ok or OL' Ob must be circles. 
 In section ac of the ray surface, Fig. 262, near but not at the 
 points E, common tangent planes can be drawn to each shell. 
 
 The directions normal to these circular sections in the indicatrix 
 and to the common tangent planes in the ray surface are the so- 
 called optic axes A A of Fig. 262. 
 
 A circular section of the indicatrix is the common ray front of all 
 the diameters determined, p. 133, by the normals to the indicatrix 
 from each point of the circle. Of these normals only one, Ob, is m 
 the circle, hence only one diameter is normal to the circle (the optic 
 
 * The directions EE are variously called Ray Axes, Secondary Axes, and Bira- 
 dials and are simply directions of equal ray velocity, the ray, however, corresponding 
 to an infinite number of planes tangent at , the normals to which diverge and form 
 a cone (outer conical refraction). 
 
 FIG. 262. 
 
136 
 
 CHARACTERS OF CRYSTALS. 
 
 axis), while the others starting from diverge and form a cone. 
 Furthermore since the normals are all in different directions the 
 rays, transmitted in the directions of the diameters; have all di- 
 rections of vibration. The same facts are evident from Fig. 262 ; 
 the common tangent plane caps the conical depression at E. 
 From each point of the circle of contact the lines to represent 
 rays with a common ray front and of which the front normal is 
 the optic axis. That is, each optic axis is the front normal for 
 a diverging cone of rays (inner conical refraction) which on emer- 
 gence are parallel to this front normal but each of which has a dif- 
 ferent direction of vibration. 
 
 The velocity in the direction of an optic axis is evidently -^- 
 
 C/b p 
 
 POSITIVE AND NEGATIVE RAY SURFACES. 
 
 In the plane of the optic axes a and c bisect the angles between 
 these axes. If the acute bisectrix is c, (Bx a = c), that is, if bisects 
 the acute angle between the optic axes the ray surface is said to be 
 positive. 
 
 If the acute bisectrix is a, (Bx a = a), that is, if a is the acute bisec- 
 trix and c the obtuse, the ray surface is said to be negative. This 
 conforms strictly to the usage in the uniaxial which is a special 
 case of biaxial with the angle between the optic axes zero, that is 
 with one optic axis in the direction c. 
 
 REFRACTION IN BIAXIAL CRYSTALS. 
 
 In general, following the Huyghens construction, p. 86, the new 
 
 ray fronts through the point E, Fig. 
 208, are tangent to the shells in points 
 which do not lie in the plane of inci- 
 dence, that is, both rays are bent out 
 of the plane or are extraordinary. 
 With normal incidence this is still 
 true. Let SURT, Fig. 263, be the 
 section of the indicatrix normal to 
 the ray. The planes ROf and TOf 
 through the axes of the ellipse and the 
 front normal contain the two rays 
 
 Oe and Ot at right angles to the surface normals RE and UN. 
 
 These normals are the vibration direction of the rays and are not 
 
 FIG. 263. 
 
OPTICAL CHARACTERS. 
 
 137 
 
 at right angles to each other, though on emergence both rays retake 
 
 the direction Of with vibrations at right angles and a phase difference. 
 Because SfJRTis normal to the incident ray it is parallel to the 
 
 plate, hence the extinction directions in the plate are OR and 
 
 OU the projections of the vibration directions. 
 
 If the plane of incidence be a principal sec- 
 tion then the points of tangency of the new 
 ray fronts will be in the plane. One will for 
 all directions show a constant refraction in- 
 dex and may be called the ordinary. 
 
 WITH PARALLEL MONOCHROMATIC LIGHT AND 
 CROSSED NICOLS. 
 
 In sections normal to an optic axis the field 
 will maintain a uniform brightness during 
 rotation of the stage, because p. 136, in the 
 emerging cylinder of rays developed by in- 
 ner conical refraction each ray vibrates in a different plane. 
 
 In all other sections the field is dark every 90 and is illuminated 
 for all other positions and most brilliantly in the diagonal posi- 
 tions. The field is also dark throughout an entire revolution for 
 
 - = 1,2, 3, etc., p. 
 
 in, and brightest for = 
 
 f, ^ etc -> m 
 
 conforming to similar uniaxial sections. 
 
 Relation between Extinction Directions and Optic Axes. As just 
 explained, the extinction directions are the diameters JTand HH, 
 Fig. 264, of that central section of the indicatrix which is parallel 
 to the plate. The two circular sections must intersect any central 
 section in equal diameters tt which from the nature of an ellipse 
 must be symmetrical to the axis of the ellipse, that is, to the ex- 
 tinction directions. 
 
 If planes be passed through the plate normal and each optic axis 
 their traces ee will be at 90 to those of the circular sections, hence 
 also symmetrical to the extinction directions. That is, the extinc- 
 tion directions must* bisect the angle between the traces of the two 
 planes, each through the plate normal and an optic axis. 
 
 WITH PARALLEL WHITE LIGHT AND CROSSED NICOLS. 
 
 In sections normal to an optic axis for light of one color there is 
 
 *Upon this principal and after a device by Prof. Groth, B5hm and Wiedemann, of 
 Munich, construct a model for graphic determination of extinction directions. 
 
138 
 
 CHARACTERS OF CRYSTALS. 
 
 constant illumination throughout rotation because of internal con- 
 ical refraction p. 1 36 ; the optic axes for other colors are, however, 
 more or less oblique to the section and double refraction takes 
 place, and as the resulting vibration directions are somewhat dif- 
 ferently oriented for each color, color tints result somewhat as in 
 circular polarization. 
 
 In all other sections. If there is no marked dispersion of the 
 principal vibration directions there will be approximately perfect 
 extinction every 90 and interference colors as in the uniaxial, but 
 with marked dispersion the vibration directions in any plate di- 
 verge for the different colors and the plate can never be perfectly 
 dark and moreover in any position the interference colors due to 
 phase difference are modified by color tints due to the partial or 
 complete extinction of certain colors. 
 
 WITH CONVERGENT MONOCHROMATIC LIGHT AND CROSSED NICOLS. 
 
 In sections normal to a or c there will be darkness whenever the 
 vibration planes of the emerging rays coincide with those of the 
 
 nicols and whenever -- is a whole number; see p. in. The 
 
 A 
 
 points of emergency of the optic axis will therefore be dark, since 
 = o, and the points corresponding to ' = I will together form 
 
 A A 
 
 a ring around each axis and similarly for values of 2, 3, 4, etc., 
 until the pair corresponding most nearly to the value for the cen- 
 tre of the field* unite at or 
 near the centre to a cross loop 
 or figure eight around both 
 axes and subsequent rings 
 form lemniscates around this 
 
 as in Fig. 265. If for the 
 
 FIG. 265. 
 
 centre is less than unity even 
 the first ring must surround both axes, giving them a figure more 
 
 *For the rays transmitted normally at the center of the field = (n, ). If 
 
 A A 
 
 the direction of transmission is c them # x n= y,3 and if direction is a then n l n 
 ft a. In either case is known and determines the number of rings between 
 the axes and the centre. 
 
OPTICAL CHARACTERS. 
 
 139 
 
 FIG. 266. 
 
 similar to that of a uniaxial crystal. As all parts of the lemniscates 
 are independent of positions of planes of vibration, there will be 
 no change during rotation of the stage. 
 
 When the stage is rotated so that principal sections are parallel 
 to the vibration planes of the nicols all the rays transmitted in 
 those sections (therefore vibrating 
 in or at right angles to those sec- 
 tions) will be extinguished and 
 there will be a sharp dark band 
 joining the poles and another some- 
 what thicker lighter band at right 
 angles to the first and midway be- 
 tween the poles. These are often 
 called brushes. For any other po- 
 sition of the stage the vibrations of 
 some other rays are parallel to those 
 of the nicols, and these are always 
 so distributed that the resultant 
 dark spots form the branches of an 
 hyperbola through each of the axes, 
 effect is as in Fig. 266. 
 
 As the stage is rotated the straight bars appear to dissolve into 
 the hyperbola, the branches of which appear to rotate in the oppo- 
 site direction to the stage, the convex side always toward the other 
 
 axis. If is nowhere equal to I there will be no dark rings, but 
 
 A 
 
 the hyperbola will appear as before. Because the two principal 
 sections are planes of symmetry the curves and brushes are at 
 all times symmetrical to the traces of the sections. 
 
 In sections normal to S the dark curves due to = I, 2, 3* etc., 
 
 are hyperbolae similar to those shown, Fig. 243. No brushes are 
 visible. 
 
 /// sections normal to an optic axis there will be the same forma- 
 
 j 
 tion of dark rays corresponding to whole numbers lor .the 
 
 A 
 
 curves, however, will be more nearly circles, Fig. 267. There will 
 be a straight black bar bisecting the curves whenever the trace of 
 the plane of the optic axes coincides with the vibration direction 
 of either nicol, and for all other angles of rotation there will be 
 
 At the diagonal position the 
 
140 
 
 CHARACTERS OF CRYSTALS. 
 
 one arm of an hyperbola through the axis, Fig. 268, the convex 
 side towards the other axis. This arm will rotate in the opposite 
 direction to the stage. 
 
 FIG 267. 
 
 FIG. 268. 
 
 WITH CONVERGENT WHITE LIGHT AND CROSSED NICOLS. 
 
 In sections normal to the acute bisectrix Because of the dispersion 
 of the optic axes described, p. 130, the centres of the ring systems 
 are not the same for the different colors. If there is also a dis- 
 persion of the bisectrices the section can only be exactly normal 
 to the bisectrix for one color. These facts, added to the general 
 one that for different colors the greater the wave-length the greater 
 the distance apart of the rings, render it possible in white light to 
 determine the crystalline system by the distribution of the colors, 
 thus obtaining Divisions V, VI and VII. 
 
 V. ORTHORHOMBIC CRYSTALS. 
 
 The color distribution must be symmetrical to the line joining 
 the optic axes and to the line through the centre at right angles 
 thereto, because these are the traces of the principal sections com- 
 mon to the ray surfaces of all the colors. Assuming that the same 
 principal section is the axial plane for all colors there may be two 
 cases : i. The axes for the longer wave-lengths may be most dis- 
 persed, that is, the angle between the optic axes for red greater 
 than that for violet, or p > o. 2. The axes for the shorter wave- 
 lengths may be most dispersed, that is, the angle between the optic 
 axes for red less than that for violet, or p < o. 
 
 Figure 269 illustrates the second class. In the interference 
 figure for the diagonal position the hyperbolae will pass through 
 the axial points for each color and in each that color will be ex- 
 tinguished, giving, if the dispersion is great, a series of colored 
 bands ranging from red (white minus violet and part of blue) 
 
OPTICAL CHARACTERS. 
 
 141 
 
 through the violet axes ; to blue or violet (white minus red and 
 part of yellow) through the red axes. Generally the intermediate 
 colors overlap, producing a dark band, except on the outer fringes. 
 That is, the colors* fringing 
 the hyperbola are in in- 
 verse position to the axial 
 points. 
 
 Similarly, if the full line 
 circles represent the first 
 extinction ring for violet 
 and the dotted those for 
 red, then as we pass from 
 the centre red is first ex- 
 tinguished and the violet 
 tint shown. 
 
 For /> < L> the red is fur- 
 ther from the centre than 
 the blue. For f>^>o the 
 red is nearer the centre 
 than the blue. Or in gen- 
 eral, the color with the 
 larger angle is nearer the 
 centre of the field. 
 
 If the axial plane is not 
 the same for all colors, that 
 is, if b for certain colors 
 becomes a or c for others, 
 then the interference fig- 
 ures will be turned 90 
 with respect to each other 
 and, if superposed in white 
 light, will give a complex 
 figure as in brookite. 
 
 FIG. 269. 
 
 VI. MONOCLINIC CRYSTALS. 
 
 There must also be considered the dispersion of the* bisectrices. 
 Three general cases have been distinguished, in all of which oio is 
 a principal section and the ortho axis b a principal vibration direc- 
 tion. 
 
 * The dotted curves should be colored blue ; the full line curves red, in Figs. 269- 
 272. 
 
142 
 
 CHARACTERS OF CRYSTALS. 
 
 V R 
 
 (a) Inclined Dispersion. The optic axes for all the colors lie in 
 the plane oio and the ortho axis b is 6, the intermediate principal 
 vibration direction, but a and c vary in position in the plane oio. 
 Evidently the color distribution will be symmetrical to oio. 
 
 Let Fig. 270 represent 
 the plane oioin a section 
 cut normal to the acute 
 bisectrix for yellow light. 
 The acute bisectrices for 
 red and violet will be ob- 
 lique to this section as 
 represented. Assume 
 P > a and construct the 
 interference hyperbolae 
 and first ring for the di- 
 agonal positions. Four 
 things are at once no- 
 ticed : 
 
 I . The color-distribu- 
 tion and shape are sym- 
 metrical to the trace of 
 oio and to this line* only. 
 
 2. The combination 
 color rings around the 
 axes due to overlapping 
 of rings for the different 
 colors are very different, 
 the one relatively small, 
 circular and intense, the 
 other larger, oval and re- 
 latively dull. The se- 
 ^ JG 2 Q quence of colors may be 
 
 the same or reversed. 
 
 3. The smaller ring is further from the centre of the field. 
 4. One hyperbola band is much broader than the other. ^ 
 
 (b) Horizontal Dispersion. If the ortho axis b is the acute bisec- 
 trix (either a or c as the crystal is -J- or ) then the planes of the 
 optic axes must for all colors pass through it and be normal to the 
 plane oio. In a section cut normal to the acute bisectrix for yel- 
 low light let Fand R, Fig. 27 1, represent the axial planes for violet 
 
 * Line connecting the axial points. 
 
OPTICAL CHARACTERS. 
 
 143 
 
 and red, rr the optic axes for red, vv those for violet. The plane 
 oio (not shown) is normal to all the axial planes. 
 
 In the interference figure it is evident that the color-distribution 
 and shape of the lemniscates are symmetrical to the trace* of oio 
 but not to any other line. Hence, for normal position the hori- 
 
 FIG. 271. 
 
 zontal dark bar will be centrally black from overlapping colors, but 
 on the edges will be red and violet or blue. 
 
 (c) Crossed Dispersion. If the ortho axis b is the acute bisectrix 
 (either c or a as the crystal is positive or negative) then will the 
 section normal to the acute bisectrix for one color be normal for 
 all and will itself be parallel to oio. The axial planes for all colors 
 will pass through b, and as b corresponds to a or c it is an axis of 
 binary symmetry, hence the color-distribution and shape of the 
 lemiscates must be such that any straight line through the centre 
 of the field passes in each direction through similar points. That 
 is, diagonally opposite parts of the field are similar. In Fig. 272 
 Fand R are the axial planes for violet and red. 
 
 It will be noticed that the chief reliance throughout is upon the 
 
 * Normal at centre to line joining the axes. 
 
144 
 
 CHARACTERS OF CRYSTALS. 
 
 colors that fringe the horizontal brush or the hyperbola, and that 
 this is in every case the other end of the spectrum to the color of 
 the light producing it. 
 
 FIG. 272. 
 VII. TRICLINIC CRYSTALS. 
 
 With triclinic crystals there is no symmetry at all in the arrange- 
 ment of the colors in the interference figure from white light. 
 Two or more varieties of dispersion may occur at once. 
 
 Plates normal to the obtuse bisectrix show similar phenomena, 
 though frequently it is impossible to obtain the interference figures 
 within the field. The color-distribution is naturally changed as 
 the dispersion of axes is in the opposite direction. 
 
CHAPTER XI. 
 
 DETERMINATION OF THE OPTICAL CHARACTERS 
 OF BIAXIAL CRYSTALS. 
 
 Some of the determinations are exactly as described under the 
 uniaxial. References may therefore be made of follows : Faster and 
 sl&iverray, p. 146 ; Retardation, p. f7f ; Strength of double refraction, 
 p. frzjT*-; Thickness of section , p. 147. V I % 
 
 DETERMINATION OF PLANES OF VIBRATION OR " EXTINCTION." 
 
 In biaxial crystals complete extinction in general is only ob- 
 tained with monochromatic light, p. 138. The methods of deter- 
 mination are as described, p. 1 37. 
 
 In sections normal to one of a, B, or c the extinction directions 
 are the other two principal vibration directions. In sections par- 
 allel to a, 6, or c one extinction direction must be a principal vibra- 
 tion direction. In sections normal to an optic axis no extinction 
 directions are obtained. The general relation between extinction di- 
 rections and optic axes is described, p. 1 37. This test is often suffi- 
 cient to determine the crystalline system, especially when an entire 
 crystal can be examined and the extinction directions determined 
 in all faces and zones. 
 
 In ortJwrhombic crystals extinction will always be parallel or sym- 
 metrical to crystallographic edges, cleavage cracks, etc. In pina- 
 coidal faces the vibration direction will be parallel to the crystal 
 axes. 
 
 In monoclinic crystals there will be parallel or symmetrical ex- 
 tinction in the zone [ooi 100] and in this zone the vibration direc- 
 tions in the base and orthopinacoid will be parallel to the axes. In 
 all other zones the extinction will be oblique. 
 
 In triclinic crystals all extinctions will be oblique. 
 
 THE ORIENTATION AND DISTINCTION OF THE PRINCIPAL VIBRA- 
 
 TION DIRECTIONS. 
 
 If the positions of the optic axes are known then all three prin- 
 cipal vibration directions a, b and c are also known, for a and c are 
 
 
146 CHARACTERS OF CRYSTALS. 
 
 the bisectrices of the angles between the optic axes and 6 is normal 
 to their plane. The indices a, ft and f may then be determined, as 
 described later, and the indicatrix and ray surface therewith con- 
 structed. 
 
 In orthorhombic crystals the pinacoids are normal to a, b and c. 
 That normal to b will show with convergent monochromatic Itght 
 hyperbolae* similar to Fig. 243 and those normal to a and c will 
 show lemniscates, Fig. 270. 
 
 In monoclinic crystals the clinopinacoid is necessarily either nor- 
 mal to b and shows the hyperbolae, Fig. 243, with convergent 
 monochromatic light, or is normal to a or c and shows lemniscates. 
 The extinction directions therein are the other two principal vi- 
 bration directions, and the faster and slower ray may be distin- 
 guished as described, p. 146. Plates may be ground normal to 
 these directions. 
 
 In triclinic crystals by trial a face is found which with convergent 
 monochromatic light shows, even eccentric, the hyperbolae or 
 lemniscates or axial image ; from this the direction is judged in 
 which a new face may be ground exactlyf normal to a principal 
 vibration direction. The extinction directions in the new face will 
 be the other principal vibration directions and the faster and slower 
 rays corresponding may be distinguished as on p. 146. Sections 
 may be ground normal to them. If there is dispersion of bisec- 
 trices accurate work will require special sections for each color, if 
 the dispersion is not large, sections for a middle color, usually soda 
 yellow, are used. 
 
 DIRECT MEASUREMENT OF THE PRINCIPAL INDICES OF RERFACTION. 
 
 (a] Prisms with a refracting edge B, Fig. 212, parallel to a 
 principal vibration direction will give for minimum deviation, p. 
 8&, a direction of transmission RS so lying in an optical principal 
 section that the ray vibrating parallelj to the edge B will yield a 
 principal index. In general the other obtainable index is not a 
 principal index. 
 
 By Fig. 259 it is seen that if B is parallel to c the transmission 
 
 * By means of one of the attachments to the polarizing microscope described later, 
 sections approximately normal to a, b or c may be tipped so that the lemniscates or 
 hyperbolae are obtained. 
 
 fit is important to leave faces of known forms so that later the ground faces may be 
 goniometrically oriented. 
 
 JThe ray which penetrates a nicol when the short diagonal is parallel to B. 
 
OPTICAL CHARACTERS. 147 
 
 is in ab and the ray vibrating parallel to B yields ^. Similarly if 
 B is parallel to b, ft results; and if B is parallel to a, a. results. 
 
 If the bisector BD, Fig. 212, is also the trace of an optical prin- 
 cipal section then the direction of transmission RS for minimum 
 deviation is also a principal vibration direction and both rays yield 
 principal indices. 
 
 (6) Prisms in which BD, Fig. 212, the bisector of the refracting 
 angle B is a principal vibration direction also give for minimum 
 deviation a direction of transmission RS lying in an optical prin- 
 cipal section. Therefore the index of the ray vibrating parallel to 
 BD, that is at right angles to the edge B, is a principal index. 
 
 By Fig. 259 it is seen that if BD is parallel to c the transmission 
 RS being normal to BD lies in the plane normal to c, that is in the 
 optical principal section ba and the ray vibrating parallel to BD 
 or at right angles to B will yield Y. Similarly if BD is parallel to 
 B the ray vibrating at right angles to B will yield ft and if BD is 
 parallel to a the ray vibrating at right angles to B will yield . 
 
 If a, b, or c, bisect the exterior angle B, Fig. 212, then is the 
 bisector BD normal to this principal vibration direction and RS 
 parallel thereto, that is for minimum deviation the direction of 
 transmission is a principal vibration direction and both rays yield 
 principal indices. The formulae under (/;), p. 103, hold for (a) and 
 
 0). 
 
 , (c] If one face AB, Fig. 228, of a prism is an optical principal 
 section and the refracting edge A is a principal vibration direction, 
 rays normally incident at AB will be transmitted in the direction 
 RS parallel to another principal vibration direction and each ray 
 will yield a principal index. If AB is ab and A is either a or b, 
 Fig. 259, then RS is c and a and ft result. The formulae under (c), 
 p. 103, hold good. 
 
 (d) In any crystal face or section parallel to a principal vibration 
 direction, by the methods and formulae of pp. 90-95, if this direction 
 is made the direction of transmission the two limit lines, as in p. 
 104, correspond to principal indices ; for instance if the transmission 
 direction is a the two limit-lines correspond to ft and f. If the 
 plate is turned 90 in its own plane the direction of transmission 
 will be normal to a principal vibration direction (to a in case given) 
 and the limit-line of the ray the vibration direction of which is at 
 right angles to this direction of transmission will correspond to 
 the third index. 
 
1 48 CHAR A CTERS OF CR YS TALS. 
 
 If the plate is parallel to a principal section* then either prin- 
 cipal vibration direction used as a direction of transmission will 
 give two of the principal indices. 
 
 DETERMINATION OF THE ANGLE BETWEEN THE OPTIC AXES. 
 
 This is usually accomplished in a plane parallel plate cut nor- 
 mal to the acute bisectrix. In orthorhombic crystals the same 
 plate will be normal for all colors, but in the other systems this is 
 not so, but if the plate be cut normal for a middle color, say yel- 
 low, the results for all colors will be approximatelyf accurate. 
 The apparent angle,;); denoted by 2, is larger than the true angle, 
 denoted by 2 V, and, since light transmitted in the direction of the 
 optic axis has the middle index /?, p. 136, then 
 
 If 2E exceed 180 the plate is usually surrounded by oil or some 
 other dense medium in a vessel with plane parallel sides. Denot- 
 ing the angle then obtained by 2H and the index of the surround- 
 ing medium by n 
 
 n sin H 
 
 - 
 
 By measuring a second plate normal to the obtuse bisectrix 
 there results 
 
 T7 . nsvsiE' 
 
 sin V 
 
 but as 2F-f2F=i8o then V+ V -=90 and sin V cos 
 whence 
 
 sin V n sin Eft sin E 
 
 r tan V '' = 
 
 n sn 
 
 *lf the plate be parallel to the optic axes the two boundaries approach as the plate 
 is turned and meet at the optic axis. In fact the shape of both sections of the Fres- 
 nel's surface may be plotted and even the optic axial angle measured. Liebisch, 
 Grundriss der Phys. Kryst., 373. 
 
 f The rays can only emerge at the true angle when the optic axes are normal to the 
 surface, as in a sphere or in a cylinder the axis of which is perpendicular to the axial 
 plane; or when the outer medium has the same index of refraction as the ray trans- 
 mitted in the direction of the optic axis. The former is rarely practicable, the latter 
 will be described later. 
 
 \ Plates normal to obtuse bisectrix yield V = 900 V. 
 
OPTICAL CHARACTERS. 
 
 149 
 
 This measurement involves actual rotation of the plate about b as 
 an axis between the lenses of a polariscope, the plane of rotation 
 being at 45 to the vibration planes of the crossed nicols and the 
 arms of the hyperbola being brought in succession into tangency 
 with one cross hair. The line joining the optic axes in the images 
 must coincide with the other cross hair. 
 
 The Groth Universal Apparatus. The optical portion of the 
 polariscope shown, Fig. 235, is placed in horizontal position in 
 another stand, Fig. 273. The nicols are crossed at 45 to the 
 horizon and just enough space left between the objectives to 
 permit free rotation of the crystal around a vertical axis. Above 
 is a horizontal graduated circle A" with concentric central rings R 
 and F which in turn support the vertical crystal carrier. The lat- 
 ter has motions of adjustment for the crystal plate which in the 
 latest type of instrument are exactly those of the Fuess Goniometer, 
 p. 67, but more frequently are simple sliding motions of a plate at 
 f and spherical segment (" mushroom joint ") at H. In this simpler 
 type the mineral is centered by drawing the front tube back until 
 the crystal is in focus, then centering as in the goniometer, then 
 
ISO CHARACTERS OF CRYSTALS. 
 
 the images are centered by the mushroom joint until both stay on a 
 horizontal line. 
 
 The closer the lenses the larger the image : that is, a small crys- 
 tal should be used, and as the best results are obtained with rather 
 narrow interference rings the crystals should be of such a thick- 
 ness (selected or obtained by grinding) that the lemniscates are 
 distinct. In measuring in a denser medium* the glass vessel J/is 
 used and the lenses touch its walls. 
 
 It is usually a sufficiently accurate selection of a plane normal 
 to a bisectrix if the center of the lemniscate coincide with the cen- 
 ter of the field. Groth givesf methods for more accurate deter- 
 mination of the plane. 
 
 The No. 2 Fuess Goniometer, Fig. 193, maybe used for the meas- 
 urement of axial angles by substituting nicols for the signal tube 
 in the collimator and for the eye piece in one of the telescopes ; 
 both nicols are adjustable. A tube containing a condensing lens 
 is fitted over the objective end of the collimator like a cap and 
 the telescope is converted into a weak microscope by the extra 
 lens used to bring a crystal into focus. The crystal section 
 is carefully mounted as previously described. The telescope 
 is set opposite the collimator, the nicols being crossed 45 to 
 the plane of rotation of the goniometer and rotation of the 
 carefully centered crystals brings the two branches of the hyper- 
 bola successively into contact with the vertical cross hair. 
 
 By a combination of the axial angle apparatus with a spectro- 
 scopej the angle for all portions of spectra can be determined. 
 
 A Polarizing Microscope may be used, the interference figure be- 
 ing seen : 
 
 (a] By placing an extra lens (Klein's lens) above the eye piece, 
 thus making visible a little image which forms there, or 
 
 (&} By placing below the eye piece but above the focal plane of 
 the objective a weak magnifying lens (Bertrand lens). 
 
 Two methods of measuring are possible : 
 
 (a) Measurement of Distances Apart of Emerging Axes. The 
 nicols are set at the diagonal positions and the distance d from the 
 
 *May use olive oil, a Brom-Napthalin, Methylene Iodide, etc., see p. 95. 
 
 \Fhysikalische Kryst., 714. 
 
 t A. E. Tutton, Philos. Trans. R. Soc. London, 1894, 185, 913. E. A. Wulfing, 
 
 in. Mitth., 1896, 15, 49. 
 
 gTschermak's Min. MittheiL, XIV., 375. 
 
OPTICAL CHARACTERS. 151 
 
 centre to either hyperbola, or the average of the two distances, is 
 
 read by either a glass- or screw-micrometer eye piece. Then 
 
 d d d 
 
 sin E = or in liquid, sin H wv> or sin V =, t in which 
 
 C is a constant for the same system of lenses and is determined 
 once for all with a crystal of known axial angle. 
 
 If the section is oblique to the bisectrix it may be tipped 
 so that the axis of the microscope coincides with the direction of a 
 bisectrix, and in this way a well defined interference figure is fre- 
 quently obtained. A simple instrument for this purpose* con- 
 sists of a clip supported 15 mm. above the stage by a ball and 
 socket joint giving free motion in all directions, through an angle 
 of about 45. Delicacy of manipulation is secured by a long key 
 fitting into the clip arm. A special lens combination is needed in 
 order to bring the condenser nearer the section. 
 
 (^) Measurement by Rotation of the Section as in tlie Universal Ap- 
 paratus. This involves some form of rotation apparatus attached 
 to the stage of the microscope, by which the angle of rotation can 
 be read. Two forms of one of the simpler devices are shown in 
 Fig. 274. The crystal is fastened by wax to the end of a glass 
 axle the rotation of which is measured on a graduated vertical cir- 
 cle. The crystal projects into a little glass trough filled with re- 
 fracting liquid and must, by trial, be adjusted so that the plane of 
 the' optic axes is vertical. The measured angle is then 2 H. A 
 similar apparatus is described by Bertrand.f 
 
 If the rotation necessary to bring the hyperbolae into tangency 
 with the cross hair cannot be secured on account of the closeness 
 of the objective to the section ; one hyperbola may be brought just 
 into the field, then the other hyperbola to the corresponding point 
 on the opposite side of the field. The angle of rotation necessary 
 to install the second point, plus the angle equivalent to 2 d, the 
 distance apart of the two points, will be 2 H. 
 
 A much more elaborate device is needed to both orient the 
 axial plane and measure the axial angle. Perhaps the best for the 
 examination of small complete crystals is the so-called Universal 
 Rotation Apparatus^ of Professor Klein, Fig. 275, which is espe- 
 
 *Described by T. A. Jaggar, Jr., Amer. Jour. Sci., III., 129, 1897, and made by 
 Bausch & Lomb Optical Co., Rochester, N. Y. 
 
 \ Hull. Soc. Min. de France, III., 96, 1890. 
 
 C. Klein, Sitzitngsber. der. Akad. Wissenschaften, Berlin, 1895, p. 91. " Der 
 Universaldrehapparat," etc. 
 
152 
 
 CHARACTERS OF CRYSTALS. 
 
 FIG. 274. 
 
 cially designed as an attachment to a No. VI. Fuess Microscope, 
 
 p. 108. The microscope is tipped 
 with its axis in horizontal position 
 and the plate G of the rotation ap- 
 paratus fastened to the microscope 
 stage by strong clips so that the axle 
 P is vertical. The crystal is attached 
 at Fby wax and centered by the cross 
 hairs of the microscope and the rect- 
 angular and rotary movements of the 
 microscope stage. Thereafter it may, 
 while still in the line of sight, be 
 rotated measured amounts in three 
 planes at right angles to each other 
 by the vertical arcs L and L lt and the 
 
 horizontal circle K. A small cubical vessel with two opposite walls 
 of glass is filled with a strongly refracting liquid and raised into posi- 
 tion between the converger and objective, and so that the crystal is 
 immersed. If by previous examin- 
 ations p. 145 or from the geomet- 
 ric properties any optical principal 
 section is known, this will be 
 placed parallel to one of the planes 
 of rotation. If by manipulating 
 the arcs L and L\ with the nicols 
 horizontal and vertical, a position of 
 the crystal is found, such that the 
 field remains dark throughout a 
 complete rotation by K\ a prin- 
 cipal section is horizontal, for in 
 no other sections would the vibra- 
 tions of the refracted rays remain 
 horizontal and vertical, p. 1 37. If 
 this principal section is the plane of 
 the optic axes, four times in the course of the rotation there will 
 be a brightening of the field due to the emergence of an optic 
 axis (internal conical refraction), p. 135. If this does not occur 
 the crystal is remounted with this principal section vertical and 
 parallel to one of arcs L, L r Then, if the arc is tipped and trial 
 is made by K, another principal section must be found and 
 
 FIG. 275. 
 
OPTICAL CHARACTERS. 153 
 
 90 away the third, and one of these must be the axial plane. 
 
 Having found the plane of the optic axes the approximate posi- 
 tions of the axes are noted, the converger is then introduced, the 
 nicols are set in the diagonal positions, the Bertrand lens inserted 
 or the Klein ocular, the branches of the hyperbola are brought into 
 tangency and the four positions of the emerging axes are read. 
 
 A special lens system is needed with this apparatus to counter- 
 act the unusual distance between crystal and objective. 
 
 MEASUREMENT OF THE TRUE AXIAL ANGLE. 
 
 If the refracting liquid used in any of the preceding methods 
 has an index of refraction equal /? then sin H= sin V. To practi- 
 cally obtain this, a liquid with n a little above ft is used and the dilu- 
 ent added drop by drop until the boundaries of the crystals fade. 
 Then for the accurate determination some coarse powder of the 
 substance is placed on an object glass, a drop of the liquid added, a 
 cover glass pressed on and the extinction directions determined in 
 a selected grain. With the condenser lowered and the analyzer 
 out, the two directions of extinction are successively made to co- 
 incide with the vibration direction of the polarizer, and the micro- 
 scope is focussed sharply on the dark boundary between the liquid 
 and the grain. The objective is then raised and the dark bound- 
 ary line appears to move towards the substance with the higher 
 index, of refraction. If the liquid is of the mean refractive index 
 of the crystal the border will for one position move towards the 
 liquid and for the other towards the grain. 
 
 A final proof of the correctness of the preparation is that if 
 = ft 2// a -f- 2H Q = 1 80, but if on measurement 2H a + 2// ft < 
 
 I So , then _< I and too much diluent has been added or if 
 2fi a -f 2// < 1 80 then- > I and more diluent should be added. 
 
 CALCULATION OF AXIAL ANGLE FROM INDICES. 
 
 This angle may be calculated from the principal indices of re- 
 fraction by the equation 
 
 _' + ' 
 
 tanF= ' "' 
 
154 CHARACTERS OF CRYSTALS. 
 
 DETERMINATION OF CHARACTER OF RAY SURFACE. 
 
 This requires in general a section normal to a or c that is show- 
 ing the lemniscates, Fig. 266, with convergent monochromatic light 
 or an oblique section tipped to show these lemniscates. 
 
 The difference between the lemniscates formed normal to the 
 acute bisectrix and the obtuse bisectrix cannot always be judged 
 and may require measurement of the axial angle. 
 
 If the section is normal to the acute bisectrix then the latter may 
 be proved to be c or a, that is, the Ray Surface proved to be posi- 
 tive or negative as follows : 
 
 (a) The line joining the optic axes is placed in diagonal posi- 
 tion between crossed nicols with parallel white light the insertion 
 of a test plate will determine as described, p. 118, whether this 
 direction is the vibration direction a of the faster ray or c of the 
 slower ray (the other extinction direction being 6), that is, whether 
 this section is normal to c or a and p. 1 36, Bx a = c, -f- ; Bx a a, 
 
 (b) With convergent light and crossed nicols the line joining 
 the axial points is placed in diagonal position and the quartz 
 wedge, p. 1 1 8, gradually inserted with the direction c parallel to 
 this line. If the crystal is positive the rings around each axis 
 will expand, moving toward the centre and merging in one curve, 
 and will contract and increase in number, if the crystal be negative, 
 the change increasing with the thickness of wedge interposed. 
 If the direction c of wedge and the line joining the axial points are 
 crossed the effects are reversed ;* that is expansion of the rings 
 when the length of the wedge is parallel to the line joining the 
 axes proves positive character and expansion when these direc- 
 tions are crossed proves negative character. 
 
 (c) If with the line joining the axial points in normal position 
 the mica plate is inserted in the diagonal position the black cross 
 is destroyed and the rings broken and new flecks developed in the 
 same quadrants as in uniaxial, but the positions of the flecks do 
 not so satisfactorily suggest the signs -f- and 
 
 If the section is normal to the obtuse bisectrix all the results are 
 reversed. 
 CRYSTALS IN THIN ROCK SECTIONS. 
 
 The proper orientation of a haphazard section may be obtained 
 with the Jaggar apparatus, but for precise measurement a more 
 
 * Hence it is always practical to develop the lemniscate, even when only the hyper- 
 bolae are visible, by inserting the wedge so as to produce contraction. 
 
OPTICAL CHARACTERS. 155 
 
 elaborate apparatus is essential. The different types of univer- 
 sal stage of von Federow* require the sections to be mounted 
 upon a special round glass with high index of refraction, while the 
 apparatus of Klein,f Fig. 276, permits the use of ordinary square or 
 rectangular object glass, the cover glass, however, being removed. 
 The metal plate G is clamped to the microscope stage. The metal 
 vessel B, which is somewhat more than a half sphere, is filled with 
 some liquid of high refractive index, usually glyerine n = 1.46. 
 
 T 
 
 FIG. 276. 
 
 The light enters through the glass plate a. The section is sup- 
 ported in the liquid by the glass plate 5 in the rim 7j and re- 
 ceives motions of rotation around the horizontal axis c by the grad- 
 uated circle 7] and in its own plane by the knob k, which turns the 
 fine-toothed wheel z in contact with teeth in the rim 7\. 
 
 ABSORPTION AND PLEOCHROISM. 
 
 The vibration directions of the rays of monochromatic light 
 which undergo greatest and least absorption and a direction at 
 right angles to these constitute the so-called absorption axes. 
 Their lengths are quite independent of the lengths of the axes of 
 the optical indicatrix, but their directions are frequently though 
 not necessarily the same. Upon these axes can be constructed an 
 absorption ellipsoid from which the absorption in all directions can 
 be deduced. 
 
 /;/ otthorliombic crystals the directions of the absorption axes 
 coincide/tfr all colors with the directions of crystal axes and the 
 axes of the optical indicatrix. 
 
 In monochnic crystals one absorption axis is parallel to the crys- 
 tal axis b, therefore to one of the axes of the optical indicatrix; the 
 
 * Zeit.f. Kryst. v. 25, p. 351. 
 
 f Sitzungsber. Berlin Akad., 1895. 
 
I 5 6 
 
 CHARACTERS OF CRYSTALS. 
 
 other axes lie in the plane oio, but for any color may or may not 
 coincide with the axes of the optical indicatrix for that color. 
 
 In triclinic crystals no coincidence is essential. 
 
 Although subject to many exceptions, the law of Babinet is 
 generally correct that "the slower transmitted ray is the more ab- 
 sorbed." 
 
 The relative absorption of the two rays for any direction of 
 transmission can be obtained by use of the spectrophotometer of 
 P. Glans, which consists, Fig. 277, of a collimator C, telescope H 
 
 FIG. 277. 
 
 FIG. 278. 
 
 and prism P. The vertical slit D in the collimator is divided into 
 halves by a cross piece of metal, and the entering light passes 
 through a so-called Rochon's quartz prism B, which gives a double 
 image, Fig. 278, of each half of the slit, the height of the slit be- 
 ing so chosen that the extraordinary image of one and the ordinary 
 of the other are in contact in the centre of the field and the other two 
 are shut out in the telescope. If now the crystal plate is placed in 
 front of the slit with its extinction directions parallel and at right 
 angles to the slit the two rays developed will pass through the 
 prism P and their spectra be seen in the telescope one above the 
 other, and by means of the horizontally movable eye plate E any 
 portion may be viewed alone and compared. 
 
 By introducing a nicol N arranged with a graduated circle, so that 
 when at zero its plane of vibration coincides the principal section 
 of the Rochon prism, either ray may be shut out at will by turn- 
 ing the nicol to o or 90 and the spectrum of the transmitted ray 
 compared with a spectrum scale sent through >S and reflected into 
 the telescope. 
 
 A quantitative comparison can be made as follows for any portion 
 of the spectrum. The crystal plate is removed and the angle of 
 
OPTICAL CHARACTERS. 157 
 
 rotation a of the nicol determined at which the two spectra of the 
 flame are equally bright; this will be somewhere near 45 but 
 not exactly. The plate is then replaced and the nicol again 
 turned through an angle /? to secure equal intensity for the desired 
 portion of the spectrum. The relative intensities are : 
 
 f = tan 2 a cot 2 /?.* 
 
 N 
 
 If only one of the halves of the slit is covered the intensity may 
 be compared with that of the incident light. 
 
 Transmission along one of the principal vibration directions a, 
 a or c will yield the intensities for vibrations parallel to the other 
 two. As pointed out these are directions of absorption axes in 
 the orthorhombic crystals but not necessarily in the monoclinic or 
 triclinic crystals. 
 
 Generally speaking, the records made are simply comparative, 
 for instance: Absorption strong a>b>c, or, Absorption feeble 
 parallel a. More commonly the record is as to pleochroism or 
 effect with white light, the color differences for vibrations parallel 
 b, b and c for some middle color being recorded, or, the colors 
 for directions referred to some definite crystal face.f 
 
 The pleochroic images are viewed either with a microscope or 
 dichroscope as described, p. 131, or the apparatus of Klein, Fig. 
 275, may be conveniently used, after the measurement of the angle 
 between the optic axes, by adjusting the crystal with the plane of 
 the optic axes horizontal. If the plane of vibration of the polarizer 
 is vertical and the analyzer is out, the color obtained throughout 
 a rotation by K corresponds to b. If the vibration plane of the 
 the polarizer is horizontal, transmission parallel to the bisector of 
 axial angle yields the color for one of a or c and by a rotation of 
 90 with K the color of the other results. 
 
 *Liebisch Grundriss der Phys. ICryst., p. 312. 
 
 f For example: Andalusite. Pleochroism a blood red to rose red, b = c olive 
 green. Dana's System, p. 496. 
 
 Titanite. a = yellowish red, 6 greenish red, c = pale yellow. Ripidolite. Yellow 
 to reddish for vibrations normal to coi, green parallel ooi. Rosenbusch liuifitabellcn, 
 Hie. 
 
 Axinite. Pieochroism strong, normal to r pale olive-green, giving with dichroscope 
 olive-green and violet blue, parallel r and normal to edge r\M cinnamon brown, giving 
 cinnamon brown and violet blue. Dana's System, p. 528. 
 
158 
 
 CHARACTERS OF CRYSTALS. 
 
 FIG. 279. 
 
 ABSORPTION TUFTS. 
 
 Plates of crystals with strong absorption, especially if cut normal 
 to an optic axis, often show in convergent non-polarized light two 
 dark tufts symmetrical to the plane of the optic axes, but sepa- 
 rated by a bright space. If white light is used the pencils are of 
 different color to the rest of the field. 
 
 In Fig. 279,^ is the optic axes to which the plate is normal ; A' 
 
 is the other optic axis. Oblique rays 
 emerging at any point m on the line 
 AA f have been transmitted in the 
 plane of the optic axes and, there- 
 fore, one of the rays vibrates in the 
 plane of the optic axis, the other 
 parallel to the principal vibration dir- 
 ection b or AN. Oblique rays from 
 any point m' on the line AN normal 
 to A A' will vibrate (see p. 1 37) par- 
 allel and at right angles to the bi- 
 sector of the angle Am'A'. As 
 explained by Mallardf the intensity 
 of the pencil of rays emerging at m 1 is necessarily less than that of 
 the pencil of rays emerging at m, the difference increasing the 
 nearer the points are to each other, that is, normal to AA' is a dark 
 tuft growing lighter on each side of A. Similar effects are ob- 
 tained for points not on AN, the total result being two tufts or 
 pencils of hyperbolic shape tangent to AA'. 
 
 When white light is used if the optic axes are dispersed, the tufts 
 are not exactly superposed and the borders show color tints. 
 
 The test may be made by holding the crystal section close to 
 the eye and looking at the bright sky or by removing the polarizers 
 from a converging polariscope. 
 
 If the plates are examined in convergent polarized light, the 
 analyzer being removed, two types may be distinguished as follows : 
 TYPE I. The dark tufts will appear normal to the plane of the 
 optic axis and in a clear field, when the plane of the optic axis 
 is normal to the vibration plane of the polarizer. Example : Anda- 
 lusite of Brazil. Plate cut normal to acute bisectrix and placed 
 on a thin plate of azurite shows black tufts on a blue ground. 
 
 * W. Voigt zur Theorie der Absorption des Lichtes in Krystallen IViedemann's 
 Ann., 1885, XXIII., 577. 
 
 f Traiet de Cristallographie, II., 361. 
 
OPTICAL CHARACTERS. 159 
 
 TYPE II. The same phenomenon results when these planes are 
 parallel. Example : Epidote of Knappemvand Tyrol. Cleavage 
 plate parallel ooi gives tufts symmetrical to oio, with white light 
 the borders towards the acute bisectrix are dark green and the 
 opposite are dark red. 
 
 In both types, with a rotation of the plate through 90 from 
 these positions, tufts also appear parallel to the plane of the optic 
 axes. 
 
 METALLIC REFLECTIONS OR METALLIC LUSTRE. 
 
 The metals do not obey the usual laws of reflection ; incident 
 plane polarized light is reflected as such, only when the plane of 
 vibration of the incident light is parallel or at right angles to the 
 plane of reflection ; and in both of these cases there is developed 
 a change of phase, which is, however, greater in the latter case 
 than in the former. For all other azimuths of vibration the vibra- 
 tion of the reflected light is the result of two unequal* compo- 
 nents at right angles and is, therefore, elliptically polarized. 
 
 The intensity! of the reflected light differs also from that obtained 
 with transparent substances. For normal incidence it is not zero, 
 but considerable, decreasing as the angle of incidence increases up 
 to a value of 50 to 60, thereafter again increasing until the 
 angle of grazing incidence is reached. 
 
 , For a given incidence certain colors are reflected by metals 
 more strongly than others, whereas, transparent substances, even 
 if colored, reflect white light. The combined effect of this intense 
 and selective reflection is called Metallic Lustre. 
 
 SURFACE COLORS. 
 
 Between metallic and transparent substances are certain which 
 are transparent for light of certain wave-lengths, but stop others so 
 completely that even with thin sections they are represented in 
 
 *Varying from o for normal incidence to A/2 for grazing incidence and for a parti- 
 cular angle, which is experimentally found to be the angle of maximum polarization of 
 common light, passing through the value A/4. 
 
 f Cauchy, Voigt and Drude have studied the phenomena and calculated the indices 
 of refraction and absorption for a number of metallic substances for different vibration 
 directions upon fresh cleavage surfaces. For gold and silver a higher velocity c flight 
 was found than in air, but very high absorption ; for lead, platinum and iron the indices 
 of refraction were high, but the absorption relatively low. Very high absorption indices 
 were obtained for stibnite and galena. Kundt discovered phenomena of double re- 
 fraction of light transmitted through extremely thin electrically deposited metallic films. 
 
160 CHARACTERS OF CRYSTALS. 
 
 the spectrum* by absorption bands. These rejected colors are 
 reflected. 
 
 When white light is used the reflected or surface color is strik- 
 ingly unlike the transmitted or body color, for instance, a drop of 
 fuchsin solution dried on a glass plate shows the surface color 
 green with metallic lustre and the body color red, or if a glass be 
 laid over the surface color is blue-green. A crystal of potassium 
 permanganate polished to a thin layer on glass gives a yellow sur- 
 face color and a dark violet body color. 
 
 The surface and body colors are only approximately comple- 
 mentary as Wiedemannf has shown. The surface colors in uniaxial 
 crystals, moreover, are non-polarized when reflected from basal 
 sections, but dichroic, that is, elliptically polarized and made up of 
 two components vibrating in and at right angles to the plane of 
 reflection for other sections. For instance, with crystals of mag- 
 nesium platinocyanide the light reflected from the basal plane is 
 violet and not to be decomposed, whereas the green color from a 
 prism face is always resolvable into two colors varying with the 
 angle of incidence. 
 
 FLUORESCENCE. 
 
 Certain organic compounds, fluorite containing dissolved organic 
 matter and uranium glass possess the property of absorbing light 
 and again emitting it as light of a different color. If sunlight is 
 focussed on a dilute solution of quinine sulphate it becomes deep 
 blue near the surface of entrance. If the spectrum of the trans- 
 mitted light is examined the violet and ultra violet rays are missing, 
 that is these have in the solution been turned into blue. A test 
 tube of the solution held in a solar spectrum beyond the visible 
 rays becomes blue. 
 
 In fluorite and uranium glass, which are isotropic, the color 
 produced is independent of the direction of transmission. In the 
 uniaxial magnesium platinocyanide crystals^ with sunlight passed 
 through blue or violet glass the green surface color disappears on 
 the prism faces, and by polarizing either the incident or emerging 
 light a dichroic fluorescence is observed, scarlet red for vibrations 
 
 ^Christiansen found that the spectrum of the red solution of fuchsin lacked green 
 and that blue was deviated more than red. 
 \Ann. de Phys. v. 151, p. 625, 1874. 
 {Lommelin Ann. de Fhys., VIII., 634, XLIV., 311. 
 
OPTICAL CHARACTERS. 161 
 
 normal to the optic axis, orange yellow for vibrations parallel to 
 the optic axis. If the violet rays are incident normal to the base, 
 polarization reveals only one color, orange yellow, whatever the 
 azimuth of the vibrations. 
 
 PHOSPHORESCENCE. 
 
 Phosphorescence, or the emission of light by a substance in the 
 dark after it has been exposed to light, heat, friction, mechanical 
 force, or an electrical discharge, has not been clearly shown to be de- 
 pendent in any way upon the crystalline structure. It is developed 
 by one process or another in a large number of minerals and salts.* 
 
 THE NORREMBERG AND REUSCH COMBINATIONS OF MlCA PLATES. 
 
 The optical phenomena of combinations of differently oriented 
 layers of a doubly-refracting substance are of great value in the 
 development of theories of structure. Muscovite in which the 
 acute bisectrix is nearly normal to the cleavage plane is especially 
 convenient for these experiments. 
 
 If two plates of muscovite of equal thickness are laid so that 
 their axial planes are at right angles, then with parallel light and 
 crossed nicols, the phase difference originated in one is exactly 
 compensated by that in the other, and the field will be dark 
 throughout rotation of the stage. With convergent light four axes 
 will emerge equally distant from the centre and between them 
 thin hyperbolic color curves, the asymptotes to which form a black 
 cross. 
 
 Norremberg discovered^ that if the plates were made so thin 
 that the phase difference of the rays developed in any plate was 
 less than a wave-length, and that if a pile of 12 to 36 of these were 
 made the axial planes in adjacent plates being at right angles, then 
 not only did the compensation of the phase differences give a dark 
 field with parallel light and crossed nicols, but with convergent 
 light the isochromatic curves produced by the oblique rays were 
 circles, that is, the effect of the pile was exactly equivalent to a 
 basal section of an uniaxial crystal, Fig. 280. 
 
 Reusch found,J by piling 12 to 36 similar plates so that the axial 
 planes of adjacent plates were at 120 to each other as in Fig. 281, 
 
 *Lommel in Ann. de P/iys., VIII., 634, XL1V., 311. 
 
 |D. Hahn, Phosphorescenz der Mineralien, Zeif. f. d. ges. Natur-wiss., Bd. 
 XLIIL, I, 131, 1874. 
 
 \ Berichte. Ak. Wissens. Berlin, July 8, 1869, p. 530. 
 
1 62 
 
 CHARACTERS OF CRYSTALS. 
 
 that with parallel incident light there emerged a plane polarized 
 ray, the vibration plane of which had been rotated in the direc- 
 tion of the piling through an angle proportionate to the thick - 
 
 FIG. 280. 
 
 FIG. 281. 
 
 ness of the pile that is, if starting at the lowest plate the piling 
 had been with a change in the direction of the hands of watch the 
 rotation was also in that direction, and the color produced with 
 white light would, by rotation of the analyzer, also in the direc- 
 tion of the hands of a watch, pass in the order, red, orange, yellow, 
 green, blue, violet. 
 
 With convergent light Fig. 281 resulted, the centre being colored. 
 
 This conforms strictly to the behavior of a basal plate of right- 
 handed quartz, except that by rotation of the stage, a slight change 
 of color takes place, this, however, decreases as the mica plates 
 are made thinner. 
 
 If the piling were in opposite direction the rotation of the plane 
 of vibration would be opposite and the color order also. Based on 
 these and similar experiments, theories as to the structure of cer- 
 tain crystals have been developed by Mallard,* Sohncke and others. 
 
 * Trait* de Cristallographie II., 262-304; Zeit.f. Kryst., etc., XIX., 529. 
 
PART III, THE THERMAL, MAGNETIC AND ELECTRL 
 
 CAL CHARACTERS, AND THE CHARACTERS 
 
 DEPENDENT UPON ELASTICITY 
 
 AND COHESION. 
 
 CHAPTER XII. 
 
 THE THERMAL CHARACTERS. 
 
 Certain substances, such as halite, are essentially transparent to 
 both heat and light ; and if sunlight be decomposed by a prism of 
 such a substance there is obtained not only the visible color spec- 
 trum consisting of the rays between the least refracted red (A = 
 760.4 /J.IJL) and the most refracted violet (A= 393.3 ////), but if each 
 ray is tested by a delicate thermo-pile the temperature will be found 
 to increase from the violet end towards the red and to a certain dis- 
 tance beyond the red where it is a maximum and thereafter to de- 
 crease, proving the existence of invisible heat rays beyond the red. 
 
 Experiments show also that these heat rays are reflected and 
 refracted and absorbed like light rays, that they may be doubly 
 refracted, as first shown by Knoblauch in calcite, that they may be 
 polarized by reflection or refraction and that when polarized their 
 power to penetrate crystals varies not only with the direction of 
 transmission but with the direction of vibration. For instance, heat 
 will not penetrate crossed nicols, but the interposition of a plate of 
 doubly-refracting substance will permit components to penetrate, 
 except at intervals of 90 when the planes of vibration of the heat 
 rays produced in the plate coincide with those of the nicols. In- 
 terference and circular polarization of heat rays have also been 
 experimentally proved. 
 
 Heat rays may, therefore, be regarded as invisible light rays 
 which are in general of greater wave-length and less refrangibility 
 than the visible rays, but are subject to the same laws. It is pos- 
 sible, though difficult, to determine experimentally a series of con- 
 stants for crystals with respect to these invisible rays, as in the 
 famous experiments of Knoblauch.* 
 
 * Knoblauch, Pogg. Ann., LXXXV, 169; XCIII, 161. 
 
1 64 CHARACTERS OF CHVSTALS. 
 
 Simpler and more convenient tests can be made with respect to 
 the conductivity or rate of transmission of heat from particle to 
 particle, and to expansion. For both a dependence is found to 
 exist upon the crystal structure. 
 
 HEAT CONDUCTIVITY. 
 
 The shape of the " surface of conductivity " or isothermal surface, 
 the radii of which are the rates of conductivity in different direc- 
 tions, is most easily determined from the surface conductivities in 
 sections of known orientation. 
 
 The relative surface conductivies are most satisfactorily obtained 
 by Rontgen's modification of the older de Senarmont method. 
 The previously cleaned and polished section is breathed upon, 
 quickly touched by a very hot metal point normally applied, and 
 instantly dusted with lycopodium powder. The section is then 
 turned upside down and tapped carefully, when the powder falls 
 from where the moisture film had been evaporated, but adheres 
 elsewhere, giving a sharply outlined figure. The entire operation 
 should take less than three seconds. 
 
 In all homogeneous substances in which the surface tested is 
 large enough to allow for irregularity of contour the resultant fig- 
 ure is either an ellipse or a circle, that is a section of an ellipsoid. 
 
 The major and minor axes of the ellipse are carefully measured 
 under the microscope with a micrometer eyepiece. 
 
 In the old method of de Senarmont the surface of the section is 
 coated with a very thin layer of wax or paraffin either applied with 
 camels' hair brush * or by melting a thin bit on the surface and 
 pouring off the excess. The wax-covered surface is placed in a 
 horizontal position and heat applied at one point either : 
 
 (a) By a wire or narrow tube of metal (silver, platinum, copper) 
 passing through a hole pierced normal to the waxed surface and 
 heated either by a lamp at some distince,f or by an electric current, J 
 or by a current of hot air. 
 
 (&) By touching with a hot pointed wire vertically applied. 
 
 (c) By a little platinum ball soldered to a platinum wire, the latter 
 being surrounded to avoid radiation and heated by an electric cur- 
 rent. 
 
 *Tyndall, Heat as a Mode of Motion, p. 189, D. Appleton, 1891. 
 
 f de Senarmont, 1847, Ann. de Chitn. et de phys., XXI., 457; XXII., 179. 
 
 JTyndall, 1. c., 190. 
 
 gjannetaz, Bull. toe. Min., I., 19. 
 
THERMAL CHARACTERS. 165 
 
 Voigt recommends* a mixture of 3 parts elaidic acid with I 
 part of wax, the mixture melting at 40 to 50 C. 
 
 A constant temperature is maintained until the coating has 
 melted far enough from the point of application of the heat. The 
 boundary of the melted patch is then the isothermal curve show- 
 ing the distances which the heat has been transmitted, and is 
 visible after cooling as a ridge which is always in the shape of 
 an ellipse or circle. 
 
 As a result of the experiments it is found that : 
 
 i In singly-refracting crystals (isometric) all sections yield 
 circular isothermal curves; that is, the isothermal surface is a sphere. 
 
 2 In optically-uniaxial crystals ( tetragonal or hexagonal ) ba- 
 sal sections yield circular curves, but all other sections yield ellip- 
 tical curves which become more eccentric as the section becomes 
 more nearly paraiiei to the optic axis. That is, the isothermal sur- 
 face is an ellipsoid of revolution around c , the optic axis. As for 
 light a division may be made into -f and . For instance, in 
 quartz the conductivity is more rapid parallel c and in calcite at 
 right angles to c, hence quartz is thermally positive, calcite 
 thermally negative. 
 
 3. In optically-biaxial crystals the isothermal curves are all 
 ellipses and the isothermal surface is a triaxial ellipsoid, the axes 
 of which are at right angles to each other. In orthorhombic 
 crystals these axes coincide with the crystallographic axes, and 
 the major and minor axes of the ellipse obtained in any principal 
 section are parallel to the crystal axes therein. In monoclinic 
 crystals one axis of the isothermal surface coincides with b and 
 the other two lie in the plane oio. In the zone [oio 100] the 
 major and minor axes of the isothermal curves should coincide 
 with crystal axes. In triclinic crystals there is no essential or 
 probable parallelism of crystal axes and axes of ellipses in any 
 section.t 
 
 * Elemente der Krystallphysik, 78. 
 
 f Jannetaz showed an apparent connection between directions of cleavage and ease 
 of conductivity. For instance, the isothermal surface for uniaxial crystals which have 
 an easy basal cleavage is usually a flat oblate spheroid. If the direction of cleavage is 
 not basal, but oblique, the longer axis will often be in the section most nearly parallel 
 to the cleavage. As will be later explained, planes of cleavage are supposed to be 
 directions of greatest closeness of molecules, and it seems, therefore, that the rapidity of 
 transmission increases as the molecules are closer together. It is curious, however, 
 that if an amorphous substance, such as glass, is subjected to compression or tension the 
 isothermal circles become elliptical, the shorter axis being in the line of greater press- 
 ure or packing of molecules. 
 
1 66 CHARACTERS OF CRYSTALS. 
 
 EXPANSION BY HEAT. 
 
 Most * substances increase in volume when heated and contract 
 on cooling, and experiments show that in crystals the rate of ex- 
 pansion is not the same for directions crystallographically unlike, 
 but varies as follows : 
 
 1 Isometric Crystals. The rate of expansion is the same for all 
 directions. A sphere heated becomes a sphere of larger diameter. 
 
 2 Tetragonal and Hexagonal Crystals. The rate of expansion 
 is the same for all directions equally inclined to the axis r, is either 
 a maximum or a minimum parallel to c, and varies regularly from 
 this to the direction at right angles. A sphere heated becomes an 
 ellipsoid of revolution around c as an axis. 
 
 3 Orthorhombic, Monoclinic and Iriclinic Crystals. There is no 
 axis of isotropy. The directions of maximum and minimum expan- 
 sion are at right angles, and a sphere heated becomes a triaxial ellip- 
 soid, the axes being these directions and a third at right angles to 
 their plane. 
 
 Orthorhombic, the axes coincide with the crystal axes. 
 
 Monoclinic, one axis coincides with b, the others lie in oio, but not 
 coincident with conductivity axes or principal vibration directions. 
 
 Triclinic, the axes have no fixed positions relative to the crystal 
 axes. 
 
 DIRECT MEASUREMENT OF LINEAR EXPANSION. 
 
 The coefficient of linear expansion, that is the increase in a 
 unit of length for a temperature change from o to i C. may be 
 
 accurately measured for any direction 
 by the method of Fizeau f which as 
 perfected by Abbe is as follows : J 
 
 In Fig. 282 is a plane parallel 
 plate of the crystal, about 10 mm. 
 thick, which rests upon three projec- 
 tions turned upon the steel plate 7. 
 Three screws, with a fine 2 mm. thread, 
 pass through T and support the glass 
 plate P, which tapers slightly so that 
 
 *A few expand on cooling below a certain point and one, at least, iodide of silver, 
 contracts when heated above the ordinary room temperatures. 
 
 f Description by J. R. Benoit, Trans, et Mem. du bur. internat. des poids et 
 wes.,1., i, 1881; VI., i. 1888. 
 
 \ Described by C. Pulfrich, Zeit. f. Instrumentkunde, XIII., 365, 401, 437. 
 
THERMAL CHARACTERS. 
 
 167 
 
 when the upper surface is horizontal the lower is inclined about 
 20 minutes. By adjusting the screws a thin wedge-like film of air 
 is left between the horizontal polished surface of the substance and 
 the lower surface of the glass. 
 
 The telescope PO, Fig. 283, is at once telescope and collimator, 
 that is the light from a Geissler tube, shown at L in the smaller 
 
 Fio. 283. 
 
 figure, enters at the side, is deflected by the prism P, made parallel 
 by the lens system O, decomposed by the two flint glass prisms P l 
 and P 2 and by varying the angles at which these are set by the screw 
 S, rays of any chosen color are made to pass vertically through R 
 and reach the interference apparatus within the brass vessel G. 
 
 The rays incident at the upper surface of the crystal plate and the 
 lower surface of the glass plate interfere on reflection producing 
 parallel dark bands wherever the thickness of the air film is -J-A, f A, 
 etc. The distance between two adjacent bands is therefore a func- 
 tion of the wave-length of the light used. This distance is meas- 
 ured by a screw micrometer M which moves a vertical double 
 hair horizontally across the held. 
 
 The vessel G containing the interference apparatus is enclosed 
 in two concentric cylinders, the outer one containing a liquid. 
 When the liquid is heated the crystal plate and the interference 
 
1 68 
 
 CHARACTERS OF CRYSTALS. 
 
 apparatus both expand and the interference bands change in dis- 
 tance apart. From this change and from the previously deter- 
 mined effect of the expansion of the apparatus the change due to 
 the expansion of the crystal is calculated.* 
 
 Voigt describesf a simpler apparatus in which the expansion 
 of the crystal tips one of two parallel mirrors. Reflected signals 
 from the two mirrors are viewed by a telescope one meter distance 
 and their divergence measured. The apparatus is heated by im- 
 mersion in paraffin oil at 60 to 70 C. 
 
 MEASUREMENT OF EXPANSION BY CHANGE IN DIEDRAL ANGLES. 
 
 The indices and therefore the zone rela- 
 tions are not changed by uniformly heat- 
 ing a crystal, for any series of points on a 
 straight line remains on a straight line and 
 at the same proportionate distances apart. 
 If, therefore, in Fig. 284 ABC is the unit 
 plane and HKL any other plane, the points 
 0, K and B will after expansion be on a 
 straight line and the same proportionate 
 distance apart, so also 0, L and C or 0, H 
 and A. Therefore, the indices of HKL 
 after expansion will be 
 
 t OL' OL 
 
 ^__ t> 
 
 OB 
 
 FIG. 284. 
 
 QA' OA 
 
 OH J ~OH 
 
 OK' OK _ 
 ' ~ ~ k; 
 
 OB' 
 
 In isometric crystals the angles are unchanged by expansion,, 
 but in all other systems all diedral angles are altered except that : 
 
 (a) Faces parallel to two expansion axes remain parallel to their 
 original position and normal to all faces in the zone of the third axis. 
 
 (&) If the rates of expansion are equal parallel to two axes the 
 faces in the zone of the third axis remain parallel to their original 
 positions. 
 
 In tetragonal and hexagonal crystals by (a) the basal planes 
 remain at 90 to the faces in the prismatic zone, so also any 
 face in the prism zone remains at 90 to all faces of the zone 
 originally normal to it and by (b) all angles in the prismatic zone 
 remain constant. 
 
 *Zeit.f. Instk., XIII., 440, etc. 
 \ Wied. Ann., XLIIL, 831, 1891. 
 
THERMAL CHARACTERS. 169 
 
 In orthorhombic crystals since a, b and c are expansion axes, each 
 pinacoid remains at 90 to every face in the zone of the third 
 axis. 
 
 In monoclinic crystals the clino pinacoid remains at 90 to faces 
 in the zone of b. 
 
 From the change in angle the linear coefficients may be calcu- 
 lated.* 
 
 The change in angle is usually small, /~ / / 
 
 requiring delicate measurement. It may 
 be demonstrated f by cutting a plate - 
 like cleavage of calcite, Fig. 285 a, / \ \ 
 
 normally and parallel to the longer di- ^- ^ ^ 
 
 agonal of the larger face, reversing one- 
 half and gluing as in b with water glass. 
 On heating, the adjacent angles expand 
 in the same direction and, as exagger- 
 ated in c, one-half becomes inclined to 
 the other about 20' for 100 C. temperature change. This bend- 
 ing is easily shown by reflection of a signal on a screen 3 meters 
 distant. 
 
 DETERMINATION OF EXPANSION BY CHANGES IN THE OPTICAL 
 
 PHENOMENA. 
 
 The expansion of a crystal produces a change in the rapidity of 
 transmission of light which may be determined by measurement of 
 the indices of refraction or indirectly by the interference phe- 
 nomena. 
 
 In singly refracting (isometric) crystals the change is alike in all 
 directions. The crystal remains optically isotropic but the index 
 of refraction may become smaller as with fluorite, or larger, as with 
 diamond. 
 
 In uniaxial (tetragonal and hexagonal) crystals the changes in 
 velocity are unequal, that is the principal indices of refraction and 
 
 * For instance if the rhombohedral angle of a calcite cleavage, which decreases almost 
 9' for 100 elevation, is 105 5' at 10 C. and 104 56' at 110 C. then by formulae, 
 sin a = cosi^X I - I 55> tan a X .866 = ^, (Moses* and Parson's Mineralogy, p. 60) we 
 have / sin a = 9.84663, a = 44. 63, tan a X -866 = c .8549, and / sin a' ir= 9.84732, 
 a' = 44 72, tan a X - 866 = c' = - 8 5/ 5- 
 
 f Voigt, Elemente der Krystalphysik, p. 49. 
 
CHARACTERS OF CRYSTALS. 
 
 a vary unequally and not necessarily in the same direction.* The 
 strength of the double refraction ? may be either increased, rais- 
 ing the interference color and contracting the rings in the interfer- 
 ence figure, or the reverse may take place. The change when y a 
 is made less may, for a certain temperature, reduce the double re- 
 fraction to zero, that is for that color of light and at that tempera- 
 ture the crystal is optically isotropic and for a further temperature 
 change in the same direction will change in optical character 
 from + to or to -f . 
 
 In biaxial crystals (orthorhombic, monoclinic and triclinic) the 
 unequal changes in a, / and y, the principal indices of refraction, 
 are indicated not only by raising or lowering the interference colors 
 and in the contraction or expansion of the rings of the interference 
 figure but by changes in the angle between the optic axes, which 
 is simply a function of the principal indices. If either a or f 
 
 at any temperature become equal to 
 /5 then for that temperature and light 
 the crystal is uniaxial and the axial 
 angle is zero and for a further change 
 in the same direction the former 
 middle index ft will become or ? t that is b will become a or c and 
 the optic axes will pass into a plane at right angles to the former 
 position of the axial plane. Fig. 286 represents successive changes 
 in such a transformation. 
 
 The following selected examples illustrate these facts. 
 
 BARITE (Orthorhombic). 2.E for sodium flame increases with rising temperature 
 from 64 i' at 20 C. to 68 51' at 100 C. and 77 16' at 200 C. 
 
 GLAUBERITE (Monoclinic). The principal vibration directions are essentially un- 
 changed, but 2.E decreases with rising temperature. Within a range of 40 C. the crys- 
 tals become successively uniaxial for all colors and the axes thereafter are in a plane at 
 right angles to the former axial plane. 
 
 FIG. 286. 
 
 Temperature 
 Centigrade. 
 1 8 
 
 22 
 
 36 
 46 
 58 
 
 Apparent Angle 2E for 
 Li Red. Na Yellow. Tl Green. 
 
 13 30' 11 8' 8 14' 
 11 i' 8 9' o 
 8 40' o 7 8' 
 o 7 14' 10 32' 
 
 Blue. 
 
 
 
 8 42' 
 11 8' 
 
 13 2 
 
 * With increased 
 
 temperature : 
 
 
 
 Calcite 
 
 a 
 increased 
 
 7 
 increased 
 
 7 a 
 increased 
 
 Beryl 
 quartz 
 
 increased 
 decreased 
 
 increased 
 decreased 
 
 decreased 
 decreased 
 
THERMAL CHARACTERS. 
 
 171 
 
 GYPSUM (Monoclinic). The angle 2E decreases with rising temperature and the 
 principal vibration directions change, the dispersion changing from inclined to hori- 
 zontal. 
 
 Temperature. 
 
 2ZTfor 
 
 Temperature 
 
 2E for 
 
 Centigrade. 
 
 Na Yellow. 
 
 Centigrade. 
 
 Li Red. 
 
 20 
 
 92 
 
 20 
 
 96 
 
 50 
 
 79 
 
 47 
 
 76 
 
 100 
 
 51 
 
 71 
 
 59 
 
 116 
 
 36 
 
 95 
 
 39 
 
 I34 9 
 
 
 
 116 
 
 
 
 The test is usually made by replacing the glass vessel M, Fig. 
 273, by a metal air bath, Fig. 287, consisting of a rectangular hol- 
 
 FJG. 287. 
 
 low box of copper, which projects on either side beyond the po- 
 lariscope, and is heated by gas burners. At the top is an opening 
 for the crystal carrier, and at the large sides are glass windows so 
 set in tubes, that by a key the distance apart can be made as small 
 as the crystals will permit. From the top of the box project two 
 thermometers T reading to 300 C. 
 
 The crystal section is adjusted, and the angle is determined'at 
 room temperature, the two burners are then lighted and the heat- 
 ing continued until a constant temperature can be maintained for 
 one-half hour, when a new reading is made ; this is repeated^ for 
 different intensities of flame. 
 
CHAPTER X11I. 
 
 THE MAGNETIC AND ELECTRICAL CHARACTERS 
 OF CRYSTALS. 
 
 MAGNETIC INDUCTION OF CRYSTALS. 
 
 All substances are either attracted or repelled in some degree 
 when in the field of a strong electromagnet. If attracted they are 
 said to be "paramagnetic" or " magnetic \" if repelled they are 
 " diamagnetic " 
 
 If a rod of any substance is suspended by a fibre so as to 
 swing freely horizontally between the vertical poles of an elec- 
 tromagnet, ab t Fig. 288, mag- 
 netic induction takes place and 
 as the lines of force between 
 the poles are essentially hori- 
 zontal, the effect of the pull or 
 thrust upon rotation is greatest 
 for the particles furthest from 
 the axis of rotation. If para- 
 magnetic, therefore, the effect 
 is to pull the rod into a longi- 
 tudinal or "axial" position 
 with its ends as near the poles 
 of the magnet as possible, and 
 if diamagnetic the rod is pushed 
 into a transverse or " equator- 
 ial" position with its ends as 
 far from the magnetic poles as 
 possible. 
 
 Fig. 288 shows the apparatus 
 of Edmond Becquerel* in which 
 A and B are the poles of a 
 large electro-magnet, C and 
 C' square soft iron pole pieces 
 and DE, D' E long narrow soft 
 FIG. 288. T V natural size. iron pole pieces placed so that 
 
 * Ann. de Chim. et de Phys. y 1850, V., 28, p. 283. 
 
MAGNETIC CHARACTERS. 173 
 
 the back face of D E and front face of D r E f lie in the same plane 
 through the torsion thread. Bars of 25 mm. long, 2 to$ mm., broad 
 are suspended in the position a b and adjusted by the torsion head 
 N until a scratch at one end of the bar coincides with the cross 
 hair of the telescope L. The current is then turned on and the 
 deviation due to attraction or repulsion observed and measured by 
 the amount it is necessary to turn the circle H H' to restore the 
 original position. 
 
 With crystals, however, the particles in certain directions be- 
 come more strongly magnetized than in others, and the para- or 
 diamagnetism is judged by hanging a thin glass tube, filled with 
 powder of the substance, between the magnetic poles, the particles 
 of the powder having all possible orientations all effect of direction 
 is eliminated. 
 
 To DETERMINE THE RELATIVE STRENGTH OF MAGNETIZATION 
 IN DIFFERENT DIRECTIONS IN A CRYSTAL. 
 
 The crystal is suspended by a silk fibre and should be of such 
 a shape that the section normal to the axis of suspension is circu- 
 lar. If, however, the crystal is small, the form is less important. 
 
 Plucker used for his experiments a large electromagnet with six 
 Groves elements, the poles of the magnet being 1.6 inches apart 
 and the space around the poles protected from currents of air by a 
 glass cage. The crystal was suspended by a double silk thread 
 from a torsion balance. 
 
 The direction, in the section normal to the axis of suspension, 
 which is most strongly affected will evidently take an axial posi- 
 tion with paramagnetic crystals and an equatorial position with 
 diamagnetic crystals. 
 
 By suspending a cube by its three rectangular axes, a, b and c, 
 successively the magnetic intensities in these directions may be 
 
 compared, for example: 
 
 Suspended by : 
 a b c 
 
 Axial direction assumed by b a b 
 
 Equatorial direction assumed by c c a 
 
 If the crystal is diamagnetic, then, since c was twice equatorial, 
 c is the axis of greatest magnetization, but if paramagnetic, then is 
 ^the axis of greatest magnetization. 
 
 The relations cannot yet be said to be well understood as very 
 
174 CHARACTERS OF CRYSTALS. 
 
 few minerals have been thoroughly tested. It does not appear 
 that isometric crystals are magnetically isotropic, for the latest in- 
 vestigations of magnetite show that prisms cut parallel to a ternary 
 axis are most strongly magnetized, those parallel to a binary axis 
 only a little more feebly and those parallel to a quaternary axis 
 much more feebly.* 
 
 The experiments of Pluckerf appear to prove that all crystals of 
 other systems are magnetically anisotropic and that a suspended 
 sphere in a uniform magnetic field is in stable equilibrium only 
 when the direction most strongly magnetized is " axial," that is, is 
 parallel to the lines of force. 
 
 In hexagonal and tetragonal crystals the direction of maximum 
 magnetization is either parallel or at right angles to the vertical 
 axis c and the crystal is said to be magnetically positive or mag- 
 netically negative. If a sphere is suspended with c horizontal then 
 four cases J result. 
 
 Paramagnetic positive, the position of c is axial. 
 
 Paramagnetic negative, the position of c is equatorial. 
 
 Diamagnetic positive, the position of c is equatorial. 
 
 Diamagnetic negative, the position of c is axial. 
 
 In orthorhombic, monoclinic and triclinic crystals the directions 
 of maximum and minimum intensity, are at right angles to each 
 other and the intensity of the third axis of the " induction ellip- 
 soid " is at right angles to their plane. 
 
 In orthorhombic crystals these axes are the crystal axes a, b, c. 
 In monoclinic crystal one axis is parallel to b t the others are in 
 oio, but not necessarily or probably parallel either to axes of 
 
 * Aimantation non isotrope de la Magnetite cristallisee. P. Weiss. C. R., CXXII. 
 1405. 
 
 f Pogg. Ann., v. 72, p. 315; v. 76, p. 576; v. 77, p. 447 ; v. 78, p. 427 ; v. 86, p. I. 
 
 \ Examples are : 
 
 -{-Paramagnetic, Paramagnetic, -f- Diamagnetic, Diamagnetic, 
 Siderite, Tourmaline, Calcite, Bismuth, 
 
 Wernerite, Beryl, Mimetite, Arsenic, 
 
 Torbernite, Vesuvianite, Wulfenite, Zircon. 
 
 g Examples are : 
 
 Strength of Magnetization. 
 
 01 > 02 > 03 
 
 Topaz, Paramagnetic, a c b 
 
 Anhydrite, Diamagnetic, a b c 
 
 Barite, Diamagnetic, cab 
 
 Epsomite, Diamagnetic, c b a 
 
ELECTRICAL CHARACTERS. 175 
 
 thermal conductivity or optical principal vibration directions. In 
 triclinic crystals there are no fixed relations. 
 
 Practically no satisfactory determinations have been made of 
 absolute values of coefficients. 
 
 TRANSMISSION OF ELECTRIC RAYS. * 
 
 Electric waves differ from light waves only in their vastly greater 
 length, they travel with the same velocity and exhibit similar phe- 
 nomena. Many substances opaque to light waves are transparent 
 to electric waves, and in this lies the hope of a series of tests 
 for optically opaque crystals corresponding to the series for optic- 
 ally transparent crystals. 
 
 Professor Bose | describes an apparatus, essentially an electric 
 polariscope, the polarizer and analyzer consisting of wire gratings, 
 made by winding fine copper wire 2 mm. diameter around a thin 
 sheet of mica (about 25 lines per cm.) ; the mica pieces are then 
 dipped in melted paraffine, after which a round disc is cut from the 
 sheet. The electric waves are produced by a small Ruhmkorfif 
 coil causing oscillatory discharges between two small (ij^ cm.) 
 metallic spheres ; beyond these in the same tube is a convex lens 
 with a spark gap at its principal focus, then follow in order the 
 grating polarizer, the crystal, the grating analyzer, a modified 
 coherer, and a connected, distant d'Arsonval galvanometer. 
 
 When the gratings are crossed no current is shown, the in- 
 troduction of the crystal produces a current which is shown by 
 the throw of the needle reflected by a mirror upon a scale. When 
 the principal vibration directions of the crystal coincide with those 
 of the gratings no current passes. Crystals of moderate size are 
 successfully tested by this apparatus. 
 
 ELECTRICAL CONDUCTIVITY. 
 
 Electrical conductivity, although varying between very wide 
 limits in different substances,f appears to be dependent much more 
 
 * The Polarization of Electric Rays by doubly refracting Crystals. ]. C. Bose, Jour. 
 Asiatic Soc., Bengal, LXIV., 291, 1895. 
 
 f No substance possesses absolute electrical resistance ; practically, however, con- 
 ductivity may be considered to be limited to the metals ; some metalloids ; most 
 sulphides, tellurides, selenides, bismuthides, arsenides and anlimonides, some of the ox- 
 ides ; and, at higher temperature, a few haloids. 
 
176 CHARACTERS OF CRYSTALS. 
 
 upon the constitution of the chemical molecule than upon the 
 crystalline structure. A certain dependence upon crystallographic 
 direction has, nevertheless, been observed in a few substances. 
 
 The principal experiments are those of Wortman,* and the more 
 recent series by Beijerinck.f Both used essentially the same 
 method, in which a prism of known dimensions was introduced 
 into a direct weak current, the strength of which was varied by re- 
 sistances and the deviation observed in a galvanometer. 
 
 Good contact was obtained by using tough copper amalgam or 
 sometimes bright sheet lead or simply graphite rubbed on with a 
 lead pencil. Natural faces were cleaned with acid, caustic soda, 
 water and alcohol, and sometimes even with hydrofluoric acid. 
 
 To study the effect of temperature the substance was heated in 
 a small air bath made with double walls of asbestos and covered 
 with copper, thus giving a very regular distribution of temperature 
 without thermo-electric effects. 
 
 The principal results bearing upon crystal structure are as fol- 
 lows : 
 Isometric Crystals. 
 
 Magnetite, the electrical conductivity is essentially alike in all 
 directions. 
 Tetragonal and Hexagonal Crystals. 
 
 Hematite,J hexagonal, the electrical conductivity parallel to c 
 is essentially twice that normal to c and for any direction making 
 the angle a with c\ the resistance is ^ = w a sin a a + w c cos 2 a in which 
 <o a and w c are the resistances normal and parallel to c. 
 
 Cassiterite, tetragonal, and zincite, hexagonal, give conformable 
 results. 
 Orthorhombic Crystals. 
 
 Marcasite and bismuthinite, show the greatest and least resis- 
 tances parallel to two of the axes a y b, c. 
 
 That is, the few tests recorded show that the electrical conduc- 
 tivity of crystals conforms to the thermal conductivity. 
 
 THERMOELECTRIC CURRENTS. 
 
 If a metallic circuit is made by soldering together one end of 
 each of two rods of different metals and connecting the other ends 
 
 * Mem.de la Soc. d. hist. Nat. de Geneve, XIII., 1853. 
 
 f Ueber das Leitungsvermogen der Mineralien fur Eiektricitat. F. Beijerinck. 
 Neues Jahrb. Min., Beil. Bd., XI., 403, 1896-7. 
 
 {H. Backstrom. Ofv. d. K. Vetenskaps-Ak. Fork,, 1894, No. 10, 545. 
 
ELECTRICAL CHARACTERS. 177 
 
 by wire, heating or cooling the junction will develop an electric 
 current the strength of which will depend upon the change of 
 temperature and upon the metals used. 
 
 In a crystal the electromotive force of the current in part de- 
 pends upon the crystallographic direction, and a thermoelectric 
 current may be produced : 
 
 (a) By coupling rods cut in different directions from the same 
 crystal and heating the junction. 
 
 (b) By inserting a rectangular parallelpipedon longitudinally in 
 a metallic circuit the sides being kept at a constant equal tempera- 
 ture but the ends differing. For example, Backstrom* placed 
 hematite between two sheet-copper boxes, through one of which 
 water was flowing and through the other steam. The boxes were 
 connected by wires with a Lippman capillary electrometer. 
 
 (c) The end surfaces may be held at the same temperaturef and 
 two opposite side faces at different temperatures. 
 
 Friedel J clamped three plane parallel plates between two plat- 
 inum wires, touching equal spaces on the crystal, connected the 
 plates with a galvanometer and placed them in a space uniformly 
 heated by a water bath. 
 
 Slight impurities appear to produce very notable changes in the 
 thermoelectric results. 
 
 DIELECTRIC INDUCTION IN CRYSTALS. 
 
 A crystal suspended in an electrostatic field develops in an al- 
 most inconceivably short period an electric polarity, and the crys- 
 tal tends to assume a position in which the lines of force and the 
 direction of maximum induction are parallel. 
 
 According to the experiments of Root|| the directions of maxi- 
 mum and minimum induction are respectively the principal vibra- 
 tion directions c and a in light transmission. In these experiments 
 circular plates of tourmaline, quartz, and calcite, about 10 II mm. 
 in diameter and cut parallel to the optic axis, were attached by a 
 little drop of glue to a silk thread and suspended between the 
 vertical plates of a condenser charged by a rapidly alternating 
 
 * Last cit. 
 
 f Liebisch, Grundrissder Phys. Kryst., 217. 
 \ Ann. de Chim. et de Phys., 1869, XVII., 79. 
 $ Less than 0.0000821 second, according to Root. 
 || Poggendorfs Annalen, CLVIII., I, 425, 1876. 
 
CHARACTERS OF CRYSTALS. 
 
 current. The vibration through an arc of 10-20 minutes thereby 
 produced in the plate was reflected to a telescope two meters dis- 
 tant, and its period determined both when the direction of maxi- 
 mum induction was vertical and when horizontal. The quotient 
 of the former by the latter gave a basis of comparison. Aragonite 
 and topaz were also tested. For instance, 
 
 
 Period of vibration wher 
 indue 
 A, Vertical. 
 
 i direction ot maximum 
 tion is 
 B. Horizontal. 
 
 Quotient of 4 
 /> 
 
 Aragonite 
 
 
 Calcite 
 
 4.1125 
 4.0500 
 2.240 
 
 3-768 
 3.668 
 
 2.175 
 
 I.OpI 
 1.105 
 1.022 
 
 The condenser Fig. 289 consisted of a ring C of gutta percha 
 serving as a frame for the two vertical brass plates A and B. 
 Through C passed the glass tube D with a torsion head to which 
 
 the discs were hung between A 
 and B by a single thread of silk. 
 Tile motions of the disc were 
 observed through the opposite 
 glass windows E. A voltaic cur- 
 rent passing through a commu- 
 tator produced alternately -f and 
 charges in A and B in some 
 instances as rapidly as 6,000 al- 
 ternations per second. 
 
 As all dielectrics have some 
 degree of conductivity the elec- 
 tric polarity is apt to be modified 
 
 FIG. 289. 
 
 by the more slowly developed 
 conductivity phenomena. 
 
 The effect of conductivity in use of the constant current was 
 shown by suspending a calcite plate with c horizontal, c was at 
 first axial but in less than two minutes turned reversing the poles, 
 whereas with alternating current c was equatorial. 
 
 The strength of the induction in any direction is indicated by 
 the so-called dielectric constant* which may be determined by 
 
 * Maxwell assumes to be proportionate to the square of the constant A in Cauchy's for- 
 
 n 
 
 mula for dispersion, n = A-\ or, since if A = oo n = A, to the square of the index of 
 
 refraction for light of infinite wave-length vibrating parallel the given direction. 
 (Groth, Phys. Kryst., III. Ed., 194.) 
 
ELECTRICAL CHARACTERS. 179 
 
 the method of Bolzmann * in which the attractions exerted are 
 measured as follows : A small sphere of the dielectric to be tested 
 is attracted by a fixed charged metal sphere and the amount of its 
 deviation h determined by the deflection of a mirror; the sphere is 
 then replaced by another sphere of the same diameter but coated 
 with tin foil and the deviation h' of this produced by the same 
 charge in the metal sphere is determined ; then, according to Bolz- 
 mann, if s is the dielectric constant 
 
 h' 
 
 either for uniaxial crystals or for biaxial crystals in which an axis 
 of dielectric symmetry is parallel to the lines of force. 
 
 The constants may also be determined by comparison ot 
 capacity.f 
 
 From the dielectric constants in the principal directions the con- 
 stants for any direction may be calculated.^ 
 
 If D a D b D c are the principal constants the constant D r for any di- 
 rection is : 
 
 D a cos 2 (a.f) + D cos 2 (b r) -*- D c cos 2 (c-r) 
 
 The conductivity on the plane surface of a dielectric crystal theoretically must con- 
 form to the dielectric constants in different directions. 
 
 Wiedemann 's$ Experiment. An insulated needle was fastened in proper holder in 
 normal contact to a crystal face, previously covered with a poorly conducting powder 
 such as lycopodium or minium, and positively charged by contact with the knob of 
 a leyden jar. The powder near the point is tossed aside in shapes which are elliptical 
 or circular according to surface. The results can be made permanent by collodion. In 
 general the longest axis is in the vibration direction of the light of greatest velocity. 
 
 de Senarmonf s\\ Experiments. de Senarmont coated the crystal with tin foil except 
 
 that a circular hole was cut in the foil covering the face to be studied. The foil was 
 
 grounded the crystal was placed in partial vacuum opposite a point of brass wire from 
 
 which positive electricity was discharged following a direction assumed therefore to be 
 
 that of easiest conductivity. The tests are made in the dark. 
 
 * Ber. Ak. Wien. (2), LXX., 342, 1874, also Ch. Borel, Arch. Soc. Fhys. et nat. de 
 Geneve (3), XXX., 131, 1893. 
 
 f Liebisch, Grundriss der Phys. Kryst., 230. 
 
 J A. Lampa, Ber. Ak. Wein., CIV., 1179. 
 
 \ p gg. Ann., 76, 404, 1849. 
 
 || de Senarmont, Ann. de Chim. et de Phys. (3), 28, 257, 1850. 
 
i8o CHARACTERS OF CRYSTALS. 
 
 PYRO-ELECTRICITY. 
 
 Equal positive and negative charges of electricity are developed 
 at different points 'or poles of certain crystals during a uniform 
 change of temperature. 
 
 A temperature change of at least 70 to 80 C. is desirable. 
 Usually the crystal is heated in an air bath to a uniform tempera- 
 ture, then drawn quickly once or twice through an alcohol flame 
 to remove any electricity occurring on the surface, and then brought 
 into a cooler place and allowed to cool. 
 
 If heating injures the crystal it rr.ay be cooled from the room 
 temperature by a freezing mixture.* 
 
 During the cooling of the crystal the positive charges collect at 
 the so-called antilogue poles, and the negative charges at the ana- 
 logue poles, f and may be distinguished by their effect on other 
 electrified bodies. For instance, a cat's hair rubbed between the 
 fingers becomes positively electrified and is attracted by the analo- 
 gous pole and repelled by the antilogous pole, or as in the method 
 of Hankel,J the poles may be touched by a platinum wire care- 
 fully insulated and worked by a system of levers, and the charge 
 conducted to an especially constructed gold leaf electrometer. 
 
 Kundt's method is, however, most generally employed and con- 
 sists in blowing upon the cooling crystal a fine well dried || mix- 
 ture of equal parts of powdered sulphur and red oxide of lead. 
 The nozzle of the bellows is covered by a muslin net and, in 
 passing through, the sulphur is negatively electrified and is attracted 
 by the positive poles coloring them yellow, while the minium is 
 positively electrified and is caught by the negative poles coloring 
 them red. By pressing the crystal upon sticky paper a permanent 
 record can be obtained. The dust should fall evenly and the bel- 
 lows be held far enough away to prevent direct action of the blast. 
 
 The figures here shown^[ represent crystals dusted with sulphur 
 and minium during cooling, the darker dotted portions showing 
 the analogue poles, the hatched portions, the antilogue poles. 
 
 * Snow or ground ice and salt ; 3 pts. snow or ground ice with I pt. H 2 SO 4 ; 2 pts r 
 snow or ground ice with 3 pts. crystals CaCl 2 . 
 
 \ With rising temperature these are reversed. 
 
 \ G. W. Hankel, Inaug. Dissertation. 
 
 \ Fogg. Ann., CXXXVI., 612, 1862; Weid. Ann., XX., 592, 1883; XXV., 145. 
 1886. 
 
 || Dry over H 2 SO 4 in a vessel from which the air has been partially exhausted. 
 
 \ Drawn from the colored plate III. in Groth, Phys. Kryst., III. ed. 
 
ELECTRICAL CHARACTERS. 
 
 181 
 
 Fig. 290 represents a tourmaline crystal, Fig. 291 a calamine 
 crystal, in both of which polarity is shown with reference to the 
 
 FIG. 290. 
 
 FIG. 292. 
 
 vertical axis. Fig. 292 represents a boracite crystal, showing 
 polarity with reference to four axes, and Fig. 293 a quartz crystal 
 with three axes of polarity. Fig. 294 shows a basal section of 
 quartz. 
 
 FIG. 293. 
 
 FIG. 294. 
 
 FIG. 295. 
 
 Quantitative examinations were made by Gaugain* by coating 
 the ends of a cylinder of the crystal with tin foil, connecting one 
 end to the ground and the other to some form of self-discharging 
 electroscope. The number of discharges are counted and serve 
 as a relative measure especially if the capacity of the electroscope 
 is small. In this way it was shown that with tourmaline the 
 amount is independent of the time of cooling, is alike for the 
 same change of temperature in either direction, and is independ- 
 ent of the length, but proportionate to the cross section and to the 
 difference between the end temperatures. 
 
 THEORY OF LORD KELVIN. 
 
 Lord Kelvin theorized f that tourmaline, in which pyroelec- 
 tricity was first observed, is internally in a state of uniform electric 
 polarity, and that, therefore, at the surface there should be an 
 
 * Ann. de Chim. et de Phys., III., 57. 
 
 fMath. phys. papers Sir. William Thomson, I., 315. 
 
1 82 CHARACTERS OF CRYS1ALS. 
 
 electric charge of uniform surface density. In a medium not 
 perfectly insulating such as damp air there is gradually formed by 
 induction an electric layer completely neutralizing the charge in 
 the tourmaline, that is making it essentially non-electric. 
 
 But if the tourmaline is heated or cooled the strength of the 
 internal polarization is altered, while that of the electrified layer of 
 outer medium is more slowly altered and the effect is electric polarity. 
 
 Riecke * confirmed this theory by showing that, in a fairly per- 
 fectly insulating medium, tourmaline remained electrified twenty 
 to thirty hours, after cooling to its normal temperature. The heated 
 tourmaline while still non-electrified was placed under an air pump 
 over a gold leaf electroscope and the air slightly rarefied. 
 
 PIEZO-ELECTRICITY. 
 
 Electric charges may be developed by pressure, for instance, 
 calcite pressed between the fingers becomes positively electrified, 
 tourmaline compressed in the direction of c shows a positive 
 charge at the antilogue end and a negative charge at the analogue 
 end or precisely the charge which would result from cooling ; 
 whereas, if the pressure is removed and the crystal allowed to 
 expand the charges are reversed conforming to rising temperature. 
 The quantity developed is always proportionate to the pressure. 
 
 If strains are developed by unequal heating electrical charges 
 may be developed as, for instance, by standing a basal section of 
 quartz upon a hot centrally placed metal cylinder, the distribution 
 of charges conforming to the effect of cooling a complete crystal, 
 whereas, with a hot metal ring the charges are reversed. 
 
 The methods for detecting the charges are the same as for 
 pyroelectricity. Fig. 295 shows the distribution of the charges in 
 a basal section of quartz produced by pressure in the direction of 
 the arrows which conforms evidently to Fig. 294, or the effect of 
 cooling. 
 
 For quantitative examination J. and P. Curie used the following 
 method.t 
 
 The crystal is cut in the form of a rectangular parallelopipedon, 
 Fig. 296, and two opposite faces, A, B, covered with tin foil. One 
 of these, A, is grounded ; the other, B, is connected with one of 
 the plates of a condenser C of known capacity, and also with one 
 
 * E. Riecke, Wied, Ann. 28, 43, 31, 889, 40, 264, ^9,421. 
 fCompte Rendu, XCL, 294, 383 ; XCII., 350. 
 
ELECTRICAL CHARACTERS. 
 
 183 
 
 of the couples 55 of a Thomson-Mascart electrometer. The other 
 
 plate of the condenser is grounded, and the couple 5' 5' of the 
 
 electrometer is connected 
 
 with one pole of a Daniell 
 
 cell, the other pole being 
 
 grounded. 
 
 The relation between the 
 pressure employed and the 
 electricity developed is de- 
 termined as follows : Let 
 D denote the potential of 
 the Daniell cell, C the ca- 
 pacity of the condenser, and 
 .c the capacity of the system, 
 
 FIG. 296. 
 
 consisting of the plate B, the couple 55 and the conductors. 
 
 For some pressure P t in the direction of the arrows, the needle 
 of the electrometer will be at zero and the entire system charged 
 to a potential D. That is, the pressure will have developed* a 
 quantity Q=(C+c)>. 
 
 If the condenser is removed the pressure necessary to bring the 
 needle again to the zero position will be P t and the quantity de- 
 veloped will be Q' = cD, hence for the pressure P F the 
 quantity developed is Q Q' = CD, that is a quantity sufficient to 
 charge a condenser of known capacity C to a known potential D. 
 
 THEORY OF PYRO- AND PIEZO-ELECTRICITY.! 
 
 Whenever the volume of a crystal is altered either by a tempera- 
 ture change or by mechanical pressure a portion of the heat 
 energy or mechanical energy may be converted into electric en- 
 ergy, which, in poorly conducting crystals, will frequently be mani- 
 fested by the accumulation of positive and negative charges at 
 different points or poles. 
 
 Conversely, as shown by Lippmann,J a section which becomes 
 for instance, positively electric by normal pressure must, if charged 
 with positive electricity, expand in the direction of the normal. In 
 other words, the electric charges are functions of the change in 
 ^volume. 
 
 * Quantity equals capacity times potential. 
 fW. Voigt, Wicd. Ann.,UV., 701, 1895. 
 J G. Lippman, Ann. (.him. et Phys,, (5) XXIV, 145. 
 
1 84 CHARACTERS OF CRYSTALS. 
 
 J. and P. Curie devised* a delicate apparatus for measuring the 
 force of such expansions and the measured results for quartz, 
 checked almost absolutely with the calculated results. 
 
 Crystals symmetrical to a central point (Groups 2, 5, 8, 13, 15, 
 21, 22, 25, 27, 29, 30, 32) cannot develop opposite charges at the 
 ends of a diameter. The apparent exceptions obtained by Harikel 
 and others have not all been explained, but may in part be due to 
 influences dependent on the method of conducting the operation, in 
 other cases, barite, for instance, they have been ascribedf to twin 
 structure. 
 
 EFFECT OF ELECTROSTATIC FIELD UPON OPTICAL CHARACTERS. 
 
 Kundt J coated two of the sides, parallel c, of a quartz parallel- 
 opiped, with tin foil, connected these with the electrodes of a Holz 
 machine and found that the circular interference rings became 
 elliptical, the minor axes of the ellipses being parallel to the direc- 
 tion of expansion produced by the charge. Pockels shows that 
 the change is too great to be simply a consequence of the expan- 
 sion and must be in part due to a direct influence of the electric 
 force on the light motions. 
 
 * Brief description Liebisch, Grundriss der Phys. Kryst., 481. 
 
 t Beckenkamp, Zeit. f. Kryst., XXVII., 85. 
 
 % Wied. Ann., XVIII., 228, 1893. 
 
 \ F. Pockels, Abh. d. k. Ges. a. Gottingen, XXXIX., 1-204. 
 
CHAPTER XIV. 
 
 ELASTIC AND PERMANENT DEFORMATION OF 
 CRYSTALS. 
 
 In an elastic substance the distance apart and relative position 
 of the particles may be changed by mechanical force, and up to a 
 certain so-called " elastic limit " the particles will, on removal of 
 the force, regain their former position. Such a change is called an 
 elastic deformation or form-alteration. 
 
 Any strain or pressure in excess of the so-called " elastic limit " 
 produces a permanent change of form. 
 
 HOMOGENEOUS ELASTIC DEFORMATION. 
 
 The only recorded experiments by pressure on all sides were made 
 upon halite by aid of a piezometer.* The cubic compression 
 coefficients have, however, been calculated by Voigt for a number 
 of species. The same five classes result as with the homogeneous 
 deformation produced by changes of temperature. 
 
 The alteration produced by a change of temperature cannot be 
 exactly compensated by uniform pressure on all sides except in the 
 case of isometric crystals. f 
 
 ELASTIC DEFORMATION DUE TO PRESSURE IN ONE 
 DIRECTION. 
 
 Observation shows that the extension or compression of any rod 
 of length /, breadth b and thickness /, produced by a weight W 
 
 WL 
 
 acting in the direction of the length, is -rr E, in which E is a char- 
 acteristic factor called the coefficient of extension. Obviously if 
 W, /, b and / are unity, the extension becomes equal E, that is the 
 coefficient of extension is the extension produced upon a unit rod 
 by a unit weight. 
 
 *R6ntgenand Schneider, Wied. Ann., XXXI., 1000, 1887. 
 f Liebisch, Phys. Kryst, 1891, 576. 
 
1 86 CHARACTERS OF CRYSTALS. 
 
 The most convenient method of obtaining the extension coeffi- 
 cient for any direction in a crystal is by cutting a thin rod R, Fig, 
 297, of rectangular cross section, in the given direction, supporting 
 
 it, as shown, upon the two rests 
 A and B, and loading by a weight 
 W connected with a centrally 
 placed knife edge. The deflec- 
 tion of the central section pro- 
 duced by the weight is deter- 
 mined by Koch as follows:* A 
 
 FIG 2 rectangular glass prism P is 
 
 placed just below and with one 
 
 face parallel to the rod, and another face vertical. Monochro- 
 matic light is reflected upon the hypothenuse face, and, very much 
 as in the Fizeau apparatus, p. 167, interference bands are pro- 
 duced by the rays reflected from the lower surface of the rod R 
 and the upper surface of the prism P, and are viewed by a horizon- 
 tal telescope through the vertical face. 
 
 As the rod is bent an interference band coincides with the cross 
 hair whenever the thickness of the air film is an odd multiple of J^A 
 for the light used, that is the depression corresponding to the inter- 
 val between two successive bands coincident with a cross hair is for 
 sodium light 589.5 millionths of a millimeter. 
 
 Denoting the central depression produced by any weight W 
 by n the coefficient of extension in the direction of the length of 
 the rod is given by the formulaf 
 
 SURFACE OF EXTENSION COEFFICIENTS. Nine classes result from 
 the measurements of extension coefficients of crystals. 
 
 1. Isometric crystals all yield an extension surface symmetrical to- 
 nine planes and thirteen axes. The four central sections parallel 
 to the octahedral planes are circles. The cubic axes are direc- 
 tions of maximum or minimum extension and the octahedral axes 
 are correspondingly minimum or maximum. 
 
 2. Hexagonal. Classes 19, 22, 23, 24, 25, 26, 27. The exten- 
 sion surface is a surface of rotation around c . Each vertical cen- 
 tral plane and the horizontal central plane are planes of symmetry, 
 
 * Wied. Ann., XVIIL, 325, 1883. 
 f Groth, Phys. Kryst., III. ed., 203. 
 
ELASTIC DEFORMATION. 
 
 187 
 
 3. Hexagonal. Classes 18, 20, 21. The extension surface has 
 one circular section horizontally through the center. In general 
 the shape is that of a rhombohedron 
 
 with rounded edges and angles symme- 
 trical to three vertical planes and to a 
 ternary vertical axis. Fig. 298 shows 
 the section of the surface for calcite made 
 by a principal section normal to a rhombic 
 face, the dotted lines being the directions 
 of greatest and least extension. Fig. 299 shows the section nor- 
 mal to this. 
 
 4. Hexagonal. Classes 1 6 and 17. Like No. 3 but without the 
 three vertical planes of symmetry. 
 
 5. Tetragonal. Classes n, 12, 14 and 15. 
 
 6. Tetragonal. Classes 9, 10 and 13. 
 
 7. Orthorhombic. The extension surface symmetrical to the three 
 
 FIG. 298. FIG. 299. 
 
 FIG. 300. 
 
 FIG. 301. 
 
 FIG. 302 
 
 pinacoidal planes. Figs. 300, 301, 302, show these sections for 
 barite. 
 
 8. Monoclinic. The extension surface symmetrical to the clino- 
 pinacoid and to the axis b. 
 
 9. Triclinic. No investigations are as yet recorded. 
 
 EFFECT OF PRESSURE IN ONE DIRECTION UPON THE OPTICAL CHAR- 
 ACTERS. 
 
 If a rectangular parallelopipedon* C, Fig. 303, is compressed be- 
 tween the parallel jaws of a screw press f in general the velocity of the 
 
 ight vlibrating parallel to the 
 pressure is increased. If the pres- 
 sure is uniform the convergent 
 light effects are studied by bring- 
 ing the specimen over the stage 
 of a polariscope, but if at all un- 
 
 FIG. 303. 
 
 * Pockels used rectangular parallelepipeds 13 mm. high, 25.5 broad and thick and 
 compressed from 30 k. g. with quartz to 2 k. g. for sylvite per sq. mm. 
 
 fFor press recording pressure exerted see Groth, Phys. Kryst., III. ed., 215. 
 
1 88 CHARACTERS OF CRYS1ALS. 
 
 equal, determinations in parallel light of vibration directions, faster 
 and slower ray and strength of double refraction are more safe. 
 
 Amorphous Substances. 
 
 Glass* gives under compression a negatively uniaxial figure the 
 optic axis parallel to the direction of pressure ; that is c parallel a, 
 or the ray vibrating parallel to the pressure is the more rapidly 
 transmitted. 
 
 Optically Isotropic Crystals. 
 
 In general become biaxial but when the pressure is applied 
 parallel to the cubic (quaternary) or octahedral (ternary) axes the 
 crystal becomes uniaxial. 
 
 Optically Uniaxial Crystals (Denoting by a, b, c the principal vibra- 
 tion directions and also velocities). 
 
 In positive crystals c is parallel to c and a = b > c, while in nega- 
 tive crystals c is parallel to a and a > b = c- Pressure parallel to 
 c, in positive crystals, will cause c to approach a and possibly to 
 equal (isotropy) or exceed it (negative) ; while in negative crys- 
 tals pressure will only make a still greater. In either positive or 
 negative crystals pressure!, parallel a, will make the three veloci- 
 ties a, b, c unequal, that is develope a biaxial structure. 
 
 Optically Biaxial Crystals. 
 
 The relations are more complex. Pressure perpendicular to 
 the plane of the optic axis decreases the axial angle in positive 
 crystals, and the crystal may be, as with heat, p. 170, for a certain 
 pressure uniaxial or the axes may pass into a plane at right angles 
 to their former position ; with negative crystals the axial angle is. 
 increased. Pressure parallel to the obtuse bisectrix will in positive 
 crystals increase the axial angle and in negative crystals decrease it. 
 
 CLEAVAGE. 
 
 In crystals the elastic limit varies with the direction, that is, in 
 certain directions there is a weaker cohesion of the particles than 
 there is in others, and many crystals tend to separate or cleave 
 
 * If the pressure is not uniformly applied at all points of the opposite surface the 
 effect may resemble biaxial lemniscates. 
 
 f The centre of each ring system thus developed in quartz is colored, that is parallel 
 to each optic axis there is a rotation, and it has been shown that in these directions 
 t wo elliptically polarized waves are transmitted. 
 
PERMANENT. DEFORMATION. 
 
 189 
 
 along more or less smooth plane surfaces normal to these direc- 
 tions of weaker cohesion. This is undoubtedly due to the fact 
 that whatever form of reg- 
 ular grouping of particles 
 may exist in a crystal, ad- 
 jacent molecular planes in 
 certain directions will be 
 further apart than adjacent 
 planes in other directions, 
 and, therefore, held together 
 with less force. For in- 
 stance, in Fig. 304, it is evi- 
 dent that the closer the par- 
 
 FIG. 304. 
 
 tides are along any direction ab t cd, ef y gh the greater is the distance 
 to the adjacent row a'&', c'd', e t f t ,g t h l , and as undoubtedly the fur- 
 ther apart the particles are the weaker their cohesion, cleavage 
 will be most likely to occur parallel to the planes in which the 
 crystal molcules are most closely* packed. 
 
 The same general relations exist for elastic and permanent de- 
 formation.t Cleavage will therefore be expected normal to the di- 
 rection of greatest elastic extension, therefore parallel to faces with 
 simple indices. The possible cleavage directions are : 
 
 System. Cleavage Form. 
 
 i Isometric. Hexahedron. 
 
 Octahedron. 
 
 Rhombic Dodecahedron. 
 Hexagonal. Basal Pinacoid. 
 
 Hexagonal Prism. 
 Rhombohedron. 
 Hexagonal Pyramid (rarely). 
 Tetragonal. Basal Pinacoid. 
 
 Tetragonal Prism. 
 Tetragonal Pyramid (rarely). 
 Orthorhombic. Pinacoid. 
 
 Prisms or Domes. 
 Pyramid. 
 
 Monoclinic. Clinopinacoid. 
 
 Basal pinacoid. 
 Orthopinacoid. 
 Orthodome. 
 Prism. 
 
 Pyramid (rarely). 
 
 In triclinic crystals there can only be equally easy cleavage parallel to one plane. 
 Nevertheless if there is a direction of nearly perfect cleavage one of the principal vi- 
 bration directions will be approximately normal to this and the two others parallel 
 thereto. 
 
 * The densest planes are usually the faces of common forms with simple indices, 
 f Groth, Phys. Kryst., III. ed., 229. 
 
 Examples. 
 Galenite. 
 
 Fluorite, diamond. 
 Sphalerite. 
 Beryl, pyrosmalite. 
 Nephelite, apatite. 
 Calcite, siderite. 
 Pyromorphite. 
 Apophyllite. 
 Rutile, wernerite. 
 Scheelite. 
 Anhydrite, topaz. 
 Barite. 
 Sulphur. 
 
 Orthoclase, gypsum. 
 Muscovite, orthoclase. 
 Epidote. 
 Epidote. 
 
 Pyroxene, amphibole. 
 Gypsum. 
 
CHARACTERS OF CRYSTALS. 
 
 GLIDING PLANES. 
 
 In those directions ef,gh, Fig. 304 in which the adjacent molecular 
 planes are closest together and therefore most strongly held to- 
 gether it sometimes happens that under tangential pressure the 
 particles glide or rotate, without separation, into a new position of 
 equilibrium. 
 
 This phenomenon has been observed in calcite, pyroxene, anhy- 
 drite, stibnite and other minerals. 
 
 In calcite for instance let ab'c'd Fig. 305 represent a section 
 through the optic axis normal to a rhombic face (see abed, Fig. 
 222) then will ac f t the optic axis, make an angle of 63 45' with ad t 
 the direction of y 2 R, and an angle of 45 23^' with ab 1 the 
 short diagonal of a cleavage face. 
 
 FIG. 305. FIG. 306. 
 
 If pressure is applied gradually in the direction of the arrows, 
 that is parallel to ad t there will be produced, about some plane ef 
 parallel to ad, a gliding or rotation of the particles until eV has taken 
 the position eb at which feb = 180 feb' = 70 5 1 y 2 ! 
 
 In this rotated portion the optic axis will be bd r making with 
 the optic axis ac' of the unchanged portion an angle of 52 30' as 
 shown. 
 
 Instead of rotation about a single plane ef there may be rotation 
 about several parallel planes as in Fig. 306 producing twin lamellae. 
 
 If a little calcite cleavage is placed, as in Fig. 307, with an edge 
 ad of a larger angle resting upon a steady support and the blade of 
 a knife is pressed steadily down at some point i of the opposite 
 
PERMANENT DEFORMATION. 
 
 191 
 
 edge the portion of the crystal between z and c will be slowly 
 pushed as indicated into a new position of equilibrium as if by rota- 
 tion about fghm, or y 2 R, until the new face^vr'/z and the old face 
 gch make equal angles with fghm. If carefully done gcfh will be a 
 perfect plane , but more frequently it is step like and the rotated 
 portion is apt to separate at the'gliding plane. 
 
 FIG. 307. FIG. 308. 
 
 In the described phenomenon of gliding the particles evidently 
 assume new relative positions. If however a rod of ice of square 
 cross section is cut with one pair of long faces parallel to the optic 
 axis the other perpendicular to the optic axis and a bending pres- 
 sure, Fig. 308, applied normal to the axis there is no change in the 
 direction of the optic axis, indicated by the arrows, nor any especial 
 point at which the movement ceases ; that is the particles have evi- 
 dently slipped* without change in orientation. This has been 
 called TRANSLATION. 
 
 PARTING. 
 
 The planes along which a slipping has occurred, although pre- 
 viously directions of maximum normal cohesion may thereafter be 
 planes of easy separation or Parting,\ differing 
 from true cleavage, however, because in parting 
 the easy separation is limited to the planes of 
 actual molecular disturbance while true cleavage 
 is obtained with equal ease in any part of the 
 crystal. 
 PERCUSSION FIGURES. 
 
 If a rod with a slightly rounded point is pressed FIG. 309. 
 
 *Mugge, Neues Jahrb.f. Min., 1895, JI - 2I1 - 
 
 f Similar planes of easy separation may be due to other causes, for instance, 
 during the growth of a crystal the planes at certain intervals may be coated with dust 
 or fine lamellae of a foreign substance and later the crystal may grow further. This 
 may be repeated several times forming thus parallel planes of easy separation, e. g. t 
 capped quartz. 
 
192 CHARACTERS OF CRYSTALS. 
 
 against a firmly supported plate of mica and sharply tapped with 
 a light hammer, three planes of easy separation will be developed 
 as indicated by little cracks radiating * from the point, Fig. 309. 
 One of these is always parallel to the ortho axis b, the others are 
 at a definite angle x thereto of 53 to 56 in muscovite, 59 in 
 lepidolite, 60 in biotite, 61 to 63 in phlogopite. 
 
 In halite, in the same way, on cube faces a cross is developed 
 with the arms parallel to the diagonals of the face, while on an 
 octahedral face a three-rayed star is developed. 
 
 CORROSION AND ETCHING. 
 
 Liquids or gases do not dissolve or attack chemically a crystal 
 with equal rapidity in all directions.*]" 
 
 This may be shown experimentally by treating a sphere of the 
 crystal in a solvent as, for instance, a sphere of quartz in hydro- 
 fluoric acid ; or plates of different orientation may be placed for 
 an equal time in the same solvent and their decrease in thickness 
 compared. 
 
 ETCH FIGURES. 
 
 The attack of any liquid or gas upon any crystal face does not 
 commence at the same time at all points but proceeds from certain 
 points first and later from others. 
 
 From each point of attack the action proceeds with different 
 velocities in crystallographically different directions and if stopped 
 at the right time the face will be found to be pitted with little an- 
 gular etch cavities of definite shape. As these increase in number 
 and size they finally reach a stage when they either touch or are 
 separated only by little elevations, etch hills, the sides of which are 
 the faces of the etch cavities. 
 
 SHAPE OF ETCH FIGURES. 
 
 Provided the same conditions exist of solvent, time and tem- 
 perature, the etch figures developed on any one face of a crystal 
 will be of the same shape and in parallel position plane for plane. 
 On similar faces the etch figures will be alike, and on dissimilar 
 faces will be unlike. 
 
 * By pressure alone three cracks diagonal to these are developed. 
 
 f Growth and solution appear to be reciprocal and the predominating faces of 
 crystals growing in a solvent are the very planes which oppose greatest resistance to 
 solution in that solvent. 
 
PERMANENT. DEFORMATION 
 
 193 
 
 The figures produced by solutions of different strength are not 
 necessarily exactly the same, and with different solvents there 
 may be a still greater difference,* as strikingly illustrated by the 
 tests of Baumhauer f upon apatite with hydrochloric and sul- 
 phuric acids. Fig. 310 represents the etch figures in general, but 
 
 FIG. 310. 
 
 FIG. 311. 
 
 the basal planes treated with hydrochloric acid show side by side, 
 dark deeply etched figures, usually consisting of a negative third 
 order pyramid truncated by the base a, Fig. 311 ; and lighter less 
 deep figures, usually a positive third order pyramid, ft, Fig. 311. 
 
 With 100 per cent, acid the dark a figures are negative, but ap- 
 proximately second order pyramid, while the lighter images ft are 
 positive and also near the second order. If weaker solutions are 
 used, the figures are differently oriented. With sulphuric acid 
 still different orientations are obtained. 
 
 The following are the average values obtained for e, Fig. 31 1, for 
 different strengths, 100% HC1 being Sp. Gr. 1. 1 30 and 100% H 2 SO 4 
 aSp. Gr. of 1.836. 
 
 ioo 60 50 20 10 5 i 
 
 per cent. per cent. per cent. per cent. per cent. per cent. per cent. 
 
 a figures HC1 ( ) 2 7 2 o' ( ) 22 57 ' ( ) 2 o 4 8' ( ) l8 44 ' ( ) i82i' ( ) i8 5 ' ( ) I 7 34' 
 
 j3 figures HCl (+)2 7 io' (-f)28s6' (+)26 4 o' ( + ) 2 8 3 i' (+)28 3 i' ( )2 7 4 i' ( ) i 7 o 34 ' 
 
 figures H S S0 4 (+)i 3 7 ' (+) 8 35 ' (+)ito S (-) i939' (-) l6 S*' (-) '3 6 ' 
 
 SYMMETRY OF ETCH FIGURES. 
 
 Whatever the solvent, the etch figures conform to the symmetry 
 of the class to which the crystal belongs, and are rarely the limit 
 
 * For this reason it is not safe to assume any relation between the shapes of the 
 etch figures and of the crystal molecule. 
 f Ber. Ak. Milnchen, 1887 , p. 457. 
 
194 
 
 CHARACTERS OF CRYSTALS. 
 
 forms common to several classes. Although the first developed etch 
 figures may be the simple forms bounded by the planes of greatest 
 resistance to solution, these are soon modified by other planes 
 with complex indices, because the saturated solution in each cavity 
 
 FIG. 312. 
 
 FIG. 313. 
 
 is replaced much more slowly near the bottom than near the 
 border, and the attack is slower. The etch figures serve, there- 
 fore, as an important means (perhaps next to geometric form the 
 most important) for determining the true grade of symmetry of a 
 crystal. For instance, Figs. 312 and 3 1 3 show the shape and direc- 
 tion of the etch figures on cubes of galenite and sylvite, respec- 
 tively. Fig. 314 shows the planes and axes proper to class 32, the 
 highest grade of symmetry in the isometric system. Evidently the 
 etch figures of galena satisfy all of these, whereas those of sylvite 
 are not symmetrical to the planes of symmetry, but are to the axes. 
 That is, galenite belongs to class 32, sylvite to class 29. 
 
 FIG. 314. 
 
 FIG. 315. 
 
 Similarly Figs. 316 and 317 show the shape and direction of etch 
 figures on rhombohedra of calcite and dolomite, respectively. Fig. 
 315 shows the planes and axes of symmetry of class 21. The etch 
 figures of calcite satisfy all of these, while those of dolomite are 
 
PERMANENT DEFORMATION. 
 
 195 
 
 not symmetrical to the planes of symmetry, but are to the axes. 
 That is, calcite belongs to class 21, dolomite to class 17. 
 
 FIG. 316. 
 
 FIG 317. 
 
 Similarly, the etch figures of wulfenite Fig. 318 show the min- 
 eral to belong to class 10, and the etch figures of pyroxene Fig. 
 319 show it to belong to class 5. 
 MANIPULATION. 
 
 No general rule can be given, it being principally a question of 
 ease of solution. The operation may consist merely in a slight 
 pressure from a rag moistened with water, or the sliding of the 
 crystal across a moistened spot in smooth filter paper, or, more 
 frequently, the crystal is immersed for various periods in one of 
 the mineral acids, caustic alkalis, or the mother liquor, hot, warm 
 or cold, dilute or concentrated. High pressure steam, water, hydro- 
 fluoric acid, a pasty solution of caustic potash at 100 to 150 C., 
 or even a red-hot fusion of acid potassium sulphate and fluorite 
 have been used. 
 
 FIG 318. 
 
 FIG. 319. 
 
 Natural faces are usually most satisfactory for etching ; cleavages 
 are sometimes successfully etched. The etch figures are usually 
 microscopic, and if large are not apt to be sharp and distinct. 
 
196 
 
 CHARACTERS OF CRYSTALS. 
 
 CORROSION FACES. 
 
 The continued action of a solvent may dissolve away certain crys- 
 tal edges, replacing them by planes conforming in symmetry to 
 that of the crystal. These planes have been called corrosion faces. 
 
 HARDNESS. 
 
 The resistance of a smooth plane surface to abrasion is called 
 the hardness, and is commonly recorded in terms of a decimal 
 scale of ten common minerals selected by Mohs. In the light of 
 the later more exact methods it is seen that there is no even ap- 
 proximately common difference in hardness between neighboring 
 members* of this scale. 
 
 More exact methods, in which greater uniformity in pressure, 
 cutting edge, condition and position of surface, etc., have been 
 used by various experimenters. Exner f and others supported the 
 crystal upon a little carriage moving on a horizontal track. The 
 face to be tested was horizontal, the pressure p on the scratching 
 point of steel or diamond was vertical and the carriage was moved 
 back and forth by a weight W. The method preferred by Exner 
 was with ^constant to determine the pressure/ necessary to pro- 
 duce a visible scratch. 
 
 From the results obtained Exner constructed the so-called 
 hardness curves, for instance, Fig. 320 represents a dodecahedral 
 cleavage face of sphalerite, radii are laid off from the center, each 
 of a length proportionate to the value of /, for that direction and 
 by connecting their ends a symmetrical figure is obtained, show- 
 ing six directions of maximum and six of minimum hardness. 
 
 * Jaggar gives the following comparison of the hardness of the minerals of the Mohs 
 scale Amer. Journ. Sci., IV., 411, 1897 : 
 
 Scale of Pfaff by Jaggar by Rosiwal by 
 
 Mohs. boring with a boring with a grinding with a 
 
 diamond point. diamond point. standard powder. 
 
 1. Talc, laminated 
 
 2. Gypsum, crystallized 12.03 .04 .34 
 
 3. Calcite, transparent 15.3 .26 2.68 
 
 4. Fluorite, crystalline 37.3 .75 4-7 
 
 5. Apatite, transparent 53.5 1.23 6.20 
 
 6. Orthoclase, white cleavable 191. 25. 28.7 
 
 7. Quartz, transparent 254. 40. 149. 
 
 8. Topaz, transparent 459. 152. 138. 
 
 9. Sapphire, cleavable 1000. 1000. 1000. 
 
 -j- Harte an Krystallflachen, 38, 60, 103, 164. 
 
PERM AN EN T DEFORM A TION. 
 
 197 
 
 Similarly in Fig. 321 a basal plane of barite shows a figure of 
 lower symmetry. 
 
 The results of Exner show that the variations in hardness ob- 
 served in any crystal are dependent upon the cleavages. Faces 
 not cut by cleavage planes have constant hardness in all directions, 
 while faces intersected by cleavage planes show minimum hard- 
 ness parallel to the intersection with the cleavage plane and if the 
 cleavages are of unequal ease the minimum hardness is parallel to 
 the plane of easiest cleavage. So absolute is this relation that it 
 may be expressed algebraically. 
 
 Very similar apparatus was used by Franz and Turner,* the 
 mineral, however, being fixed and the point moving. 
 
 FIG. 320. 
 
 FIG. 321. 
 
 PfafTf drew a diamond splinter of definite shape 100 times back 
 and forth in one place, then shifted and repeated, widening the 
 groove. The loss in weight of the crystal for the same number of 
 movements of the diamond over the same area and with a constant 
 pressure serve as approximate values for hardness, i. e., hardness is 
 inversely as the loss in weight. PfafT also used a revolving diamond 
 point. For equally deep penetration the hardness was as the num- 
 ber of revolutions. 
 
 Jaggar J designed an attachment to the microscope, in which 
 the point of a cleavage tetrahedron of diamond is rotated at a uni- 
 form rate and under uniform pressure until a cut of uniform depth 
 is obtained (measured by focusing on the rulings of a Zeiss mi- 
 crometer glass, which is slightly inclined and follows the downward 
 movement of the diamond point). The number of rotations of 
 the point varies as the resistance of the mineral to abrasion by 
 diamond. 
 
 * Proc. Phil. Soc. Birmingham, 1886. 
 \ Ber. Ak. Munchen, 1883, 1884. 
 
 \ Microsclerometer, for determining Hardness of Minerals. T. A. Jaggar, Jr. 
 Amer. Jour. '., IV., Vol. IV., 399, 1897. 
 
198 CHARACTERS OF CRYSTALS. 
 
 The effect of grinding has also been used as a test for hardness, 
 for instance, Jannetaz and Goldberg* employed a so-called " Us- 
 ometer," consisting of a rotating grinding disc, upon which four 
 plates, the hardness of which is to be determined, were pressed by 
 normally acting weights, the loss of weight of each giving the rela- 
 tive hardness. 
 
 THE METHODS OF STATIC PRESSURE. 
 
 Following the definition of hardness given by Hertz, Auerbach 
 devised a method in which a piano convex lens of any mineral is 
 pressed against a horizontal plate of the same mineral. Both are 
 bent and touch throughout a circular space, and for some pressure 
 P the elastic limit is reached, at which there is produced, if the 
 mineral is brittle, a circular fissure, or, if the mineral is tough, a 
 circular permanent indentation. 
 
 According to Auerbach f the limit pressure P l upon a square 
 mm. of surface varies with the radius p of the lens, but the product 
 of P l into the cube root of this radius p is essentially constant and 
 may be called the Absolute Hardness, that is 
 
 On this basis he determines J the absolute values of the Moh's 
 scale to be: corundum, 1,150; topaz, 525 ; quartz, 308; orthoclase, 
 253; apatite, 237; fluorite, no; calcite, 92 ; gypsum, 14. 
 
 * Ass. franc, p. /. avanc. d. sc., IX., Aug., 1895. 
 
 fF. Auerbach. Wied. Ann., XLIIL, 61 ; XLV., 262, 277. 
 
 I Wied. Ann., LVIIL, 357. 
 
APPENDIX. 
 
 SUGGESTED OUTLINE OF A COURSE IN PHYSICAL CRYSTALLOGRPHY. 
 PRELIMINARY EXPERIMENTS. 
 
 In order to secure systematic work the following outline of experiments 
 has been prepared : 
 
 A. GEOMETRICAL CHARACTERS. 
 
 Preliminary practice should be given with crystal models in the study 
 of the thirty two classes and the Miller indices, also in the use of the 
 zonal equations and in stereographic projections. The hand goniometer 
 and the 743 model set of Krantz furnish excellent practice. 
 
 1. MEASUREMENT OF CRYSTALS. (Pp. 62-75.) 
 
 Practice should first be given in accurate adjustment and measurement 
 of a known angle, for instance the cleavage angle of calcite. 
 
 Crystals of known system may then be measured, the elementary planes 
 (p. 13) being chosen and, by a preliminary examination, the zones noted 
 and the angles selected which it will be necessary to measure in order to 
 determine the elements of the crystals and the indices of the faces. 
 
 Practice should, if possible, be given : 
 NJ (a) With horizontal circle goniometer. Babinet or Fuess. 
 
 (b) With vertical circle goniometer. Wollaston, Mallard or Groth. 
 
 *(^) With two circle (theodolite) goniometer. 
 
 2. DETERMINATION OF ELEMENTS, in part by spherical triangles (p. 6), in 
 part by special formulas. (Pp. 13-15, 34, 38, 45, 56.) 
 
 3. DETERMINATION OF INDICES AND ANGLES. In so far as possible by 
 zonal equations (pp. 1 7-20) but also by spherical triangles and spe- 
 cial equations. 
 
 4. CRYSTALS PROJECTION OR DRAWING. 
 
 In addition to the free -hand perspective and stereographic projections 
 needed in the measuremeut, accurate drawings by the various methods 
 should be prepared, especially : 
 
 (a) Stereographic projection. (Pp. 20-24.) 
 
 (b) Clinographic prospective. (Pp. 76-84.) 
 
 * Chas. Palache. Am. Jour. Sci. S. 4, Vol. II., p. 279, 1896. 
 
200 CHARACTERS OF CRYSTALS. 
 
 B. OPTICAL CHARACTERS. 
 >/5. DOUBLE REFRACTION AND POLARIZATION. (Pp. 97.100.) 
 
 With mounted calcite rhombs. (P. 97.) 
 
 (#) A ray of common light through one rhomb, is split into two rays of 
 approximately equal intensity which remain always in a principal section. 
 
 (b) If one of these rays is sent through a second calcite rhomb, the 
 two resultant rays vary in intensity as if due to a resolution of a vibration 
 parallel (or perpendicular) to the principal section of the first rhomb, 
 into components parallel and perpendicular to the principal section of the 
 second rhomb. 
 
 V 6. DETERMINATION OF INDICES OF REFRACTION, with monochromatic 
 
 light. 
 
 Graphic determination (pp. 86-90) of the direction of the refracted 
 ray and of the ray incident at the critical angle should be made with as- 
 sumed indices, after which measurements should be made both upon 
 singly and doubly refracting crystals by : 
 
 (a) Prism method. (Pp. 88-90, 103, 146-147.) 
 
 () Total reflection methods. (Pp. 90-95, 104, i47d.) 
 
 (c*) Displacement method. (Pp. 120.) 
 
 If singly refracting, the substance will yield the same index for all di- 
 rections. If doubly refracting the principal indices will only be yielded 
 in certain directions. 
 
 v 7. PROPUCTION OF POLARIZED LIGHT. (Pp. 105-110.) 
 
 Examination of Nicol and Hartnack prisms (double refraction and 
 total reflection), tourmaline pincers (double refraction and absorption), 
 glass plate polarizers (reflection or refraction). Polariscopes (combina- 
 tions of two polarizers), including older types Norremberg, and later 
 types of Universal Apparatus, and Polarizing Microscopes. 
 
 N 8. DETERMINATION OF EXTINCTION (VIBRATION) DIRECTIONS. (Pp. 117, 
 
 J 450 
 
 With polariscope (or polarizing microscope) and monochromatic par- 
 allel light, sections between crossed nicols are black when, Fig. 237, 
 tp = 90, 180, 270, 360, and midway between show brightest illumi- 
 nation. 
 
 With white light use a test plate (gypsum red) or special eye piece. Re- 
 sults are less accurate if there is marked dispersion. 
 
 The section, if too thick, will admit rays at directions not normal, pre- 
 venting perfect extinction. 
 
 Sections of enstatite and pyroxene illustrate respectively parallel and oblique ex- 
 tinction. 
 
APPENDIX. 201 
 
 9. INTERFERENCE OF PLANE POLARIZED RAYS. (Pp. 110-115.) 
 
 (a) With monochromatic parallel light and crossed nicols, extinction 
 (throughout revolution) takes place when J = A, 2/1, 3^, etc., and for 
 J = i/l, |-A, |^ there is the brightest illumination. 
 
 (b) With white parallel light Newton's colors of i, 2, 3, etc., 
 order result. 
 
 Wedges of quartz, gypsum, or mica, or plates of different thickness of any mineral 
 serve for illustration. 
 
 ,/ 10. DETERMINATION OF A AND n^ n. (Pp. 118-119.) 
 
 By the v. Federow mica wedge, quartz wedge or Babinet compensator 
 
 J is found and checked by color chart. Obtain n l n by formula J = 
 
 t(n^ ;/) , / being measured. 
 Use the sections of tests 8 and 9. 
 
 \/ ii. DETERMINATION OF VIBRATION DIRECTION OF FASTER RAY. (Pp. 
 118.) 
 
 By test plate of mica or gypsum or by wedge of quartz or mica. 
 Use the sections of tests 8 and 9. 
 
 12. OPTICALLY ISOTROPIC CRYSTALS. 
 
 Between crossed nicols, with either parallel or convergent light all 
 sections remain dark throughout rotation. The index of refraction is in- 
 dependent of the direction. 
 
 Use sections garnet, haiiynite, nosite, etc. 
 
 13. OPTICALLY UNIAXIAL CRYSTALS. 
 
 Models of positive and negative ray surfaces and indicatrices should be 
 studied. (Pp. 100-102.) 
 
 (a) Sections normal c between crossed nicols. In parallel light the sec- 
 tion (if not too thick) remains dark throughout rotation. In convergent 
 monochromatic light the arms of the cross remain fixed during rotation of 
 the plate, the rings are wider apart in thinner sections, and with white 
 light are colored in the order of Newton's colors. (Pp. 110-117.) 
 
 (b) Sections parallel c : interference hyperbolae in monochromatic light 
 and approximate indices by displacement. (Pp. 116.) 
 
 (c) Sections oblique to c. (Pp. no, 112, 116.) 
 
 (d) Determine character of ray surface by different methods. (Pp. 
 
 I2O-I2I.) 
 
 Sections of zircon, quartz, calcite, beryl, and apatite form a good series. 
 
 14. UNIAXIAL CYSTALS IN WHICH c is A DIRECTION OF CIRCULAR 
 POLARIZATION. (Pp. 122-130). 
 
 (a) Circular and elliptical polarization, with Fresnel rhombs. 
 () Sections normal c, with parallel light. 
 
202 CHARACTERS OF CRYSTALS. 
 
 Monochromatic, determine the direction and amount of rotation of 
 analyzer to produce extinction. 
 
 White, note the sequence of colors on rotation of the analyzer. 
 
 (V) In section normal c, with convergent light. (Pp. 127.) 
 
 Monochromatic, interference figure, giving the inter-wound spirals with 
 *^ undulation plate, or with superposed right and left sections. 
 
 White light, interference figures with colored center, changing on rota- 
 tion of analyzer in the same sequence as for parallel light. 
 
 (d) Sections parallel or oblique to c essentially conform to 13 () and 
 (c). 
 
 Use sections of quartz and cinnabar. Compare thickness of quartz wedges cut re- 
 spectively parallel and normal to c, which give the same color. 
 
 15. BIAXIAL CRYSTALS. 
 
 Models of positive and negative ray surfaces and indicatrices should be 
 studied. Given three principal indices of refraction for yellow light 
 construct the three optical principal sections. (Pp. 132-137.) 
 
 (a) In sections normal to the acute bisectrix. 
 
 1. In parallel light essentially as in tests 8 and 9. The relation of the 
 vibration (extinction) direction to any cleavage or crystalline outline 
 should be noted. (Pp. 137, 138.) 
 
 2. In convergent monochromatic light leminiscates depend upon thick- 
 ness, but the distance apart of the axial points does not. (Pp. 138, 139.) 
 
 3. In convergent white light determine character of dispersion. (Pp. 
 140-144.) Examine models of dispersion. 
 
 4. Determine character of Ray Surface. (Pp. 154.) 
 
 Sections of aragonite, niter, cerussite, barite, calamine, topaz, gypsum, titanite, di- 
 opside, orthoclase, borax, heulandite, muscovite and phlogopite illustrate all phases. 
 
 () In sections normal to an optic axis in parallel monochromatic light 
 there is a uniform illumination of the field, in convergent monochromatic 
 light will be seen rings and one dark arm, which rotates in opposite di- 
 rection to the stage. (Pp. 137-139.) 
 
 #16. ORIENTATION OF a, b and c, IN BIAXIAL CRYSTALS. (Pp. 145, 146.) 
 By extinction directions and interference figures in variously oriented 
 
 pairs of parallel planes (natural faces, cleavages or planes obtained by 
 
 grinding). 
 
 Use suite of sections or crystals of one substance. 
 
 17. DETERMINATION OF ANGLE BETWEEN OPTIC AXES. (Pp. 148-154.) 
 (a) By measurement of apparent angle in sections normal to the bisec- 
 trices either by rotation about b, with universal apparatus or its equiva- 
 lent, in air or in a liquid of known index (pp. 148-150, 152), or by 
 measurement of the distance apart of the emerging axes in a microscope 
 or pclariscope. (Pp. 151.) 
 
APPENDIX. 203 
 
 (&) By measurement with entire crystal in a liquid of approximately 
 or exactly the same index as the crystal. (Pp. 151.) 
 Use sections of muscovite and complete crystals. 
 
 18. ABSORPTION AND PLEOCHROISM. .(Pp. 96, 130, 131, 155-159-) 
 
 In isotropic crystals there is no pleochroism, but there may be either 
 color (unequal absorption) or no color (equal absorption). The absorp- 
 tion is independent of the direction. 
 
 In uniaxial crystals there is no pleochroism for transmission parallel 
 c, but there may be pleochroism in other directions with greatest differ- 
 ences for transmission normal to c . 
 
 In biaxial crystals the greatest differences in color are for rays vibrating 
 parallel to certain directions which in orthorhombic crystals are a b c, in 
 monoclinic crystals, one is b and in triclinic they are not related to the 
 crystal axes. 
 
 Sections of vesuvianite, tourmaline, penninite, iolite, andalusite, chrysoberyl, epi- 
 dote, titanite and axinite. 
 
 Absorption tufts in epidote and andalusite. 
 
 ^19. MICA COMBINATIONS. (Pp. 161-162.) 
 
 Production of uniaxial interference figure and rotation of plane of vi- 
 bration by piling mica plates. 
 
 C. THERMAL CHARACTERS. 
 20. CONDUCTIVITY. (Pp. 164-165.) 
 
 Illustrate surface conductivity by methods of Rontgen and Voigt, obtain- 
 ing circular figures in isotropic or in uniaxial normal to c , but in other 
 sections ellipses in which note relation of axes of ellipse to crystal axes. 
 
 *2i. EXPANSION. (Pp. 166-171.) 
 
 Indirect determination by observing the effect of heat upon the optical 
 characters, especially the effect upon the position of the optic axes. 
 (P. 170.) 
 
 Sections of gypsum. 
 
 Indirect determination by observing change in diedral angle with cal- 
 cite cleavage. (P. 169.) 
 
 D. ELECTRICAL AND MAGNETIC CHARACTERS. 
 22. MAGNETIC INDUCTION. (Pp. 172, 175.) 
 
 Suspension between the poles of an electromagnet of a small mass no 
 too unequal in dimensions. 
 
 Isometric Crystals indifferent equilibrium for all positions. 
 
204 CHARACTERS OF CRYSTALS. 
 
 Tertragonal or Hexagonal Crystals. 
 
 With c vertical all positions are in indifferent equilibrium, but with c 
 horizontal it takes either axial or equatorial position. 
 
 Note -j- and paramagnetic or diamagnetic character. 
 
 Crystals without any axis of magnetic isotropy (orthorhombic, mono- 
 clinic, triclinic). 
 
 Determine position and relative strengths of magnetization axes. 
 (P- 1730 
 
 Experiments illustrating electrical transmission (p. 175), conductivity 
 (p. 175), thermoelectric currents (p. 176), or dielectric induction (p- 
 177), may be devised when apparatus is available. 
 
 23. PVROELECTRICITY. (Pp. 180-182.) 
 
 Qualitatively by Kundt's dusting with sulphur and minium upon crystal 
 while undergoing uniform change of temperature. 
 Quantitatively by self -charging electrometer. 
 Crystals of tourmaline and quartz. 
 
 24. PIEZOELECTRICITY. (Pp. 182-183.) 
 
 Qualitatively by Kundt's method, developing the strain either by un- 
 equal heating, or by direct pressure. 
 Basal Sections of Quartz. 
 
 E. CHARACTERS DEPENDENT UPON ELECTICITY AND COHESION. 
 
 27. ELASTIC DEFORMATION FROM PRESSURE IN ONE DIRECTION. (Pp. 
 185-188.) 
 
 Effect of pressure in one direction upon the optical characters of cubes 
 of glass and of different minerals. 
 
 28. CLEAVAGE ANG GLIDING PLANES. (Pp. 189-191.) 
 In Calcite, Pyroxene, Stibnite and the Micas. 
 
 29. ETCH FIGURES (192196.) 
 
 Examination of suite of Apatite, Calcite, Calamine, etc. 
 
 SYSTEMATIC EXAMINATION* OF THE CRYSTALS OF ANY 
 
 SUBSTANCE. 
 
 The objects are to obtain a record of the physical characters for differ- 
 ent directions, and to determine, from the physically equivalent direc- 
 tions, the grade of structural symmetry. 
 
 The best crystals (p. 69), are chosen, the symmetry judged by exami- 
 nation with a hand glass or microscope and checked by an approximate 
 
 * Based upon Groth, Phys. Kryst., III., 537-543. 
 
APPENDIX. 205 
 
 determination of extinction directions (p. 117, 145), and possibly of in- 
 terference figures (115, 127, 138). Great assistance may here be given 
 by a rotation apparatus (p. 151, 152). 
 
 The crystals are sketched and letters assigned to the faces. Elemen- 
 tary planes (p. 13) are chosen parallel' to cleavages (p. 189) or according 
 to prominence. 
 
 GEOMETRIC CHARACTERS OR SYMMETRY OF GROWTH. 
 
 The crystals are mounted, centered and measured (pp. 63-70), by 
 zones, precedence being given to the angles between the elementary 
 planes. A record is kept, as described (p. 71), and also upon a free 
 hand stereographic projection (p. 20). A quality mark is assigned 
 (p. 70), to each reflection and used in the averaging. 
 
 The elements are calculated (pp. 30, 34, 38, 45, 56, 61) by the for- 
 mulae given or by spherical triangles (p. 6). The indices are determined 
 in general by intersecting zones (p. 17-19), and all angles are calcula- 
 ted from the indices and elementary angles by solution of the spherical 
 triangles (p. 6), in the stereographic projection, or by special formulae, or 
 by zonal equations, and compared with the measured angles. 
 
 One or more perspective drawings are then made. (Pp. 76-84.) 
 
 ETCH FIGURES, OR THE SYMMETRY OF SOLUTION. 
 
 Experience shows that the symmetry of the etch figure is almost in- 
 variably the true structural symmetry of the crystal. More than one sol- 
 vent should be tried and the conditions varied. 
 
 OPTICAL CHARACTERS AND SYMMETRY. 
 
 For transparent crystals the optical characters are the safest proof of 
 symmetry, often proving apparently simple crystals to be complex. 
 
 The optical constants for transmission are a, ft and y (133) for light of 
 different wave-length (89) and their orientation (145) and direct meas- 
 urement by the prism method (88, 103, 146) or by total reflection (90, 
 104, 147) for light of each of Frauenhofer's lines is sometimes made. 
 
 More frequently the determination is limited to the orientation and to 
 constants dependent on a, ft and ^, such as retardation (118), strength of 
 double refraction (ii9),character of ray surface (120, 154), and, if biaxial, 
 angle between optic axes (148). These determinations are for sodium 
 flame and possibly for lithium and thallium (89). If biaxial the interfer- 
 ence figure should also be observed in white light for dispersion (132-144). 
 
 Absorption and pleochroism are determined (96, 130, 155). 
 
 If the crystals show a rotation of the plane of polarization (123) the 
 amount (129) and direction (128) is determined. 
 
206 CHARACTERS OF CRYSTALS. 
 
 THE REMAINING PHYSICAL CHARACTERS. 
 
 The most frequently used tests remaining are the effects of heat, elec- 
 tricity and pressure upon the optical characters. In addition, however, 
 and especially in minerals opaque to light the following may be determined 
 without very elaborate apparatus. 
 
 (#) The heat conductivity, Rontgen or Voigt methods (p. 164). 
 
 (b) The pyroelectric charges, Kundt's method (p. 180). 
 
 (<:) Cleavage (p. 189), gliding planes (p. 190), percussion figures 
 (p. 192). 
 
 Thermal expansion, Magnetic Induction, Electrical conductivity, 
 Thermoelectric currents, Dielectric induction, Elastic deformation rarely 
 form part of an investigation. 
 
 The ascertained facts may be conveniently recorded as suggested by 
 Groth* and constitute a Crystallographic description. 
 
 (#) Class of Symmetry. 
 
 () Geometric Characters. 
 
 Elements, Enumeration of forms and description of habit with per- 
 spective drawings, Twinnings, Conditions of temperature solution, etc., 
 in production of crystals. 
 
 (V) Cleavages, Tabulation of measured and calculated angles, Gliding 
 Planes and Etch Figures. 
 
 (d} Optical Characters. 
 
 (i) Other physical characters. 
 
 * Phys. Kryst, III., 543. 
 
INDEX. 
 
 Abbe, total reflectometer, 95. 
 
 Absolute hardness, 198. 
 
 Absorption, 96, 130, 155. 
 
 Absorption bands, 1 60. 
 
 Absorption axes, 155. 
 
 Absorption, measurement'of relative, 156. 
 
 Absorption tufts, 148. 
 
 Adjustments, Fuess' goniometer, 68. 
 
 Airy's spirals, 128. 
 
 Amplitude, 85. 
 
 Analyzer, 107. 
 
 Analogue pole, 180. 
 
 Angle alteration by expansion, 1 68. 
 
 Angles between crystal axes, determining, 
 
 *3- 
 
 Angles between axial planes, 13. 
 Angles between optic axes, 148. 
 Angles between any two planes, 46. 
 Angles, constancy of interfacial, 3. 
 Angles and indices, relations between, 29, 
 
 39, 46, 62. 
 Antilogue pole, 1 80. 
 Auerbach, absolute hardness, 198. 
 Apparent axial angle, 148. 
 Axes, 8, ii. 
 Axes, changing, 19. 
 Axes, equivalent, II. 
 Axes of elasticity, 101. 
 Axes, optic, 97, 122, 124, 131, 133, 148. 
 Axial angle, measurement, 148. 
 Axial angle, calculation, 153. 
 Axial angle changed by heat, 170. 
 Axial angle apparatus, 149. 
 Axial cross, 80. 
 Axial image, 115, 127, 140. 
 Axial plane, 135. 
 
 Babinet, compensator, 118. 
 
 Basal pinacoid, 32, 37, 44, 50, 54. 
 
 Basal plane, 32, 37, 44, 50, 54. 
 
 Basal section, superposition, 121. 
 
 Becke, method indices of refraction, 120. 
 
 Becquerel, magnetism, 172. 
 
 Beijerinck, electrical conductivity, 176. 
 
 Bending rods of crystals, 186. 
 
 Bergman, Torbern, 3. 
 
 Bernhardi, 7. 
 
 Bertrand, eyepiece, 117. 
 
 Bertrand, lens, 150. 
 
 Bertrand, prism, 105. 
 
 Biaxial crystals, 132. 
 
 Biot, quartz plate, 130. 
 
 Bipyramids, 44, 50, 54. 
 
 Bipyramidal classes, 36, 41, 42, 48, 49, 
 
 .52, 53- 
 Biquartz, 130. 
 
 Biradials, 133. 
 Bisectrices, 136, 148. 
 Bisphenoids, 44. 
 Bisphenoidal classes, 35, 40. 
 Body colors, 160. 
 
 Bolzman, method dielectricity, 179. 
 Bose, electrical polariscope, 175. 
 Brightest illumination, 73. 
 Bromnaphthalin, a, 95. 
 
 Calculation of crystals, 28, 33, 38, 45, 55, 
 
 6l. 
 
 Centering, 65, 71. 
 Character of ray surface, 120, 154. 
 Circular polarization, 95, 122, 126. 
 Cleavage, 5, 188. 
 
 Clinographic parallel perspective, 79. 
 Collimator, 65. 
 Cohesion, 188. 
 
 Color distribution in lemniscates, 140, 143. 
 Color rings, 115. 
 Color tints, 96. 
 Colors, interference, 113. 
 Colors, surface and body, 160. 
 Compensators, 118. 
 Conical refraction, 135, 136. 
 Constants, optical, 103, 145. 
 Composite crystals, 73. 
 Contact goniometer, 3, 63. 
 j Conductivity, electrical, 175. 
 Conductivity, thermal, 164. 
 Convergent light, 107. 
 Corrosion faces, 196. 
 Critical angle, 91. 
 Cross and rings, 115. 
 Crossed dispersion, 143. 
 Crystal, definition, I. 
 Crystals in rock sections, 154. 
 Crystal carrier, 67. 
 Curie, piezoelectric method, 182. 
 
 Decretion, 7. 
 
 Deformation, elastic, 185. 
 
 Deformation, permanent, 188. 
 
 Deformation, effect on optical characters 
 187. 
 
 Derivation, II. 
 
 Diamagnetism, 172. 
 
 Dichroscope, 131. 
 
 Dielectric coefficients, 179. 
 
 Dielectric induction, 177. 
 
 Dihexagonal bipyramid, class of, 53. 
 
 Dihexagonal prism, 50, 54. 
 
 Dihexagonal pyramid, class of, 53. 
 | Diploid, class of, 58. 
 I Dispersion of axes, 132, 140. 
 
208 
 
 INDEX. 
 
 Dispersion of the bisectrices, 132, 141, 144, 
 Displacement method, 120. 
 Ditetragonal bipyramid, class of, 42. 
 Ditetragonal prism, 44. 
 Ditetragonal pyramid, class, 44. 
 Ditrigonal bipyramid, class, 49. 
 Ditrigonal prism, 50. 
 Ditrigonal pyramid, class, 48. 
 Ditrigonal scalenohedron, class, 48. 
 Dogtooth spar, 3. 
 Domes, 32, 37. 
 Dome class, 31. 
 
 Double refraction in calcite, 97. 
 Double refraction by pressure, 188. 
 Double refraction of electric rays, 175. 
 Double refraction of heat rays, 163. 
 Drawing, 76. 
 Dull faces, 73. 
 
 Edges, 82. 
 
 Elastic deformation by pressure, 185. 
 
 Elastic limit, 188. 
 
 Electric polariscope, 175. 
 
 Electrical conductivity, 175. 
 
 Electro-optical phenomena, 184. 
 
 Elementary planes, 13. 
 
 Elements determination, 30, 34, 38, 45, 56,61. 
 
 Elements of a crystal, 13. 
 
 Elliptical polarization, 122, 123. 
 
 Enantiomorphs, 125, 126. 
 
 Etch figures, 192. 
 
 Ether, 85. 
 
 Exner's hardness curves, 197. 
 
 Expansion by electric charge, 183. 
 
 Expansion measurements, 167, 168, 169. 
 
 Expansion by heat, 166. 
 
 Extension coefficients, 186. 
 
 Extension surfaces, 186. 
 
 Extinction, 117, 145. 
 
 Extinction directions and optic axes, 137 
 
 Fizeau, expansion measurements, 167. 
 Fluorescence, 160. 
 Foucault, prism, 105. 
 Fresnel, rhomb, 123. 
 Fuess, goniometer, 66. 
 Fuess, microscope, 108. 
 Fundamental form, 9. 
 Fundamental law of crystals, II. 
 
 Gahn, 3. 
 
 Gaugain, pyroelectricity, 181. 
 Glans, spectrophotometer, 156. 
 Gliding planes, 190. 
 Goldschmidt, projection, 76. 
 Goniometer, application, 3, 63. 
 Goniometer, reflection, 63, 64. 
 Goniometers with horizontal axes, 64. 
 Goniometers with vertical axes, 66. 
 Goniometer, two-circle, 74. 
 Glycerine, 95. 
 Graphic method, 9. 
 
 Grazing incidence, 91. 
 Grinding new faces, 104. 
 Groth, goniometer, 66. 
 Groth, universal apparatus, 149. 
 Gulielmini, Dominico, 3. 
 Gypsum test plate, 1 1 8. 
 Gypsum red, first order, 121. 
 Gyroid, class of, 57. 
 
 Haidinger, dichroscope, 131. 
 
 Hankel, pyroelectric method, 180. 
 
 Hardness, 196. 
 
 Hardness absolute, 198. 
 
 Hardness curves, 197. 
 
 Hartnack, prism, 105. 
 
 Hausmann, 8. 
 
 Hatty, Abbe, 5. 
 
 Heat conductivity, 164. 
 
 Heat rays, 163. 
 
 Heat expansion, 166. 
 
 Hemihedrism, 25. 
 
 Hexagonal bipyramid, 50, 54. 
 
 Hexagonal bipyramid third order, class 
 of, 52. 
 
 Hexagonal crystals, projection and calcu- 
 lation, 55. 
 
 Hexagonal prism, 50, 54- 
 
 Hexagonal pyramid, 50? 54- 
 
 Hexagonal pyramid, third order, class of 
 the, 52. 
 
 Hexagonal system, 46. 
 
 Hexagonal trapezohedron, class of, 52. 
 
 Hexahedron, 60. 
 
 Hexoctahedron, class of, 58. 
 
 Hextetrahedron, class of, 58. 
 
 Historical introduction, I. 
 
 Horizontal dispersion, 142. 
 
 Huyghen's construction, 80, 87. 
 
 Homogeneous deformation, 185. 
 
 Hyperbolae, interference, '116. 
 
 Imbedded crystals, 74. 
 
 Inclined dispersion, 142. 
 
 Index of refraction, 86. 
 
 Index of refraction by prism method, 88, 
 
 103, 146. 
 
 Index of refraction by total reflection, 90, 
 
 104, 147. 
 
 Indicatrix, optical, IOI, 133. 
 
 Indices, see index. 
 
 Indices of planes, 12. 
 
 Indices of zones, 17. 
 
 Indices and axes, equation between, 29, 39, 
 
 46, 62. 
 
 Induction dielectric, 177. 
 Induction, magnetic, 172. 
 Inner conical refraction, 136. 
 Integrant molecules, 6. 
 Intensity of rays, 112, 157. 
 Intercepts, 9, II. 
 
 Interfacial angles, constancy of, 3. 
 Interference, 106, no, in. 
 
INDEX. 
 
 209 
 
 Interference colors, 112. 
 Interference figures, 115, 127, 138. 
 Interference phenomena uniaxial crystals 
 
 no. 
 Interference phenonmena, biaxial crystals 
 
 137- 
 Isometric crystals, projection and calcula 
 
 tion, 61. 
 
 Isometric system, 57. 
 I so tropic crystals, 85. 
 
 Jaggar, apparatus, 154. 
 Jamitzer, W., 2. 
 
 Kepler, 2. 
 
 Kelvin, theory of pyroelectricity, l8l. 
 
 Kernel, 7. 
 
 Klein, lens, 150. 
 
 Klein, Universal Apparatus, 151, 155. 
 
 Koch, measurement extension, 186. 
 
 Kohlrausch, total reflectometer, 91. 
 
 Kundt, method pyroelectricity, 180. 
 
 Law of Babinet, 156. 
 
 Law of rational indices, 7. 
 
 Law of symmetry, 3. 
 
 Lemniscates, 138. 
 
 Least deviation, 89, 103, 146, 147. 
 
 Liebisch, total reflectometer, 93. 
 
 Light rays, 85. 
 
 Linear expansion, 167. 
 
 Linear projections, 76. 
 
 Magnetic induction, 172. 
 
 Magnetization, relative, 173. 
 
 Mallard, goniometer, 65. 
 
 Measurement of angles, 63, 70. 
 
 Metallic lustre, 159. 
 
 Metallic reflections, 159. 
 
 Methylene iodide, 95. 
 
 Mica test plate, 118, 120. 
 
 Mica wedge, v. Federow, 118. 
 
 Mica combinations, 161. 
 
 Microscope, Seibert n A, 107. 
 
 Microscope, Fuess VI, 108. 
 
 Miller, W. H., 9. 
 
 Minimum deviation, 89, 103, 146, 147. 
 
 Mitscherlich, goniometer, 66. 
 
 Mohs, 9. 
 
 Molecular net structure, 189. 
 
 Monochromatic light, 89. 
 
 Monoclinic crystals, extinction directions, 
 145- . 
 
 Monoclinic crystals, interference figures, 
 141. 
 
 Monoclinic crystals, projection and calcu- 
 lation, 33. 
 
 Monoclinic dome, class of, 31. 
 
 Monoclinic sphenoid, class of, 31. 
 
 Monoclinic system, 30. 
 
 Narrow faces, 72. 
 
 Negative ray surface, IOI, 104, 121, 136. 
 
 Negative crystals, 101, 104, 121, 136. 
 Neumann, F. C., 9. 
 Nicol, prism, 105. 
 
 Norremberg, mica combination, 161. 
 Norremberg, polariscope, 109. 
 Octahedron, 60. 
 Optic axes, 97, 133. 
 Optic axes, angle between, 148. 
 ! Optical characters during pressure, 187. 
 Optical characters during heating, 170. 
 Optical characters in electrostatic field, 184. 
 Ordinary ray, 98. 
 Orientation of principal vibration directions, 
 
 145- 
 
 Orthographic parallel perspective, 77. 
 
 Orthorhombic crystals, extinction, 140. 
 
 Orthorhombic crystals, projection and cal- 
 culation, 38. 
 
 Orthorhombic system, 35. 
 
 Outer conical refraction, 135. 
 
 Parallel polarized light, 106. 
 
 Paramagnetism, 172. 
 
 Parameters, 12. 
 
 Parameters, changing, 20. 
 
 Parameter ratios, 14. 
 
 Parting, 191. 
 
 Pentagonal dodecahedron, 60. 
 
 Percussion figures, 191. 
 
 Permanent deformation, 1 88. 
 
 Permutations of letter and sign, 36, 42, 54. 
 
 Phosphorescence, 161. 
 
 Piezoelectricity, 182. 
 
 Pinacoids, fig. 79, 27, 32, 37. 
 
 Pinacoid, class of, 36. 
 
 Plane, or pedion, 27, 32. 
 
 Plane-parallel plates, 87. 
 
 Plane polarized light, 105. 
 
 Plane of polarization, 100. 
 
 Plane of vibration, 98. 
 
 Pleochroic images, 157. 
 
 Pleochroism, 130, 155. 
 
 Pliicker, magnetization, 173. 
 
 Polariscopes, 106. 
 
 Polarizers, 105. 
 
 Polarization apparatus, 1 06. 
 
 Polarization colors, 112. 
 
 Polarization of electric rays, 175. 
 
 Polarization of light in calcite, 98. 
 
 Polarization of heat rays, 163. 
 
 Polarized light, 105, 122. 
 
 Polarizing microscope, 107, loS. 
 
 Pole of a face, 16. 
 
 Pole, position of any, 29, 34, 39, 45, 56, 
 
 62. 
 
 Poles, arc between, 30, 35, 39, 56. 
 Positive crystals, IOI, 104, 121, 136. 
 Positive ray surface, IOI, 104, 121,^136. 
 Pressure figures, 192. 
 Pressure, uniform, 185. 
 Pressure in one direction, 185 
 Pressure, effect on optical characters, 187. 
 
210 
 
 INDEX. 
 
 Primitive form, 4, 5. 
 
 Primitive circle, 20. 
 
 Principal indices of refraction, 103, 132. 
 
 Principal optical sections, 132. 
 
 Principal vibration directions, 132. 
 
 Prism, 32, 37, 44, 50, 54. 
 
 Prism methods indices refraction, 88, 103, 
 
 146. 
 
 Prismatic class, 3 1 . 
 
 Projections in parallel perspective, 77. 
 Projections, stereographic, 9, 20, 28, 33, 
 
 38, 45, 55, 61. 
 
 Pulfrich, total reflectometer, 94. 
 Pyramids, 44, 50, 54. 
 Pyramidal classes, 35, 40, 44, 47, 48, 52. 
 Pyroelectricity, 180. 
 
 Quadrant in drawing axial cross, 84. 
 
 Quality mark, 70. 
 
 Quarter undulation mica plate, 118, 120. 
 
 Quartz wedge, 118. 
 
 Quenstedt, linear projection, 76. 
 
 Rationality of parameters, 7. 
 
 Ray axes, 133. 
 
 Ray front, 85. 
 
 Ray surfaces, 85, 86, 95, 100, 125, 133. 
 
 Ray surfaces, character of, 1 20, 154. 
 
 Reflection goniometer, 63, 64. 
 
 Refraction in biaxial crystals, 136. 
 
 Refraction for normal incidence, 87. 
 
 Refracting liquid, 95, 153. 
 
 Retardation, 118. 
 
 Reusck, mica combination, 161. 
 
 Rhombic bipyramid, class of, 36. 
 
 Rhombic bisphenoid, class of, 35. 
 
 Rhombic dodecahedron, 60. 
 
 Rhombic pyramid, class of, 35. 
 
 Rhombohedral crystals, 47. 
 
 Rhombohedral class, 47. 
 
 Rhombohedron, 50. 
 
 Rontgen, conductivity, 164. 
 
 Rohrbach, solution, 95. 
 
 Rome Delisle, 3. 
 
 Root, dielectric induction, 177. 
 
 Rotation of polarization plane, 123, 128, 129. 
 
 Rotation by Reusch mica combination, 161. 
 
 Scalenohedral class, 41, 48. 
 
 Sclerometric tests, 196. 
 
 Secondary forms, 4, 5. 
 
 Senarmont, electrical conductivity, 175. 
 
 Senarmont, thermal conductivity, 164. 
 
 Sensitive tint, 129. 
 
 Signals, 68. 
 
 Snellius', construction, 87, 90. 
 
 Soleil, quartz plate, 130. 
 
 Sorby, displacement method, 1 20. 
 
 Spectroscope in index measuring, 92. 
 
 Spectrum photometer, 156. 
 
 Sphenoid, 31. 
 
 Sphenoidal class, 31. 
 
 Spherical projection, 6. 
 
 Spherical trigonometry formulae, 28. 
 
 Static pressure tests, 198. 
 
 Steno, N., 2. 
 
 Stereographic projection, 9, 20. 
 
 Strength of double refraction, 119. 
 
 Striated faces, 72. 
 
 Structure, 7, 162. 
 
 Surface colors, 159. 
 
 Symmetry axes, 10. 
 
 Symmetry, composite, II. 
 
 Symmetry grade, IO, 25. 
 
 Symmetry of etch figures. 
 
 Symmetry plane, lo. 
 
 Symbol, II, 17. 
 
 Symbols, see indices and index. 
 
 Systems, 9. 
 
 Tangent principle, 39, 56. 
 
 Tetartoid class, 57. 
 
 Tetragonal bipyramid, 44. 
 
 Tetragonal bipyramid 3 order, class, 41. 
 
 Tetragonal bisphenoid, 44. 
 
 Tetragonal bisphenoid 3 order, class, 40. 
 
 Tetragonal crystals, projection and calcu- 
 lation, 45. 
 
 Tetragonal prism, 44. 
 
 Tetragonal pyramid, 44. 
 
 Tetragonal pyramid 3 order, class, 40. 
 
 Tetragonal system, 40. 
 
 Tetragonal trisoctahedron, 60. 
 
 Tetragonal tristetrahedron, 60. 
 
 Tetrahedron, 60. 
 
 Tetrahexahedron, 60. 
 
 Theodolite goniometers, 74. 
 
 Thermal characters, 163. 
 
 Thermal conductivity, 164. 
 
 Thermal expansion, 1 66. 
 
 Thermoelectric currents, 176. 
 
 Thirty-two classes, crystals, 25. 
 
 Thickness of section, 119. 
 
 Thoulet, solution, 95. 
 
 Tint of passage, 129. 
 
 Total reflection, 90, 104, 147. 
 
 Total reflectometers, 91. 
 
 Tourmaline distribution of charge, 182. 
 
 Translation, 191. 
 
 Transparent crystals, measuring, 72. 
 
 Traube, attachment goniometer, 73. 
 
 Trapezohedron, 47. 
 
 Trapezohedral class, 41, 47. 
 
 Triclinic crystals, projection and calcula- 
 tion, 28. 
 
 Triclinic crystals, extinction, 145. 
 
 Triclinic system, 26. 
 
 Trigonal bipyramid, 5- 
 
 Trigonal bipyramid, 3 order, class, 48. 
 
 Trigonal prism, 50. 
 
 Trigonal pyramid, 50. 
 
 Trigonal pyramid 3 order, class, 47. 
 
 Trigonal trapezohedron, class, 47. 
 
 Trigonal trisoctahedron, 60. 
 
INDEX. 
 
 211 
 
 Trigonal tristetrahedron, 60. 
 True axial angle, 153. 
 Twin crystals, 73, 83. 
 
 % 
 
 Unsymmetrical class, 26. 
 Uniaxial crystals, 97. 
 Universal stage of v. Federow, 155. 
 Universal apparatus, Groth, 149. 
 Universal rotation apparatus, Klein, 151, 
 155- 
 
 Vibration, direction of faster ray, Il8. 
 Vibration plane, 98, 117, 145. 
 Vibration, principal directions of, 132, 146. 
 Vibration of ordinary and extraordinary 
 
 ray, 98. 
 Voigt, measurement expansion, 167 
 
 Wave-lengths, 85, 89, 113. 
 
 Weiss, 8. 
 
 Whewell, 9. 
 
 Wiedemann, surface conductivity, 179. 
 
 Wollaston, goniometer, 64. 
 
 Zone, 1 6. 
 
 Zone circle, 1 6. 
 
 Zone plane, 1 6. 
 
 Zone axis, 17. 
 
 Zone control, 17. 
 
 Zone during expansion, 168. 
 
 Zone, fourth face in a, 18. 
 
 Zone indices or symbols, 17. 
 
 Zone through one pinacoid, 19. 
 
 Zone of two pinacoids, 19. 
 
 Zone projection, 23. 
 
 Zonal relations, 55. 
 
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