CRYSTALLOGRAPHY 
 
 CRYSTALLO-PHYSICS.
 
 NOTES 
 
 CRYSTALLOGRAPHY 
 
 CEYSTALLO-PHYSICS. 
 
 BEING THE SUBSTANCE OF LECTURES DELIVERED AT TEDO 
 DURING THE YEARS 1876-77. 
 
 JOHN MILNE, F.G.S., 
 
 PROFESSOR OF GEOLOGY IN THE IMPERIAL COLLEGE OF ENGINEERING, 
 YEDO, JAPAN. 
 
 LONDON: 
 TRUBKER & CO., 57 AND 59, LTJDGATE HILL. 
 
 1879.
 
 
 
 HERTFORD: 
 
 PRINTED BY STEPHEN AUSTIN AND SONS.
 
 LUuarj 
 
 CONTENTS. 
 
 Note by the Editor vi 
 
 Introduction ... ... ... ... ... ... ... vii 
 
 PART I. 
 
 Determination of Symbols ... ... ... ... ... 1 
 
 Determination of Elements ... ... 28 
 
 To find the position of a Pole and the distance between any 
 
 two Poles 33 
 
 Rules for the changing of Axes and Parameters 35 
 
 PART IT. 
 Projection of Poles 37 
 
 PART III. 
 
 
 Crystal Symmetry and the Classification of Crystals into 
 
 Six Systems 45 
 
 PART IV. 
 
 Notes on Crystallo-Physics 58 
 
 Errata 71
 
 NOTE BY THE EDITOR. 
 
 IN the latter part of 1877, Prof. J. Milne sent home from 
 Japan lithographed copies of his written Lecture-Notes 
 on Crystallography and Crystallo-Physics to Prof. N. 
 S. Maskelyne, F.R.S., Dr. H. Woodward, F.R.S., Prof. 
 J. Tennant, F.Gr.S., to the Editor, and other friends, 
 with a request to, me to publish the same in the 
 GEOLOGICAL MAGAZINE, or elsewhere. 
 
 Owing to the absence of the Author and from other 
 causes, a long delay has occurred in presenting them to 
 the scientific public in their present form ; and it is only 
 due to Prof. Milne to state that these notes (as now 
 printed] were completed, and lithographed by his Japanese 
 assistant, in 1877. 
 
 I have to thank Prof. J. Morris, M.A., F.G.S., and 
 my colleague, Dr. H. Woodward, F.R.S., for kindly 
 assisting me in reading over and correcting the proofs of 
 these Notes on behalf of the Author. 
 
 THOMAS DAYIES, F.G.S. 
 
 DEPARTMENT OF MINERALOGY, 
 BRITISH MUSEUM. 
 
 69, LAWFORD ROAD, N.W. 
 26<A February, 1879.
 
 INTRODUCTION. 
 
 THE following notes on Crystallographical calculations 
 have been written for those students who wish to know 
 the general principles which these calculations involve, 
 rather than for those who wish actually to employ them. 
 The system that has heen followed is that of Prof. Miller. 
 In this system the symbols of a face consist of three whole 
 numbers, each of which invariably refer to the same axes ; 
 whilst the calculations to determine these symbols are 
 concise, being usually of such a nature that they may be 
 determined by making one or two angular measurements, 
 and the observation of those faces which have parallel 
 intersections. In all respects there is a simplicity which 
 recommends it before all others. 
 
 The few demonstrations which have been given have 
 been treated of by those methods which appeared to be 
 the simplest for the student's comprehension ; the first 
 portion, referring to the determination of symbols, being 
 treated by Analytical Geometry and Spherical Trigono-
 
 Vlll INTRODUCTION. 
 
 metry; whilst the portion on projection is dealt with 
 by reasoning altogether geometrical. 
 
 For a full development of Miller's system, together 
 with a series of examples illustrating the more special 
 methods of calculation, the student may refer to Prof. 
 Miller's " Treatise on Crystallography," 1839. As a 
 general book, in which an outline of the various systems 
 which have been employed by different crystallographers, 
 Quenstedt's "Grundriss der bestimmenden und rechnen- 
 den Krystallographie " may be consulted. The nature of 
 Crystal Symmetry will be found fully worked out in the 
 lectures on the " Morphology of Crystals," by Prof. Nevil 
 Story Maskelyne, F.R.S., published in the " Chemical 
 News," vol. xxxi, 1875. 
 
 To the authors of these and other treatises on Crystal- 
 lography and Physics I acknowledge my indebtedness 
 for the greater portion of the substance collected in the 
 following few pages. 
 
 JOHN MILNE. 
 
 YEDO, JAPAN, 1878.
 
 NOTES 
 
 ox 
 
 CRYSTALLOGRAPHY 
 
 AXD 
 
 CRYSTALLO-PHYSICS. 
 
 PAET I. 
 
 CRYSTALLOGRAPHY. 
 
 1. Amongst some of the common minerals, simple forms 
 are often to be met with, the faces of which are at once 
 seen to have geometrical, and consequent symmetrical, 
 relations to each other. 
 
 In the more complex forms, however, without the aid of 
 mathematical investigation, these relations remain hidden, 
 and the generality of crystals are regarded as being little 
 more than is implied in the definition, which describes them 
 as natural polyhedral forms bounded by plane surfaces. 
 There are, however (governing the inclinations which these 
 plane surfaces have to each other, and the manner in 
 which it is possible for them to be repeated), certain 
 laws which give rise to various relationships among their 
 forms. 
 
 In any particular crystal, one way in which these 
 relations may be illustrated is to imagine a figure to
 
 X CRYSTALLOGRAPHY. 
 
 be constructed, the faces of which are parallel to those 
 of the given crystal, but instead of having an unequal 
 development, are balanced in size and position, giving 
 rise to a solid having a greater or less amount of 
 symmetry. 
 
 Another method of realizing these hidden relations is 
 to conceive the given crystal to be placed within a hollow 
 sphere, and from its centre, which is supposed to coin- 
 cide with the centre of the sphere, perpendiculars to be 
 drawn through its faces. If these perpendiculars be con- 
 tinued until they meet the surface of the sphere, which 
 they will do in a series of points, it will be found that 
 these points have an orderly arrangement ; although, 
 from unequal development of the faces of the crystal, 
 we might have imagined that there would have been 
 an utter want of order in their arrangement. 
 
 This is the result of observation, and it is the dis- 
 tinguishing mark of a figure which has accreted according 
 to some natural laws of attraction, which may be unknown 
 to us, from a figure which has been carved at random. 
 And it must be observed that such a result as this must 
 flow from the relative inclinations which the planes have 
 to each other, and not from their size, the law concerning 
 which has not yet been discovered. 
 
 2. Reference of Planes to Axes. In order to show the 
 relations which have been spoken of, it is necessary to have 
 some means of defining the positions of the various planes 
 upon any given crystal. As an introduction to this it will 
 be well to repeat the generally well-known method by 
 which the situation of planes are referred to by means 
 of axes, these axes being three intersecting lines not in 
 the same plane.
 
 CRYSTALLOGRAPHY. 
 
 3 
 
 FIG. 1. 
 
 Let the three lines XX, YY', ZZ', be three such 
 axes intersecting in 
 the point 0, which is 
 called their origin. 
 Any plane like the 
 plane ABC, which 
 intersects these three 
 axes in the points A, 
 B and C, might be 
 defined by its inter- 
 cepts, that is, the 
 lengths OA, OB,OC, 
 because no other plane 
 but ABC can have 
 similar intercepts. 
 
 It will be observed that in thus denning a plane with 
 regard to the point in which three given axes intersect, 
 at least three measurements must be given. Another 
 point to be observed is, that if it were required to 
 construct a face from its three given intercepts OA, OB, 
 C, upon three given axes, each of these distances might 
 be measured from the point in either of two directions. 
 
 As, for example, OA, which might be measured on OX 
 or on OX'. It is, therefore, necessary to distinguish 
 between distances measured in one direction from those 
 measured in another. The distances measured in the 
 directions OX, OY, and OZ, are considered as being 
 positive, and those in the direction OX', Y', and OZ', 
 are considered negative, and these latter are distinguished 
 from the former by placing above them a negative sign. 
 Thus, if it were intended that OA should be measured 
 along OX', the resulting plane would be OA, OB, OC.
 
 4 CRYSTALLOGRAPHY. 
 
 If OB were also to be measured in a negative direction, 
 that is, along Y', the resulting plane would be OA, OS, 
 OC, and so on; altogether, with three intercepts each of 
 the same length, viz. OA, OS, OC, it being possible to 
 produce the following eight planes: 
 
 OA, OS, OC. OA, OS, 'OC. 
 
 OA, OB, 0~C. OA, OB, OC. 
 
 OA, OS, OC. OA, OS, 0& 
 
 OA, OS, OC. OA, OB, OC. 
 
 3. It sometimes happens that a plane may be parallel to 
 one or to two axes. This condition is expressed mathe- 
 matically by supposing the plane to have an intercept upon 
 these axes which is infinitely long, or briefly by the 
 sign co . Thus a plane with intercepts on OX and OP", 
 respectively equal to A and OB, and which was parallel 
 to OZ, would be indicated as the plane A, OB, co ; or 
 again, if a plane had an intercept on X equal to A 
 and was parallel to Y and Z, it would be indicated 
 as the plane A co co . 
 
 4. Parallel Planes. If we take any plane ABC with 
 intercepts A, B, C, and multiply these three quanti- 
 ties by the same number m, it is evident that a plane 
 m A, tn OB, m C, will be obtained, which will be 
 parallel to, or have the same direction as the plane ABC. 
 And as in crystals we have only to deal with the relative 
 inclinations which the planes of crystals hold to each 
 other, the plane m A, m B, and m C, would be 
 regarded as being equivalent to the plane A, OB, C, 
 because both of them would make the same angle with any 
 third plane which they might intersect. 
 
 5. Application to Crystals. Having properly selected 
 three axes in any given crystal, and imagined all its faces
 
 CRYSTALLOGRAPHY. 
 
 produced until they either show themselves parallel to one 
 or two of the axes, or else intersect all three of them, it 
 will be seen that a series of planes corresponding in direc- 
 tion to those of the given crystal have been obtained which 
 are determinable by means of their intercepts. But as in 
 treating of crystals, all the geometrical relations connecting 
 together the various faces of crystals are due to their 
 relative inclinations, and not to their linear dimensions, 
 planes parallel to the actual planes may be taken when 
 treating of the subject, and consequently the relative 
 lengths, and not the actual lengths of the intercepts as 
 have hitherto been taken, will now be spoken of. 
 
 6. Law of Rational Indices. If we take any point 0, and 
 through it draw planes parallel to those of a given crystal, 
 and select the intersection of three of these planes as axes, 
 we shall find that if any of the planes of the crystal or 
 planes parallel to them have intercepts on their axes 
 
 A, OB, 00 
 
 OH, OK, OL 
 
 OP, Q, OR 
 
 &c. &c. &c. 
 
 these planes are such that if 
 
 OA : OB : OC = a : b : c 
 
 then OH : OK : OL = ~ a : -r b : c 
 
 n K l 
 
 OP : OQ, : OR = -a : -b : -c 
 
 &c. &c. &c. == &c. &c. &c. 
 when h k I, p q r, etc., are some simple numbers like \, |, 
 i, f , 1, 2, etc., being seldom greater than 6. 
 
 This law, which, it must be observed, is only one of 
 experience, might be more briefly expressed by saying
 
 6 CRYSTALLOGRAPHY. 
 
 that the ratio of the intercepts of a series of crystal planes 
 upon given axes are simple multiples or submultiples of 
 the ratios of the intercepts of any one plane selected as 
 a type or standard. 
 
 7. The quantities a : b : c expressing the ratios of the 
 intercepts of the standard plane, are called the parameters 
 of the crystal. 
 
 They might also be called the axial units of the 
 crystal, as they are unit lengths used in the com- 
 parison of the lengths of the intercepts of the other faces 
 of the crystal. 
 
 The quantities h k I, p q r, etc., are called the indices 
 of the faces which have the intercepts OH, OK, OL, 
 OP, OQ, OR, etc. From the fundamental relation 
 
 OH : OK : OL = -r a : -r b : c, it is seen that 
 hi /c l 
 
 OH OK OL 
 
 h = k 7 = / ; and as a, b, and c are fixed 
 
 a o c 
 
 and each of these expressions is equal to a finite quantity 
 as OH increases, h must become less, from which it is 
 evident that the indices h k I are inversely proportional 
 to the intercepts. 
 
 When these indices of a face are inclosed in round 
 brackets like (hkl), it is a single face which is referred to, 
 and (h k I) is called the symbol of that face ; but if they 
 are inclosed in braces like {hkl}, it is a collection of 
 faces which are dependent for their co-existence upon 
 certain laws of symmetry. Such a collection of faces is 
 called a form, an example of which is illustrated at the 
 end of paragraph 2. 
 
 8. Indices of Planes parallel to one or two Axes. 
 
 1. If a plane HKL is parallel to one axis, say the
 
 CRYSTALLOGRAPHY. 7 
 
 axis OX, then its intercept in this direction, 
 viz. OH, = oo , but from the fundamental law 
 
 OH = a, and as a is constant, therefore 
 h=0 ; and such a face, if it intercepted the 
 axes OT and OZ, would be indicated by the 
 symbol (OKI). Similarly a face parallel to 
 Y and intersecting OX and OZ would be 
 indicated by the symbol (h o I) and so on. 
 Collections of faces like these, where one of the 
 indices is zero, give rise to forms known either 
 as domes or prisms. 
 
 2. If a plane H KL is parallel to two axes, say 
 the axes X and F, then OH=*> and also 
 K=. GO , and in order to satisfy the funda- 
 mental equations OH= , and OK= b, 
 the indices h and k must be each equal to zero. 
 That is, such a plane would be indicated by the 
 symbol (ool). Similarly we may have the 
 planes (o k o) and (h o o) . 
 
 Such a collection of planes, when two of the indices are 
 equal to zero, are called pinacoid planes. 
 
 9. Illustration to show the meaning of Indices and a method 
 for their determination. As an illustration of what has 
 been said about the relation which the indices of the 
 faces of crystals hold to one of the FIG. 2. 
 
 number which has been selected as a 
 type, we give the following crystal of 
 Witherite (Fig. 2) taken from p. 697 
 of Dana's " System of Mineralogy." 
 
 In Fig. 2 let A B C D E F repre- 
 sent a series of planes with parallel intersections. In
 
 CRYSTALLOGRAPHY. 
 
 FIG. 3. 
 
 Fig. 3 these are seen in a section taken at right angles 
 
 to these intersections. 
 In this figure the 
 planes A B C D E F 
 are seen to have their 
 edges parallel. By 
 actual measurement 
 the interfacial angles 
 between these planes 
 are found to be as 
 follows : 
 Between A and B=. 
 
 155 18'| 
 Between A and (7 
 
 145 24'i 
 Between A and D = 
 
 125 57' 
 Between A and E = 109 55' 
 
 A F = 90 
 
 If it is now assumed that the axes to which these 
 faces are to be referred are rectangular, and that two 
 of them, X and Z, are respectively parallel to 
 the planes F and A, which are at right angles, and 
 that some such face as C shall be taken as the standard, 
 the indices of the remaining planes may be readily 
 determined. 
 
 Continue the profile of the plane B until it intercepts 
 the axes X and Y in the points H and b. 
 
 Through H draw planes parallel to (7, D, and E, 
 meeting the axis Z in c, d, and e. 
 
 The angles which the planes BCD and E make with 
 X are respectively b' c d' and e.
 
 CRYSTALLOGRAPHY. 
 b , C . ,, d 
 
 i tan czz: tan a =- tai 
 
 o b o c o d o e 
 
 tan b' tan c tan <f tan ?' 
 o b o c 
 
 2 1747 ~ 1-4499 -7254 
 because 
 
 tan 6'=tan (155 18'-90)=tan 65 18*1=2-1747 
 tan c'=tan (145 24' - 90)= tan 55 24'|= 1-4499 
 tan <f =tan (125 57'i-90)=tan 35 57'%= -7254 
 tan '=tan (109 55^-90)=tanl955'|= -3624 
 If the ratio of the intercepts of the plane on the axes 
 X and Z, viz. a : b, be taken as parameter, that is, as 
 units for the particular axes to which they refer 
 that is H : o c =. a : b 
 
 then H: o b = a : 
 3 
 
 H:od=a: A 
 
 OH:oe = a:- 
 4 
 
 The indices of these four planes C B D E are therefore 
 (1. 1) (1. I) (1. i) (1. J). 
 
 If a third axis of reference, viz. Y at right angles to 
 X and Z, were introduced, it would be parallel to these 
 four planes, and therefore their indices in reference to it 
 would each be equal to zero, and the full symbols of such 
 planes as referred to these axes would be (101), (10J), 
 
 "We have here illustrated a method which may be used 
 in the determination of a series of planes, when such a
 
 10 
 
 CRYSTALLOGRAPHY. 
 
 series have parallel intersections and lie between two 
 planes at right angles. 
 
 10. Normals. If a series of planes as represented by 
 the straight lines A, B, C, D, E, and F, be taken, and 
 
 FIG. 4. from any point 0, we draw 
 
 perpendiculars through them 
 or their continuations, these 
 perpendiculars are called the 
 normals of the planes. 
 
 And it is evident that the 
 angle between any two suc- 
 cessive normals is equal to the 
 supplement of the interfacial 
 angle of the planes to which 
 
 they are perpendicular. For example, the angle between 
 
 the planes A and B = IT A B. 
 
 11. Poles. Next if be taken as centre and with any 
 radius a sphere be described, and then normals be con- 
 tinued until they intersect the surface of this sphere, which 
 they will do in a series of points called poles, the arc 
 between any two poles would measure the angles between 
 the two normals which they represent, and this arc would 
 also be supplementary to the angle between the planes 
 through which the normals had been drawn. For 
 example TT the arc a b = the angle between the planes 
 A and B. 
 
 12. Sphere of Projection. The sphere on which the 
 poles of a crystal may be thus projected is called the 
 sphere of projection. 
 
 13. Fundamental Relation connecting Angular Measure- 
 ments with Indices, etc. An expression which is of great 
 value in calculations for the determination of relations
 
 CRYSTALLOGRAPHY. 11 
 
 existing between indices, axial units, or parameters, and 
 angular measurements, is given by Miller as follows : 
 " Let the axes meet the surface of a sphere described 
 round as a centre on X Y Z; and let P be a 
 normal to the plane h k I drawn towards it from 
 O, meeting the plane in p and the surface of the 
 sphere on P. Then, if the plane h k I meet the 
 axes in H K L" 
 
 OH ' OK 
 
 cos YP, ^~ = cosZP 
 OL 
 
 .'. OHcosXP = OK cos 
 YP = OLcosZP; and from 
 the relation between inter- 
 cepts, parameters and in- 
 dices given in par. 6 ; 
 
 7 
 
 /. -r COSXP= 
 
 h k 
 
 As these expressions form the connecting link between 
 angular measurements, indices and parameters, they receive 
 in crystallographic calculation a wide application. Re- 
 garding them by themselves it will be observed that they 
 involve two equations, from which two unknowns or three 
 ratios might be deduced. Thus, for example, if P X, 
 P Y, P Z, and a, b, c, were given, the ratios h : k : I would 
 be determined ; or if h k I, and a b c, were given, the 
 ratios of P X, P Y, and P Z might be found. 
 
 From this expression we also see that the ratios of the 
 indices h : k :l are directly proportional to the cosines of 
 the angles made by the normal to h k I and the axes.
 
 12 CRYSTALLOGRAPHY. 
 
 14. Zones. When, as in the illustration just given, a 
 series of faces have their poles lying in a great circle, such 
 a collection of poles are said to form a zone, and the 
 planes which they represent are said to lie on the same 
 zone, or to be tautozonal. On consideration, it will be 
 evident that such planes will be those that intersect each 
 other in parallel lines, and any plane at right angles to 
 these will be either identical with or parallel to the plane 
 of the great circle of the sphere of projection which 
 contains their poles. 
 
 As a simple case of tautozonal planes, the lateral faces 
 of a hexagonal prism may be taken, when the parallelism 
 may be seen by mere inspection. In many cases, however, 
 if we have the intersection of any two faces given, and it 
 is wished to know if any other planes would, on being 
 produced, intersect either of the given planes in a line 
 parallel to the observed intersection, the crystal must be 
 tested by means of a reflecting goniometer. To do this 
 the given crystal is so adjusted in the instrument that the 
 given edge is parallel to its axis. On turning the crystal 
 round, those faces which reflect a ray of light in the same 
 direction will have parallel intersections, and will be in 
 the same zone. 
 
 15. Use of Zones. The method in which zones are em- 
 ployed in determining the indices of a face is as follows : 
 
 In any crystal it is evident that every face belongs at 
 least to two zones; that is, if we take any face a, not only 
 does it belong to a series of faces b, c, d, e,f, etc., which 
 have parallel intersections with it, and with each other, but 
 it also belongs to another series of faces b f , c' t d', e ', f, etc., 
 also with parallel intersections. Now, having determined 
 indices for any two planes, say b and c in the first zone A,
 
 CRYSTALLOGRAPHY. 13 
 
 it will be shown possible from the indices of b and c to 
 calculate quantities which are analogous to indices called 
 symbols for A, which represents a plane at right angles to 
 the planes a, b, c, d, e, f, etc. 
 
 Similarly, from the indices of any two faces in the 
 second zone as b' and c, symbols may be calculated for the 
 zone B, representing the plane at right angles to b' c' d' 
 e' /', etc. 
 
 A has now known symbols, and it is at right angles to 
 the plane a. 
 
 B has also known symbols, and it is also at right angles 
 to the same plane a. 
 
 The symbols of two planes A and B being now given, it 
 is required to calculate the indices of a plane a at right 
 angles to them, and just as A was calculated from b and 
 c, or as B was calculated from b' and c', so it will be shown 
 that a may be calculated from A and B. 
 
 Here two problems have been considered ; the first to 
 determine the symbols of a zone from the indices of two 
 planes in that zone ; and the second from the symbols of 
 two zones to determine the indices of a plane which is 
 common to them both. 
 
 16. Relation between the Indices of Planes and the 
 Symbols of Zones. If, instead of considering tautozonal 
 planes as they are exhibited upon an actual crystal, a 
 series of parallel planes, which crystallographically will be 
 identical with the actual ones, be drawn through the 
 origin, the intersection of these planes will be coincident, 
 and will be represented by a line drawn through the origin. 
 
 As each set of tautozonal planes will have its own 
 particular line of intersection, such a line may be taken to 
 represent a zone.
 
 14 CRYSTALLOGRAPHY. 
 
 But just as a certain line may be used to particularise 
 a zone, so may a plane be used, and the one which will 
 be here employed will be one at right angles to the zone 
 planes. This plane may evidently be considered as 
 crystallographically identical with the plane of the great 
 circle upon the sphere of projection containing the poles 
 of the zone which it indicates. 
 
 As the relation that the indices of this plane bear to 
 the parameters of the crystal and its own intercepts 
 are different from the ratios which the indices of an 
 ordinary plane in the crystal bear to their intercepts and 
 the parameters, for the sake of distinction the indices 
 of a zone plane have been called the symbols of that 
 plane. 
 
 The nature of these symbols and the relations they bear 
 to the indices of the tautozonal faces they characterise, 
 and to the parameters of the crystal, will be seen as 
 follows : 
 
 17. Given the indices (h k I) and (p q r) of any two 
 planes P and R in a zone, it is required to find the indices 
 and symbols of their zone plane. 
 
 The equations to these two planes in terms of their 
 intercepts, which we will suppose are a b' c and a" b" c" 
 respectively, will, when passing through the origin, be, 
 
 -. + 4, + 4 = o 
 
 a b c 
 
 X V Z 
 
 - + * + - _ o 
 a b c 
 
 but the intercepts a b' c and a" b" c" are by the funda- 
 
 a b c a b c 
 
 mental law (6) proportional to -jr TT r and 
 
 and therefore these two equations may be written
 
 CRYSTALLOGRAPHY. 15 
 
 X V Z 
 
 h- + k^r + I- = 
 
 a b c 
 
 If these two planes intersect, then x y and z may be taken 
 as coordinates of points on the intersection, and each of 
 these points x y and z will be the same for the two 
 planes. 
 
 By elimination of x and subtraction, we obtain 
 
 (hq-kp) f = (Ip-hr) j 
 By elimination of y and subtraction, we obtain 
 (hq-kp) ~ = (kr-lq) ^ 
 
 x I y 1 z 1 
 
 hence 7 - = -r - - = - - that is, 
 a krlq b Ip hr c hqkp 
 
 the coordinates of any point in the line are proportional to 
 aU = Vb = We 
 
 when U= (kr-lq) V= (Ip-hr) W=(hq-kp.} 
 It will be here observed that U V and W characterise a 
 line parallel to the intersection of the planes P and R y 
 and consequently to all other planes in the same zone 
 and therefore U V W characterise the zone and it is in 
 this sense that U V W are usually used as symbols of a 
 zone. We will now proceed to show that U V W also 
 characterise a zone plane. 
 
 Now the zone plane perpendicular to P and Q will also 
 be perpendicular to this line of intersection, the ratio of 
 the coordinates of any point in which have been found.
 
 16 CRYSTALLOGRAPHY. 
 
 It will now be shown that this zone plane will have its 
 FIG. 6. intercepts inversely 
 
 proportional to these 
 coordinates. (Fig. 6.) 
 Let P be any line 
 with coordinates pro- 
 portional to X, Y 
 and Z, or to a.U, 
 b. V, and c. W, passing 
 through the origin 
 of the axes X, Y, Z. 
 
 Let ABC be a 
 plane at right angles 
 to OP. 
 
 On P as a diagonal construct the parallelepiped with 
 its edges equal to the coordinates of P. Join PA and 
 L, L being the corner of the parallelepiped on the 
 plane Z Y, adjacent to P. 
 
 In the triangles OPL and POA, the angle OPL 
 equals the. angle P OA, and the angle OLP equals the 
 angle P A, each of them being equal to a right angle. 
 .*. because these triangles are similar L P : OP '.'. OP 
 : OA. 
 
 .*. L P is inversely proportional to O A, that is, the 
 coordinate X is inversely proportional to the corresponding 
 intercept of the plane A C on the axis X. 
 
 Similarly Y and Z can be shown to be inversely pro- 
 portional to the intercepts on Y and Z. 
 
 The intercepts of the zone plane are therefore pro- 
 portional to j but as indices of a plane equal 
 
 the parameters a, b and c divided by the intercepts of that
 
 CRYSTALLOGRAPHY. 17 
 
 plane, the indices of the zone plane must be proportional 
 to a. 2 U b 2 V c 2 W Q, E. D. 
 
 The quantities used as symbols of the zone will be 
 U V W 
 
 18. Note. The result which has just been given shows 
 the meaning of U V W as usually given as compared 
 with their meaning in connexion with a zone plane. 
 
 This latter signification might have been arrived at 
 more readily as follows : 
 
 Let k !L + t !L + ,L = 
 
 a b c 
 
 x y z 
 
 *-a + T + r ~ = d ' 
 be the equations to the planes P and Q in terms of their 
 
 intercepts, and let m + -y + s = d" be the 
 a o c 
 
 equation to a plane R with indices (m n s) at right angles 
 to P and Q. 
 
 If P and R are at right angles, then 
 mh nk si 
 
 w r + T. + .- = 
 
 and if Q and R are at right angles, then 
 
 (2) "If + " * + " = 
 
 a 2 b 2 c 2 
 
 Multiplying (1) by - \ and 
 
 * ' J (hrps) (krlq) 
 
 krlq ~ b 2 ps-hr c 2 hqpk 
 
 2
 
 18 
 
 CRYSTALLOGRAPHY. 
 
 That is, m : n: s or the ratio of the indices of the plane 
 R, which relatively to P and Q is their zone plane, is as 
 
 a 2 kr-lq : b^ps-hr : c*hq-pk 
 a* V : b 2 V: c* W 
 
 19. Although U Fand JFdo not hold the same relation 
 to the parameters of the crystal and to its intercepts as p q r 
 and h k I hold to the parameter and to the intercepts, it 
 is, nevertheless, convenient to use U V W as the symbol of 
 a plane such a plane being always a zone plane. To 
 distinguish a symbol from those referring to the planes in 
 a crystal it is inclosed in a square bracket, thus \_TTV W"]. 
 
 Sometimes a zone is indicated by inclosing in square 
 brackets the indices of two planes belonging to that zone, 
 thus [h k I, p q r], two planes not parallel to each other 
 being sufficient data from which the zone may be 
 determined. 
 
 20. The respective values of U V W may be remembered 
 by observing that they are the determinants of h k I and 
 p q r, or of the indices of any two planes in the same zone. 
 
 A convenient mnemonic for the determination of these 
 values is the process called cross- multiplication. The 
 nature of this will be seen from the two following 
 illustrations. 
 
 Given the plane P or (h k I) and the plane R or ( p q r), 
 to find the symbol \_UVW~] of the zone to which they 
 belong. 
 
 First write down the indices of h k I twice over in a 
 horizontal line and beneath these write the indices p q r 
 in a similar manner. Connect k I and h respectively 
 with r p and q with lines which will run downwards 
 towards the right, and multiply these three connected
 
 CRYSTALLOGRAPHY. 19 
 
 quantities together, which give as a result k r, Ip, 
 and h q. 
 
 Next connect I h and k respectively with q r and p by 
 lines which will run downwards towards the left, and 
 multiply these three quantities together, which will give 
 as a result I q, h r and k p. 
 
 From each of the three quantities produced by multi- 
 plication to the right, subtract the corresponding quantity 
 produced by multiplication to the left. The three results 
 will be krlq; Iphr; h qk p, and these three 
 quantities are the values of U Fand W which were sought. 
 
 The following is the operation described : 
 h k I h k I 
 
 XXX 
 p q r p q r 
 
 krlq; Iphr; hqkp. 
 or, U V W . 
 
 As a numerical example, the zone corresponding to the 
 planes (100) and (010) is 
 
 100100 
 XXX 
 
 010010 
 
 00-01; 00-10; 11-00 
 or, 01 
 
 [001] 
 
 21. It will now be shown that just as the symbol of a 
 zone has been produced by cross-multiplying the indices 
 of two planes, that the indices of a plane can be obtained 
 by cross-multiplying the symbols of two zones. 
 
 Let the symbols of the two zone planes be [7"F JF] 
 and [Zr V W'~\\ it has been shown that the indices of 
 these two planes are respectively proportional to
 
 20 CRYSTALLOGRAPHY. 
 
 a 2 U, 6 2 F, c 2 W, 
 and a 2 U', V V, c 2 IT. 
 
 The cross-multiplication of these indices will give the 
 three symbols b 2 c 2 ( V W- W V'}, a* c*(WU'-U W'\ 
 a 9 b* (IT V V U') corresponding to the three quantities 
 Z7, V and W obtained by cross-multiplying h k I and p q r 
 (see par. 20). But just as U, V and JFhad respectively 
 to be multiplied by a 2 b 2 and c 2 to obtain the ratio of the 
 indices of the plane they represent, so must J 2 c 2 (V W 
 -WV'\aic*(W~O'--UW'\a*W(WV'-VW'} be 
 multiplied by a 2 J 2 -and c 2 to obtain the ratio of the indices 
 of a plane atj_to V V JFand U 1 V W, which ratio will 
 be (VW- W F'), (WU'-UW'},(U V - V U'}, quan- 
 tities which it will be observed are the result of cross- 
 multiplying the symbols of the two zone planes \_U V W~\ 
 and[ZT V W'}. 
 
 For this reason two zone planes being given by their 
 symbols, it is possible to calculate by cross-multiplication 
 indices of a plane which is common to the zones thus 
 indicated, that is, of a plane at J. s to the two zone planes. 
 The pole of such a plane, it will be observed, will, upon a 
 sphere of projection, occur at the intersection of two zone 
 circles. 
 
 NUMERICAL ILLUSTRATION. 
 
 Given the symbols of two zones [111] and [110], by 
 cross-multiplication 
 
 111 111 
 
 X XX 
 
 110 110 
 
 10-11, 11-10, 11-11 
 112
 
 CRYSTALLOGRAPHY. 21 
 
 .'. (112) being the indices of the plane at _L s to [111] 
 and [110]. 
 
 22. Condition that a Plane belongs to a Zone. Sometimes 
 it is desired to know whether a given plane x y z belongs 
 to a given zone [Z7TJF]. The test of this is whether 
 the indices xyz satisfy the following equation. 
 
 Ux + Vy + Wz = 
 The truth of this may be seen by supposing another 
 plane p q r also to belong to the zone \_UVW~\. From 
 what has gone before (par. 20), the cross-multiplication of 
 (x y z) and (p q r) will give the values of U V and W 
 such that 
 
 U = qz ry 
 V rx pz 
 W = py qx 
 and it is evident that 
 
 (qz-ry}x + (rx-pz] y + (py-qx)z - 0. 
 that is, Ux + Vy + Ws = 0. 
 
 Q. E. D. 
 
 23. By means of this result we are often enabled to 
 determine the general character of the indices of a face 
 belonging to any particular zone which may be presented 
 to us ; for example, 
 
 1st. Let the given zone be one which passes through 
 two pinacoids. For illustration, these pinacoids 
 may be (001) and (100), that is, [010] will be the 
 symbol of the given zone. If a plane (h k I) 
 belongs to this zone, then 
 
 Oh + Ik + 01 = 
 
 that is, k = 0. 
 
 Therefore all faces belonging to this zone will have 
 their symbols of the form (h I). The value of
 
 CRYSTALLOGRAPHY. 
 
 this fact will be at once realized by observing that 
 we have now only two indices h and /, or the ratio, 
 
 that is, the ratio to determine for each of the 
 
 faces in this zone, instead of three indices or two 
 ratios. 
 
 2nd. Let the given zone be one which passes through 
 a pinacoid and a face. For illustration these may 
 be respectively (010) and (hkl), that is, the 
 symbol of the zone is [I A]. 
 If a plane p q r lies on this zone, then 
 I'p 4. Qg hr = 
 
 . P_ 1 
 ' r ' I 
 
 From which we see that all the faces in the given 
 zone have their two indices, which correspond to 
 the two zeros in the pinacoid in a constant ratio, 
 and this ratio it will be observed is given to 
 us in the indices of the given face. In this par- 
 ticular case it being the ratio -r 
 3rd. Because any two faces (h k I) (p q r), where the 
 
 ratio = the ratio can be expressed in the 
 I y 
 
 form of (s q r) and (p q r), the remaining faces in 
 the zone will have indices of the form (m q r). 
 The zone corresponding to (s q r) and ( p q r) by cross- 
 multiplication = [0 r (p s), q (s p}~\ = [0 r q~\. 
 If (x y z) is in this zone, then 
 Q x + r y qz = Q 
 V <1
 
 CRYSTALLOGRAPHY. 23 
 
 By the application of these three results, it is evident 
 that much cross-multiplication may be avoided, and 
 also by using this result in conjunction with the 
 one which is next to be given, indices of faces may 
 be determined by general rules which it would be 
 otherwise impossible to obtain. 
 
 24. In conclusion, we will now proceed to state a last 
 result which is of importance in determining the indices 
 of faces by general methods. 
 R/pqr 
 
 I If four poles P Q 8 and R whose symbols are 
 
 S'uvw respectively (efg), (hkl), (uvw), and (pqr), 
 
 are in one zone, Professor Miller has shown 
 
 Q\ ^ fc i that the following relation must exist, viz. 
 
 V that 
 
 P\efg 
 
 sin Q R sin P 8 krlq f w gv 
 sin PQ sinSR ~ flgk vrwq 
 Iphr gueto _ hqkp evfu 
 ~~ ghel wp ur e kf h uqvp 
 A mnemonic for these three equations here indicated 
 is obtained by observing that the two sides of these 
 equations have corresponding terms ; thus, for example, 
 corresponding to sin Q R, we have either kr Iq, Iphr, 
 or h qkp, which are the cross-multiples of the indices 
 of Q and R, thus 
 
 h k I h k I 
 
 XXX 
 p q r p q r 
 
 krlq, Iphr, hqkp. 
 
 Similarly sin P 8 corresponds to the cross- multiples 
 of the indices (efg} and (uvw) or ftcgr, guew, 
 or e vfu.
 
 24 CRYSTALLOGRAPHY. 
 
 It must also be observed that when the first component 
 of the three products produced by cross-multiplication, as 
 krlq, which is produced by cross-multiplying the 
 indices of Q and R, this quantity krlq must be com- 
 bined with the first quantities produced by the cross- 
 multiplication of the indices of P S, P Q, and 8R. 
 
 Similarly, the second component produced by the 
 cross-multiplication, as Iph r, must be combined with 
 other second components, as guew, and so on for the 
 third components, which are also combined together. 
 From this last result it will be observed that if 
 
 1st. The angles P Q and P R, together with the 
 symbols of P, Q, S, and R, are given, the angle 
 P S can be determined. 2nd. If the angles P Q, 
 P R, and P S, together with the symbols of P, Q 
 and R, are given, the symbol of S can be determined. 
 3rd. If the angles P Q, P R and P S are given, 
 the symbol u v w of S can be determined by com- 
 bination with the equation Uu + Vv + Ww= 0, 
 where W v w are the symbols of the zone plane 
 PQR. 
 
 25. Examples to Illustrate the Development of Indices. 
 Four general results have now been given with which it 
 is usually possible to determine the indices and often also 
 the position of the poles of any given crystal. 
 
 So long as the given poles each belong to two deter- 
 minable zones, the proper cross-multiplication of the 
 symbols of these zones will give to us the indices which 
 are required. 
 
 I. As an illustration of determining planes solely by 
 their zone combinations, the following crystal of felspar 
 may be taken as an example (Fig. 7).
 
 CRYSTALLOGRAPHY. 25 
 
 First let the zones which have been determined by 
 inspection or experiment be FIG. 7.* 
 
 (1) a m z b &c. 
 
 (2) c q x y a &c. 
 
 (3) c n b n &c. 
 
 (4) x o b o &c. 
 
 (5) c o m c &c. 
 
 (6) yon m &c. 
 
 (7) q o z n' &c. 
 Symbol of the zone (1) amzb,, 
 
 etc., is obtained by the cross- 
 multiplication of the indices (100) of a, and (010) of 6, 
 and is therefore [001], 
 
 Similarly the symbol of the zone (2) c q x y a, etc., is 
 obtained by the cross-multiplication of the indices (100) 
 of a and (001) of c, and is therefore [010]. 
 
 Similarly the symbol of the zone (3) c n b ', etc., be- 
 comes [100], and the symbol of (4) xobo', etc., is [101], 
 and the symbol of (5) come' [110]. 
 
 As we only know the indices of one face in each of the 
 remaining zones (6) and (7), which in this case is the face 
 o, it is evident that before we can proceed with one zone 
 determination, a second face must be determined for each 
 of these zones. 
 
 Because x is in the two zones [010] and [101], by cross- 
 multiplication of these symbols, the indices of x will be 
 found to be (101). 
 
 Similarly because m is in the two zones [001] and [110], 
 by cross-multiplication we find its indices to be (110). 
 
 By inspection we therefore see that m' will be (110). 
 
 We can now determine the symbol of the zone (6) 
 * See Brooke and Miller's Mineralogy, page 365.
 
 xJb CRYSTALLOGRAPHY. 
 
 yon m', etc., from the indices of the two faces and m ' . 
 This symbol is therefore [112]. 
 
 Since n is in the zone [100] and [112], by cross-multi- 
 plication the indices of n will be (021). 
 
 By inspection n' will be (02l). 
 
 The symbol of the zone (7) q o z n', etc., can now be 
 determined from the indices of the faces o and n. 
 
 This will be [312]. 
 
 Since y is in the zones [010] and [112], its indices will 
 be (201). 
 
 Since z is in the zones [312] and [001], its indices will 
 be (130). 
 
 Since q is in the zones [312] and [010], its indices will 
 be (203). 
 
 The indices for the faces of the given crystal may, 
 therefore, be written down as being 
 
 =(100) &=(010) c=(001) o=(lll) z=(101) 
 =(110) =(021) y=(201) 2 = (130) <?=(203) 
 
 II. Sometimes, however, it happens that the faces of 
 a crystal cannot be altogether determined by its zone 
 combinations. 
 
 In such cases the results obtained in paragraph 24 will be 
 required, as is illustrated in the following crystal of cuprite. 
 
 Let the zones which have been observed be b y c d, etc. 
 FIG. 8.* a0 fi f e tc. 
 
 Knowing, or having discovered 
 the mineral to be cubical, the three 
 planes a b and c, at right angles 
 to each other, may be chosen as 
 parallel to the planes of the axes, 
 and they will therefore be respec- 
 tively denoted by the indices (100), (010) and (001). 
 * See Brooke and Miller's Mineralogy, page 223.
 
 CRYSTALLOGRAPHY. 27 
 
 The plane o, which is equally inclined to the planes 
 a b and c, may be taken as the standard or parametral 
 plane to give the axial units. The indices of this plane 
 will therefore be (111). 
 
 Because the planes b and c are in the zone bydc, etc., 
 the symbol of this zone, as obtained by the cross-multipli- 
 cation of the indices (010) and (001), will be [100]. 
 
 Similarly, by the cross-multiplication of the indices 
 (111) of o and (100) of a, the symbol [Oil] of the zone 
 a o d is obtained. 
 
 By the cross-multiplication of the symbols [100] and 
 [Oil] of the two zones just obtained, the indices (Oil) of 
 the plane d, which is common to the two zones, is 
 obtained. 
 
 The indices of similar planes d' d", etc., may be written 
 down from inspection, or be obtained in a manner similar 
 to that already employed. 
 
 There are, however, remaining, a number of like planes 
 marked y y ', etc., which, as they only belong to one zone, 
 do not admit of being solved by the method hitherto 
 adopted. 
 
 Because the plane y belongs to the zone [100] b y d c, 
 etc., (by paragraph 23) it must have one of its indices 
 equal to zero. 
 
 In this particular case it will be the index corresponding 
 to the axis X, which equals zero. 
 
 Having measured the arcs, b y = 11 !#, b d = 45, and 
 b c = 90, we have now the case of four poles lying in a 
 zone circle, where, (as compared with the equation in 
 paragraph 24), the pole M is in this case equivalent to the 
 pole of b (010), 8 is equivalent to y, which we have deter- 
 mined as having its indices of the form fO k I), Q is
 
 28 CRYSTALLOGRAPHY. 
 
 equivalent to the pole d (Oil), and P is equivalent to the 
 polec (001). 
 
 The equation then becomes : 
 
 sin db sin cy _ (1-Q-l-l) (Q-/-1-&) _ k_ 
 sin cd smyb ~ (Ol-l'l) (*-0-H) ~ : 7 
 
 sin 45 sin 7841 / _ k 
 sin 45 sin 1119' ~ 7 
 
 k_ -_98 5 
 *' 7 ~ -176 " : I* 
 /. the indices of the face y are (015). 
 
 From this example it will be observed that the right 
 hand side of the equation given in paragraph 24, under 
 certain circumstances becomes much simplified. 
 
 26. Determination of Elements. In the foregoing 
 examples, which were given to illustrate the general 
 methods by which the indices of any given combination 
 could be determined, it will have been observed that three 
 planes were chosen to represent the planes of the axes, 
 and another plane to determine the axial unit. 
 
 As, in the description of any crystal, it is necessary to 
 define these axes by their relative inclinations and the 
 axial units of their ratios, we will now proceed to show 
 the general method by which such fundamental elements 
 may be determined. 
 
 Case I. The most general case will be when there 
 are three inclinations, viz. XT, YZ, and ZY> of the 
 
 a b 
 
 axes, and two ratios -r and of the parameters to 
 
 be determined. 
 
 We have here five unknown quantities, and to find these
 
 CRYSTALLOGRAPHY. 29 
 
 it is necessary to have at least five relations established 
 between them. 
 
 27. 1st. To Determine the Inclinations of the Axes. 
 The obvious way of finding these is to determine experi- 
 mentally certain angular measurements, and from these 
 to compute the required ones by the ordinary rules of 
 Spherical Trigonometry. 
 
 Three of the faces between which it will be found most 
 convenient to measure angles will be three to which it is 
 assumed the plane of the axes shall be parallel. This will 
 give us three angles. 
 
 Two more angles may be measured from two of those 
 planes to a fourth plane. 
 
 For example, let X YZ be the points in which the three 
 axes meet the sphere Fio 
 
 of projection. 
 
 Let A B and C be 
 the poles of three faces 
 of a crystal parallel 
 to the assumed axes, 
 which are represented 
 as meeting the sphere 
 of projection in X YZ, 
 and P be the pole of 
 a fourth face (hkt). 
 Join these points and 
 poles as indicated in 
 Figure 9, producing the arc X P to meet B C in x. 
 
 The angles most convenient to be measured would be 
 AS, B C, C A, and two of the three angles P A, P B, 
 or P C, all of which are really supplementary to the 
 angles between the corresponding faces.
 
 30 CRYSTALLOGRAPHY. 
 
 Supposing then that these arcs have been determined, 
 
 cos B C cos C A, cos A B 
 then cos CAB - - . n A . ' p 
 sin C A, sin A B 
 
 But because the triangle X Y Z is polar to the triangle 
 ABC, 
 
 :. the angle Y Z - TT- C A B. 
 
 Similarly X Y and Z X may be found. And these 
 measure the required inclinations. 
 
 28. To find the Ratios of the Parameters or Axial 
 Units. These must be determined from the general equa- 
 tion (par. 13) ^ cos P X = j cos P T = j cos P Z, 
 
 where h k and I are the known indices of a face P, and 
 P X, P Y, and P Z are determined either by direct 
 measurement or else have been computed from the other 
 angular measurements. 
 
 Taking, for example, the same conditions as were given 
 in the last figure, these ratios might be determined as 
 follows :* 
 
 (1) First determine the angle A B C and B C A as 
 CAB was determined in the last article (27). 
 
 cos BP cos PA, cos AB , 
 
 (2) cos P A B = - ^ PA> ^ AB - *"" 
 
 the angle P A B, and C A B-P A B = P A C. 
 Similarly P B A and P B C can be found. 
 
 (3) cos P C = cos P A, cos C A + sin P A, sin C A, 
 cos P A C, whence P C. 
 
 * In all these illustrations, to avoid repetition, similar letters denote 
 similar points ; thus, X T and Z are always used to indicate the points in 
 which the three axes meet the sphere of projection. A B and C represent 
 the poles of the planes of these axes, A being the pole of the plane Z; B 
 the pole of the plane ZX; and C the pole of the plane X Y. P usually 
 denotes any pole with indices (p q r).
 
 CRYSTALLOGRAPHY. 31 
 
 (4) cos PX=sin PX=sin Pff = sin 
 
 P B, sin P B C; whence P X, P Y, and P Z may be 
 
 found in a similar manner. 
 
 a b c 
 
 In the equation cos P X = -r cos P Y = cos P Z. 
 
 fl rC I 
 
 cos P X cos P Y cos P Z 
 
 Let - - - = A - - - = B : - = (7 and (7=1. 
 h k I 
 
 c e 
 
 Then A = b B = C, whence b = jr = and (7=1. 
 
 Jj A. 
 
 The case which has been here illustrated is that which 
 would occur with forms belonging to the Triclinic System. 
 
 The other cases which could occur are illustrated by the 
 remaining systems, and are as follows : 
 
 29. This case is where one of the planes chosen as being 
 parallel to one of the planes of the axes, is at right angles 
 to the other two planes which have been chosen. Let this 
 plane be the plane Z X. In this case the pole B and the 
 point Y would coincide, and the arcs A B and C B would 
 each equal 90. 
 
 It remains then for us only to measure such arcs as 
 C A, PA, and P B, and compute as before. This case, 
 where there is only one axial element and two ratios to be 
 determined, and which may be solved by making three 
 angular measurements, is represented by the Monoclinic 
 System. 
 
 30. In this case the three planes which are chosen as 
 planes of the axes are at right angles. There are, there- 
 fore, only two ratios remaining to be found. It will be 
 observed that the points X Y and Z will coincide with the 
 poles A B and C, and each of the arcs A B, B C, and (7 A
 
 32 CRYSTALLOGRAPHY. 
 
 will equal 90. We have, therefore, only to measure the 
 arcs P A and P B, and to determine P C as before, by 
 substituting these three quantities for P X, P Y, and P Z, 
 with which they are identical, in the general equation, 
 
 a b 
 
 the required ratio -r- and may be determined. 
 o c 
 
 The conditions here involved are exemplified in minerals 
 belonging to the Trimetric System. 
 
 31. In this case three planes maybe chosen to represent 
 the planes of the axes which are equally inclined to each 
 other. In such a case it must be observed that, because 
 the arcs X Y, Y Z, and Z X are equal to each other, the 
 arcs A B, B C, and C A, will also be equal, and the 
 measurement of any one of them will determine the 
 remainder. 
 
 Finding the three sides A B, B C, and C A in this 
 way, the angles A, B and C, can be determined, and 
 {irA)=X Y, Y Z, orZX. 
 
 When such an axial system as this exists, it will be 
 found that a plane may be chosen which will give equal 
 intercepts upon each of these axes, and consequently the 
 axial units will be in the ratio of 1 : 1 : 1. 
 
 Forms illustrative of these conditions, where there is 
 one angular element to be determined, are found in the 
 Rhombohedral System. 
 
 32. In this case the planes chosen as parallel to the 
 planes of the axes may, as in (par. 30), be at right angles 
 to each other. 
 
 A, B and C will, as before, coincide with -X", Y and Z, 
 and the arcs A B, C, and C A be each equal to 90. 
 
 Here, however, as the standard plane may be such that 
 it has two of its intercepts equal, and consequently two
 
 CRYSTALLOGRAPHY. 33 
 
 & 
 
 of the axial units also equal, there is only one ratio 
 to be determined. 
 
 This may be done by measuring some angle like P A, 
 from which P C and P B can be determined, in terms of 
 some unknown angle PAY, and the results substituted in 
 
 the general equation, from which can be determined. 
 
 C 
 
 The forms illustrative of these conditions will be repre- 
 sented by those in the Tetragonal System. 
 
 33. In this case the three planes parallel to the axial 
 planes may be supposed to be at right angles, and the axes 
 so formed intercepted by a standard plane, which will give 
 the axial units in the ratio 1 : 1 : 1. 
 
 Here all the elements are determined. The forms illus- 
 trating such conditions belong to the Isometric or Cubical 
 System. 
 
 34. To find the Position of a Pole. This is to find the 
 angular distance of a pole from the points in which the 
 axes meet the sphere of projection. 
 
 By referring to paragraph 28 (No. 4) it will be seen 
 that the distances P X, P F, FlQ< 10 
 
 and P Z have been numerically 
 determined by processes of or- 
 dinary spherical trigonometry, 
 and these are the distances of 
 any pole Q from X, T and Z, 
 which might be determined. 
 
 35. To find the Distance be- 
 ticeen the Poles of any two 
 Faces. Let the given poles be 
 P and Q. 
 
 Having given or having measured P A and Q A, and 
 
 3
 
 34 CRYSTALLOGRAPHY. 
 
 computed the angles P A B and Q A B as in paragraph 28 
 (No. 2), the angle QAP or the difference of these two 
 will be known. We have now in the triangle P A Q 
 two sides and the included angle, knowing from which 
 the side P Q or distance between P and Q may be readily 
 deduced. 
 
 The last two results will evidently be of great use in 
 projecting the faces of any given crystal upon the sphere 
 of projection. 
 
 36. The methods which have been suggested for the 
 solution of three problems, although obvious ones, are 
 neither so concise nor so useful as if the determinations 
 had been made in terms of the indices of the faces to 
 which they refer, and certain special angular measure- 
 ments which are known as the angular elements of the 
 crystal.* 
 
 The value of such a series of formulae or equations is 
 that angular measurements may be used in the determina- 
 tions of the indices of a face, or vice versd. Having 
 determined the indices of the faces of a crystal by means 
 of some of the results in paragraphs 20, 23, and 24, which 
 generally afford a sufficient and simple means, we may 
 determine the angular measurements when it would be 
 inconvenient, and perhaps impossible, to find them by 
 direct experiment. 
 
 If any actual sphere was given to us, and it was required 
 to distribute the poles of a given crystal over its surface, 
 little difficulty would be experienced because they would 
 altogether lie either at the intersection of great circles or 
 
 * For all these expressions see Miller's Crystallography, or, what is better, 
 the epitome of the same, given in Brooke and Miller's Mineralogy.
 
 CRYSTALLOGRAPHY. 35 
 
 else at such distances apart upon such circles as could 
 either be directly measured or else computed. 
 
 37. Rule for changing the Axes or the Parameters of a 
 Crystal It will be evident that as the positions of axes 
 have hitherto been considered as the intersections of any 
 three planes not in the same zone, it may be sometimes 
 necessary to change the three planes which were first 
 selected for three others. 
 
 This procedure will entail the changing of the symbols 
 of the remaining faces, which may be accomplished as 
 follows : 
 
 First determine the symbols of the three zone planes 
 which will pass through the poles of the three axial 
 planes in terms of the original indices of these planes. 
 Let these be [efg\ t [h k J], and [p q r\. 
 
 Then if the indices of a face referred to the old axes be 
 u v w, and where referred to the new axes, u' v' to', 
 
 then u = e u + fv + g w 
 
 v =. hu + kv + I w 
 w' = p u + qv + r w 
 
 38. Rule for changing the Parameters or Axial Units. 
 Similarly it may be required to change the parameters, 
 that is, to select a new face as the standard or type. 
 
 The general rule, as given by Professor Miller, is ex- 
 pressed in the following relations : 
 
 a' a b' be c 
 
 ^ ~h' ~ ~h, ~k' " ~k t 7 '' 7 
 
 where h k I are the original indices of a face P referred 
 to parameters a b c, and h' k' I' are the new indices of the 
 same face P as referred to a new set of parameters a' b' c' .
 
 36 CRYSTALLOGRAPHY. 
 
 If the face P with indices h k I be selected as the 
 parametral face, its new indices will be 111, and the above 
 relations may be written : 
 
 a b c 
 
 (2) "' = * v = T ' = 7 
 
 which gives to us the values of the new parameters. 
 
 Having in this way found the values of the new para- 
 meters a b' c', the new indices p q r' of any remaining 
 face Q, the original symbol of which was p q r, will evi- 
 dently from (1) be * 
 
 p a' q b' re 
 
 p = - q = ~1~ r = ~ 
 
 a b c 
 
 * For the proof of these rules see Miller's Crystallography.
 
 37 
 
 PART II. 
 
 PROJECTION OF POLES. 
 
 39. In order to obtain a graphic representation of a 
 crystal, it is necessary to have either some projection of its 
 planes or of its poles. As the projections of the latter are 
 the most capable of showing the angular and consequently 
 symmetral relations which are more or less hidden in any 
 projections that can be made of faces, it will be only these 
 which are here referred to. 
 
 The general way in which the poles of the faces of any 
 crystal may be imagined in the first place to be produced, 
 and afterwards to be distributed over the surface of a 
 sphere, has already been referred to, and it only now 
 remains to show how from zone determination and angular 
 measurements such a curved surface may be represented 
 upon a sheet of paper. 
 
 40. Of the various forms of projections by which this 
 may be accomplished, such as orthographic, gnomic, and 
 stereographic, the stereographic method of Prof. Miller 
 is the one which presents the greatest advantages, and is 
 therefore the one which will be here illustrated. 
 
 41. To illustrate the nature of this projection the polea 
 of a crystal may be imagined as being distributed over the 
 surface of a globe, the equatorial plane of which is the
 
 38 CRYSTALLOGRAPHY. 
 
 plane on which the figure of the sphere is to be represented. 
 If the south pole of the globe be taken as the point of 
 sight, and lines be drawn joining it with the poles of the 
 faces represented upon the surface of the sphere, the 
 points in which these lines intersect the equatorial plane 
 will be the stereographic projections of the poles. 
 
 42. In such a projection it is evident that all the poles 
 of the northern hemisphere will lie on the equatorial plane 
 proper, that is, within the circular area bounded by the 
 equator ; those on the equator of the globe will retain their 
 position ; whilst those on the southern hemisphere will lie 
 on the extension of the equatorial plane outside the sphere. 
 These latter, however, are seldom represented, as the poles 
 upon the two half-hemispheres usually present a similar 
 distribution. 
 
 The plane upon which the poles are projected, which for 
 an illustration has here been called the equatorial plane, 
 is usually named the primitive or picture plane, one of the 
 poles of such a plane being the point of sight. 
 
 This plane is generally so chosen that it shall coincide 
 with some important zone circle, so that the projection of 
 poles which such a zone circle may contain will simply 
 be laid down upon its periphery at the same angular 
 distances from each other as they occupied on the actual 
 spheres. 
 
 43. Characteristics of Stereographic Projection. It will 
 be observed that all zone circles at right angles to the 
 picture plane will be projected upon that plane as dia- 
 metral lines. 
 
 Another important feature in this projection is that all 
 other great circles will be projected as arcs of circles, the 
 chords of which are diameters on the picture plane.
 
 CRYSTALLOGRAPHY. 
 
 39 
 
 The truth of this will be seen from the following geo- 
 metrical construction : 
 
 FIG. 11. 
 
 Let N E 8 W\>& a sectional view of the sphere of pro- 
 jection, S the point of sight, E W the equatorial or picture 
 plane, and T R a plane at right angles to the paper, the 
 plane to be projected. The required projection will be 
 T' R r . This represents the section of a conical envelope 
 of rays radiating from the point S, and therefore such a 
 section must be either an ellipse or a circle. The propo- 
 sition is then reduced to the following lemma : 
 
 Given a cone R S T on a circular base It T, the length 
 of whose axis equals the radius of its base, and whose slant 
 sides are at right angles to each other, it is required to 
 prove that a section at right angles to the axis is a circle. 
 Draw SKS' at right angles to H 0, and draw MKL as
 
 40 CRYSTALLOGRAPHY. 
 
 a section parallel to WE, that is, at right angles to the 
 axis S 0. 
 
 1st. Because 8 P L is a right-angled triangle 
 
 /. P L 8 is the complement of P S L. 
 But OS Mis the complement of P S L 
 
 .'. OSR = PLS. 
 2nd. Because KR S = S R 
 and because KR 8 is a right-angled triangle 
 
 /. KR S is the complement of K S R. 
 But KSLis the complement ofKSR 
 .'. KSL = KRS = OS.R 
 .'. KLS = KSL 
 .'. KS - KL. 
 But MS L is a right-angled triangle 
 = KS = JTJf 
 
 and jK" $ is the dimension at right angles to the paper in 
 either direction from K, and it is=^KL=^KM. 
 
 Hence in the elliptic section made by M P we have two 
 chords bisecting each other at right angles and equal to 
 one another. Hence the ellipse is a circle, and any other 
 section parallel to WE or at right angles to S is also 
 a circle. Q.E.D. 
 
 44. In any given crystal, after having selected the zone 
 plane which shall represent the picture plane, and plotted 
 round its circumference the poles belonging to that zone, 
 the remaining poles of the crystal may usually be plotted 
 by the application of the following propositions : 
 
 45. Having given two poles P and Q (Fig. 12) situated
 
 CRYSTALLOGRAPHY. 41 
 
 in the zone of the picture plane, to find a pole R which 
 is from P and <j> from Q. 
 
 On the periphery of the picture plane set off P R' = 
 from P and Fio 12 
 
 Q R" = </> 
 from Q. Draw 
 the zone circles 
 PP'andQQ' 
 through the 
 centre 0. The 
 point 72 lies 
 at the inter- 
 sections of the 
 peripheries of 
 these circles. 
 
 By the re- 
 batement of 
 these two zone- 
 circles on their 
 
 respective diameters, it is seen that R will describe (if it 
 were projected vertically downwards, that is, orthographi- 
 cally) the paths m R' and n R", giving the point r as the 
 point vertically below R, or the intersection which is 
 sought. Also as R will lie on the zone circle n r, the 
 stereographic projection of which is obtained by joining 
 r with S, the pole, and so determine R which was sought. 
 
 46. To find the Zone Circle containing two points 
 P and R. As the projection of this zone circle will be a 
 circle, all that is required is to find out an additional point 
 P' which will lie in the same zone as P and R, and 
 through the three points P, R andP' to describe a circle ; 
 the portion of the arc intercepted by the periphery of
 
 42 
 
 CRYSTALLOGRAPHY. 
 
 the picture plane will represent half of the zone circle 
 required. 
 
 In the last diagram (Fig. 12), if it were required to de- 
 scribe a zone circle through P and R, a third point in this 
 zone is evidently the opposite pole of P, which lies on the 
 extremity of the diameter POP'. 
 
 In Fig. 13, where neither P nor R lie on the periphery 
 of the zone plane, the opposite pole of either P or R, say 
 of P, must be found thus. Draw a diameter through 
 P 0, then P' wiU lie at 180 distant from P in the zone 
 circle which this diameter represents. By the rebatement 
 of this zone circle we obtain S, the point of sight, and by 
 joining and producing 8 P, the point P is obtained, which 
 is the position of P on the sphere of projection. Measuring 
 180 from p to p and drawing S p', the point P' is 
 obtained, which is the pole of P. All that now remains 
 is to draw a circle through P, R and P', and the arc of 
 FIG. 13.
 
 CRYSTALLOGRAPHY. 43 
 
 this which is intercepted by the picture plane will give 
 the required projection. 
 
 47. Having given the projection of a great circle E P W 
 and a pole P in that circle, to find on the same circle a 
 point Q, distant from P. (Fig. 14.) 
 
 Since P lies on a great circle passing through 0, and 
 which has its pole at 8, the height of D above the picture 
 plane must equal Pp, and its vertical or orthographic pro- 
 jection upon that plane be the point p. 
 
 The orthographic projection of the circle on which P 
 lies will be an ellipse which may be imagined to be de- 
 scribed through the points E p W, etc. 
 
 If the plane of the great circle containing P be turned 
 down into the picture plane, the point P will coincide 
 with p'. 
 
 Laying off 6 from p', the point q' is obtained. The 
 orthographic projection of which is q, the point where 
 the perpendi- 
 cular from q 
 intersects the 
 orthographic 
 projection of 
 the given great 
 circle. The 
 stereograp hie 
 projection of 
 this point 
 being in the 
 great circle 
 through 0, is 
 therefore at 
 
 Q.
 
 44 
 
 CRYSTALLOGRAPHY. 
 
 Because = " 
 mp n q 
 
 the ellipse E p q W need not 
 be drawn, but only the con- 
 struction shown in Fig. 15, 
 where m p and q n make any 
 angle, and are respectively 
 equal to the corresponding lines in Fig. 14, n q being 
 FIG. 16. obtained 
 
 by drawing 
 through the 
 given point 
 p a line p q 
 parallel to the 
 line p q. 
 
 48. The 
 usual method 
 of solving the 
 same proposi- 
 tion is as fol- 
 lows : 
 First find the 
 pole of a given 
 great circle 
 
 EPW. This will evidently be in the great circle n S, 
 and at 90 from it, and therefore at 90 from any point 
 like m. By construction it will be therefore at R. 
 
 Secondly, through R draw R P P' and from P ' lay off 
 the point Q', 6 distant from P. Join R P', and this will 
 give the stereographic projection of the pole Q which 
 was sought.
 
 45 
 
 PART III. 
 
 CLASSIFICATION OF CRYSTALS INTO SIX SYSTEMS. 
 
 49. It is found that the multitudinous number of forms 
 in which minerals are presented to us are capable of being 
 classified in six groups or systems, each of which has 
 characteristics more or less special, which serve as a means 
 for its identification. 
 
 The most important of these characters is that of 
 symmetry. 
 
 50. Symmetry as Determined by Inspection. On ex- 
 amining any set of crystals, or, what is better for our 
 present purpose, a set of crystal models, it will be observed 
 that all crystals, excepting certain hemihedral forms, have 
 faces existing in similar opposite pairs. The faces are, in 
 fact, diametrically balanced about a centre. 
 
 Some crystals possess symmetry only of this kind. In 
 other crystals it would be observed that the faces are not 
 only symmetrical to a centre, but they have also sym- 
 metral relations to each other ; for example, faces may 
 exist in opposite similar pairs, each member of these 
 pairs being equally inclined to the same diametral plane 
 which may be imagined as being drawn between them. 
 
 Proceeding in this way it would be found that the col- 
 lection of models might be divided up into groups, each 
 characterised by its peculiar relations of symmetry. 
 
 51. Symmetry as Observed by Measurement. As we 
 are now in a position to determine the position of poles of 
 any given crystal upon a sphere of projection, we have
 
 46 CRYSTALLOGRAPHY. 
 
 evidently a means of demonstrating crystal symmetry by 
 a method equivalent to direct measurement. 
 
 For example, out of any collection of models illustrating 
 the various forms in which minerals crystallize, we should 
 find that there would be a certain group which, when 
 their poles were distributed over a sphere of projection, 
 would have these poles symmetrically balanced about no 
 less than nine great circles. 
 
 This arrangement would be such that if we 'measured 
 the distance of any particular pole from one of these 
 great circles or circles of symmetry, we should always 
 be able to find a second pole at a similar distance but 
 oppositely placed. 
 
 In other groups the symmetry would not be so perfect, 
 that is, the poles of any form would not be arranged 
 symmetrically to so many zone circles, and in some cases 
 not only would the circles of symmetry be fewer in number, 
 but they would also be placed differently with regard to 
 each other. 
 
 In this way it would be found that crystals might be 
 divided up into groups, the members of each group having 
 their poles symmetrically placed with regard to a par- 
 ticular set of zone circles. 
 
 52. It will now be shown that the peculiar grouping of 
 sets of zone circles, or, what is the same thing, the group- 
 ings of the planes they represent, is a natural consequence 
 of the fundamental law of Rational Indices. This will be 
 done in the following five paragraphs. 
 
 53. Condition that four Planes in a Zone may be Faces of a 
 Crystal. Because the indices of all planes in a crystal are 
 
 sin Q R, sin P 8 
 rational numbers, the quantity - _^ ; -g-= given m
 
 CRYSTALLOGRAPHY. 47 
 
 paragraph 24 must also be equal to a rational number, and 
 any planes whose normals satisfy this condition will be 
 possible faces of a crystal. 
 
 54. Nature of Planes of Symmetry. Let A B C DE F 
 in Fig. 17 represent the faces of a hexagonal prism. 
 
 If any plane like m n makes equal angles with two 
 planes like A F and FIG. 17. 
 
 B C, it would be said 
 to be symmetral to 
 them. If the faces of 
 this prism were un- 
 equally developed, the 
 symmetrical relations 
 would be best seen by 
 observing their poles, 
 which would be balanced 
 in position with regard 
 to the great circle in 
 which m n would meet the sphere of projection. 
 
 From this it will be observed that a plane of symmetry 
 must be at right angles to the zone to which it is sym- 
 metrical, and if it is symmetrical to a series of zones, that 
 is, to a crystal combination, it must be at right angles to 
 the series, and its pole will consequently lie at their 
 intersection. 
 
 Because m n is equally inclined to A F and B C, we 
 see that any plane like A B at right angles to m n will 
 also be equally inclined to A F and B C, and therefore 
 essentially also a plane of symmetry. 
 
 However, as the poles of A F and B C would not be 
 balanced in regard to the great circle of symmetry which 
 would be produced by a plane parallel to A B, drawn
 
 48 CRYSTALLOGRAPHY. 
 
 through the centre of the sphere of projection, A B would 
 not be regarded as being actually a plane of symmetry 
 unless A F and B C were represented by parallel and 
 opposite faces as FE and CD, under which circumstances 
 there would be a series of poles balanced about the great 
 circle of symmetry, produced by a plane parallel to A B 
 and at right angles to m n. 
 
 The next point to be observed is, that if normals P, 
 Q, OS, OR, be drawn to these four planes, these 
 four normals will form a series of angles, the sines of 
 which, taken in the order given in paragraph 24, viz. 
 
 sin P S sin Q R 
 
 T> ^ .~~CY T> will be equal to the rational quantity 2. 
 
 sin P Q sin S R 
 
 These planes, therefore, according to paragraph 53, are 
 possible faces of a crystal. 
 
 From this it is seen that planes of symmetry are usually 
 repetitive at right angles to each other, and also that they 
 are possible planes of a crystal, and must therefore be 
 FIG. 18. governed by the same 
 
 laws which regulate 
 the planes of a crystal. 
 55. Arrangement of 
 Planes of Symmetry. 
 Let A and B (Fig. 18) 
 be two planes of sym- 
 metry making with 
 each other an angle 
 of <. 
 
 Because A may be 
 considered as a plane 
 of the crystal to which 
 it is supposed to be symmetral, and B being a plane of
 
 CRYSTALLOGRAPHY. 49 
 
 symmetry of that crystal, B will require a plane a to exist 
 making with it <, and placed on the opposite side from A. 
 
 Similarly by regarding B as a plane of the crystal, A 
 will require that the plane b exists making with it an 
 angle of <f> and placed on the opposite side from B. 
 
 In other words, because planes of symmetry may be 
 regarded as planes of a crystal, they must be placed 
 symmetrically with regard to each other. 
 
 We thus see that if in any zone we have two planes of 
 symmetry making an angle of with each other, we 
 shall have in that zone a series of planes also making an 
 angle of <f> with each other. 
 
 .sin A a sin B b 
 
 .Now from Art. 24 - ~ : j-.=m, a rational quantity. 
 
 sm B a sin A b 
 
 sin 2 < sin 2 <j> 
 
 : : : =m 
 
 sin </> sin 
 
 cos 2 < = ~r a rational quantity, 
 
 and the only angles which have the squares of their 
 cosines equal to rational quantities are 90, 4 5, 60, or 30, 
 therefore the angles between two planes of symmetry must 
 be equal to 90, 45, 60, or 30, i. e. the angles between 
 planes of symmetry are commensurate with TT. 
 
 56. Combination of Planes of Symmetry. As the planes 
 of crystals can only be symmetrical to planes which make 
 angles of 90, 45, 60, or 30 with each other, which at 
 the same time must be arranged symmetrically to each 
 other, there are here conditions given for grouping crys- 
 tals under various systems corresponding to the various 
 combinations that can be made with groups of planes sub- 
 ject to the conditions which have been given. 
 
 In forming these combinations it must be observed that
 
 50 CRYSTALLOGKAFHY. 
 
 planes which form angles of 90 and 45 cannot be com- 
 bined with those forming angles of 60 and 30. 
 
 For example, let A A', B B', C C' y D D', (Fig. 
 
 F IG .i9. 19 ) be . four 
 
 planes inter- 
 secting each 
 other at 45, 
 and therefore 
 being a possi- 
 ble collection 
 of planes of 
 symmetry. If 
 the plane A A' 
 be intersected 
 by a plane 
 A B C D at 
 an angle of 
 45, and also 
 the three other 
 
 symmetrically arranged planes which are shown in dotted 
 
 lines, this plane will be a plane of symmetry because it will 
 
 make angles of 90, 45, 60, with all the planes it intersects. 
 
 Thus, for example, cos B C=coa D C, sin BO C 
 
 =cos 45, sin 45= $. 
 
 .*. the angle B (7=60, an angle which it is possible 
 for a plane of symmetry to make. 
 
 If, however, A B C D had been inclined at 30 or 60 
 to A A ', the resultant planes, which would have been 5 or 
 2 in number, would have made angles with some of those 
 planes which they intersected which were not commen- 
 surate with TT, and therefore they would not have been 
 admissible as planes of symmetry. In a similar way it
 
 CRYSTALLOGRAPHY. 51 
 
 can be shown that crystallo-symmetric planes inclined at 
 30 and 60 must always lie in the same zone. 
 
 57. In forming these combinations of planes of sym- 
 metry and in considering the distribution of poles which 
 will consequently result, the fact that all fully formed or 
 holohedral crystals have similar faces existing at the 
 extremities of a diameter must not be overlooked. The 
 poles of these crystals are, in fact, symmetrically placed 
 with regard to the centre of the circumscribing sphere of 
 projection. 
 
 Observing these facts, it will be found that in the sym- 
 metry of the six crystallographic systems already referred 
 to, we have exhibited the combinations which we are able 
 to produce. 
 
 SYSTEMS OP CRYSTALLIZATION. 
 
 58. Commencing with more simple forms of symmetry, 
 these may be FIG. 20. 
 
 epitomized as 
 follows : 
 
 I. Tnclinic 
 Syxtem. - This 
 system in- 
 cludes all those 
 crystals which 
 have no plane 
 of symmetry. 
 The only kind 
 of symmetry 
 remaining for 
 such a group 
 to exhibit is 
 symmetry to a centre, that is, for similar faces to exist in
 
 52 CRYSTALLOGRAPHY. 
 
 opposite pairs, or faces which have poles 180 distant from 
 each other. 
 
 Here, as in all the remaining systems, the three lines of 
 intersection of three planes are chosen for axes. In this 
 case they are so selected that the angles ZY and. ZXin 
 the positive or right-hand upper quadrant are greater 
 than 90. 
 
 On such a system of axes the closed form produced by a 
 set of planes, the general indices of which are h k /, will 
 have eight faces. These eight planes, although having 
 Fm. 21. indices identi- 
 
 cal as tonume- 
 rical value, 
 will only be 
 similar to each 
 other in oppo- 
 site pairs, and 
 therefore such 
 a general form 
 must be re- 
 garded as be- 
 ing made up of 
 four indepen- 
 dent forms. 
 
 Such a sys- 
 tem which has 
 
 here been indicated will be one where the axes are oblique 
 to each other, and the parameters are unequal. There 
 are therefore five elements to be determined, three incli- 
 nations of the axes ZX, X Y and Y Z, and two ratios of 
 
 the parameters an d
 
 CRYSTALLOGRAPHY. 53 
 
 II. Monoclinic System. In this system there is sym- 
 metry to a single plane B. Therefore, if we have given 
 to us a single face h k /, it is necessary that a second face 
 also with indices numerically equal to h k I should exist 
 on the opposite side of B. And if we satisfy the condi- 
 tions of central symmetry, there will be two more faces 
 corresponding to these, but diametrically placed with re- 
 gard to them. (Fig. 21.) 
 
 That is, in such a system the faces of a form {h k 1} will 
 exist in equal opposite pairs. 
 
 It must here be observed that by taking axes as far as 
 possible coincident, or at least symmetrically arranged 
 with respect to the plane of symmetry, the faces of a form 
 will become balanced about the axes, and therefore have 
 indices which differ from each other only in their arrange- 
 ment and signs, and not in their numerical values. 
 . For this reason the plane of symmetry B is chosen as 
 the plane of the axes Z X. The planes of the axes X Y 
 and YZ being at right angles to ZX. 
 
 If upon such a system of axes we construct the general 
 form {h k I}, we shall find that it will have eight faces 
 which are equal to each other in opposite pairs. 
 
 These conditions, it will be observed, are required by the 
 existence of a single plane of symmetry, and the fulfilment 
 of the conditions of central symmetry. 
 
 In this system we therefore see that one axis is perpen- 
 dicular to the other two, and the parameters are unequal. 
 
 There are therefore three elements to be determined, 
 
 a b 
 
 one inclination ZX, and two ratios and 
 
 c c 
 
 III. Orthorhombic System. In this system there are 
 three planes of symmetry A, B and (7, and these are at
 
 54 CRYSTALLOGRAPHY. 
 
 right angles to each other. These are selected as the 
 planes of the axes. 
 
 /. X Y = 90, YZ = 90, ZX = 90. 
 
 If any single plane h k I exists, this will require to be 
 FIG. 22. repeated, first 
 
 with regard to 
 each of the 
 planes of sym- 
 metry, and 
 secondly dia- 
 metrically 
 with regard to 
 itself. This 
 general form 
 {h k 1} will 
 have eight 
 similar faces, 
 and may be 
 observed by 
 building it up upon the given axes. (Fig. 22.) 
 
 This system is therefore one where the axes are at right 
 angles, and the parameters are unequal. There are there- 
 fore only two elements to be determined, viz. the ratios 
 
 a b 
 
 and 
 c c 
 
 IV. Rhombohedral and Hexagonal System. In this 
 system the planes of symmetry are tautozonal. These are 
 three in number and they are inclined to each other at an 
 angle of 60. They are shown by the three lines in Fig.
 
 CRYSTALLOGRAPHY. 55 
 
 23. In Prof. Miller's system three axes are selected 
 parallel to the 
 edges of a 
 rhombohedron. 
 These lie in the 
 three planes of 
 symmetry, and 
 are equally in- 
 clined to each 
 other. 
 
 A plane par- 
 allel to the 
 zone plane of 
 the three planes 
 of symmetry 
 will give the 
 parametral ratio a : b : c = 1: 1 : 1. There is, therefore, 
 only one element to be determined, namely, the inclination 
 of the axes to each other. 
 
 The general form {h k 1} may either be a twelve-faced 
 scalenohedron or a dihexagonal prism, the latter being the 
 case when h + k + / = 0. 
 
 If we consider the plane at right angles to the three 
 planes of symmetry to be an actual plane of symmetry, 
 then the general form (A k 1} will be a twenty-four- 
 faced figure. 
 
 And if we were now to intercalate three more planes of 
 symmetry as shown in dotted lines in Fig. 23, bisecting 
 the angles between the three original planes, we should 
 have a combination of seven planes of symmetry, a series 
 to which the general twenty-four-faced scalenohedral form 
 {h k 1} would also be symmetrical.
 
 56 CRYSTALLOGRAPHY. 
 
 Such conditions are represented by the Hexagonal 
 System. 
 
 V. Tetragonal System. In this system the planes of 
 symmetry which are tautozonal are four in number, and 
 are inclined at angles of 45 to each other. There is a 
 fifth plane at right angles to these which may also be 
 considered as a plane of symmetry. (Fig. 24.) 
 
 From these five planes three which are perpendicular 
 to each other are selected as axes. 
 
 .*. X Y = 90 TZ= 90 Z X = 90. 
 
 The general form [h k 1} of this system will have 
 FIG. 24. sixteen faces, 
 
 being a double 
 eight - faced 
 pyramid. In 
 this system the 
 axes are at 
 right angles, 
 and two of the 
 parameters 
 are equal. 
 
 There is, 
 therefore, only 
 one element 
 to be deter- 
 mined, viz., 
 
 a 
 
 the ratio . To determine this parametral ratio a 
 c 
 
 plane at right angles to one of those planes of symmetry 
 which is intermediate to those to which the axes are 
 selected as being parallel, is usually selected.
 
 CRYSTALLOGRAPHY. 57 
 
 VI. Isometric System. In this system there are nine 
 planes of sym- FIG. 25. 
 
 metry. Four 
 are tautozonal 
 and are in- 
 clined at 45 
 to each other. 
 The fifth one 
 is at right 
 angles to the 
 four which are 
 inclined at 45. 
 The remaining 
 four are tauto- 
 zonal in pairs, 
 and are in- 
 clined at 45 to the fifth one. (Fig. 25.) 
 
 Three of those planes which are at right angles to each 
 other are selected as axes. 
 
 A plane equally inclined to the planes of the three axes 
 is selected to give the parametral ratio a : b : c, which 
 must therefore be equal to 1 : 1 : 1. 
 
 The general form {h k 1} will have forty-eight faces. 
 In this system the axes are at right angles and the 
 parameters are equal. 
 
 In these six systems the various combinations which it 
 is possible for crystal planes of symmetry to admit of, have 
 been illustrated, and therefore these combinations may be 
 taken as a basis from which the various systems may 
 be considered to spring.
 
 58 
 
 PAHT IV. 
 
 NOTES ON CETSTALLO-PHYSICS. 
 
 PHENOMENA CONNECTED WITH MOLECULAR STRUCTURE CONSIDERED IN 
 RELATION TO THE VARIOUS CRYSTALLINE SYSTEMS. 
 
 59. In considering the structure of any body like a 
 crystal in relation to the phenomena it presents, it is 
 necessary that some hypothesis as to the nature of this 
 structure should be assumed, and more especially should 
 it be attempted to correlate such phenomena. For the 
 latter purpose the most valuable assumption would ap- 
 parently be that of molecular vortices. For the working 
 out of this however, high mathematical analysis would be 
 required. 
 
 A more popular hypothesis, and one which will more 
 easily enable us to perceive without analysis the probable 
 relations which exist, is to conceive a crystal as the result 
 of the mutual attractions of its molecules, and in addition, 
 when considering phenomena like those of light, to add 
 the idea of an elastic medium separating these molecules. 
 
 60. From what has already been demonstrated, it is 
 seen that different groups of crystals possess differences in 
 their symmetry. For the present we shall confine our 
 attention to that symmetry or arrangement of parts in 
 axial directions. 
 
 In the cubical system the arrangement was the same 
 in all three directions, but in other systems there was 
 more or less variation ; for example, the arrangement of
 
 CRYSTALLOGRAPHY. 59 
 
 planes round the lateral axes * of a hexagonal crystal is 
 different to the arrangement exhibited at the extremities 
 of a principal axis. This latter fact might lead us to 
 suppose that during the formation of such a crystal, the 
 crystalline force of attraction amongst the constituent 
 molecules had been different in the one direction to what 
 it had been in the other ; and as a consequence of this the 
 resulting physical structure in the two directions would 
 also be different, and the phenomena presented to us by 
 light, heat, and other physical agents, which depend upon 
 the medium they traverse, would in these different struc- 
 tural directions be differently affected. 
 
 For example, a crystal of calcite would be expected to 
 exhibit different elasticity and density in the direction of 
 its lateral axes to that which it exhibited along its princi- 
 pal axis. The way in which heat and light would be 
 transmitted in such directions would also be different. 
 On the other hand, if we took a cube of rock salt, the 
 molecular arrangements of which along its axial directions, 
 as inferred from its external appearance, are similar, the 
 phenomena exhibited in such directions would also be 
 similar. 
 
 61. In comparing similar forms as exhibited in different 
 substances, as for example the hexagonal prisms of Quartz 
 and Calcite, we must bear in mind that although the 
 attractive force which has drawn the molecules of these 
 substances into the forms they assume is apparently iden- 
 tical, so far as its directional influence was concerned, it 
 will, on account of the differences of material between 
 
 * The axial system of the hexagonal system which is here referred to, is 
 that of Naumann and Weiss, where we have one principal axis coinciding 
 with "the optical axis, and three lateral axes. These latter form a plane at right 
 angles to the principal axis, and intersect each other at angles of 60.
 
 60 CRYSTALLOGRAPHY. 
 
 which it has acted, have acted in different degrees. For 
 example, the constraint placed upon the molecules of 
 Quartz in the direction of the principal axis will, in all 
 probability, be different from the constraint or attraction 
 existing between molecules of Calcite in the same direc- 
 tion, that is, the ratio of the attractive force in the 
 direction of the principal axis as compared with the 
 lateral axis in the case of Quartz will be different from a 
 similar ratio in the case of Calcite. 
 
 62. -These surmises, based upon the external symmetry 
 which crystals exhibit, are apparently borne out by 
 appealing to experiment, as will be seen in the following 
 summary of their physical characters. 
 
 63. When we consider the various phenomena which 
 crystals having unequal axial elements present to us, it 
 would seem that the main differences which have been 
 produced along these unequal directions, which, of course, 
 means directions along which there is a difference in sym- 
 metry, is one of difference in the intermolecular spaces or 
 one of elasticity the " packing " together of the mole- 
 cules or their density having probably an intimate con- 
 nection with their elasticity. 
 
 64. Starting with such an hypothesis that in any par- 
 ticular substance the molecules are drawn more lightly 
 together in one direction than they are in another, it 
 would be easy to conceive why in a direction at right 
 angles to such a line we should have the direction of 
 easier separation or of cleavage. 
 
 65. Considering heat and electricity as being propagated 
 from molecule to molecule, the reason that these should be 
 conducted with greater rapidity in one direction than in 
 another would also be intelligible.
 
 CRYSTALLOGRAPHY. 61 
 
 And if we conceive such agents as light, electricity, and 
 magneto-electric induction to have the medium they tra- 
 verse affected by the molecules of the solid it may be 
 pictured as permeating, they also will be affected by the 
 molecular structure of the material they thus pervade. 
 Thus, if a large water area is covered pretty equally with 
 logs of timber, if these logs are in greater proximity to 
 one another in directions North and South than in direc- 
 tions East and West, we see that water vibrations will be 
 transmitted unequally in different directions. 
 
 66. It would also be equally easy to understand how the 
 coefficients of elasticity should be different in different 
 directions, and therefore, although we have no method of 
 direct proof, we may expect that sound is propagated with 
 different velocities in these directions. Also when any 
 causes acting equally through the material, such as heat, 
 tend to produce a general dilatation or contraction, this 
 also should have a directional variability. 
 
 67. These remarks, although not correlating the various 
 physical phenomena presented to us by crystals, will 
 nevertheless help to explain some of the relations which 
 will be seen to exist in the following summary of the 
 physical characters presented by crystals, more especially 
 drawn up to show additional characteristics of the several 
 systems which flow from the laws of crystal symmetry. 
 
 68. Cleavage. It is found that crystals will often split 
 in certain directions more readily than they will in others. 
 As these planes are always parallel to some of the actual 
 or possible faces of a crystal, we may often be enabled 
 to produce faces, which are of great assistance in the 
 determination of the system to which the mineral under 
 examination belongs.
 
 DZ CRYSTALLOGRAPHY. 
 
 The "direction at right angles to that along which 
 cleavage takes place is evidently one in which the co- 
 herence of the individual particles of which a crystal may 
 be supposed to be built is least, or, as some writers state 
 it, the directions in which the particles are the least 
 "packed." 
 
 This being the case, the relations of the molecular struc- 
 ture, as exhibited by cleavage and diamagnetic pheno- 
 mena, is understood. 
 
 69. Elasticity. The chief experiments which have 
 hitherto been made, directly illustrating the elasticity of 
 crystals as being different in different directions according 
 to the symmetry which the crystals exhibit, are those by 
 Savart, who showed that the axes of the figures formed 
 upon vibrating plates of crystals were connected with 
 their optical axes. 
 
 ; From these experiments the directions of greatest and 
 least elasticity would be observed and also the positive or 
 negative character of a crystal. 
 
 "We have here our first illustration of the mutual rela- 
 tion between the physical structure of a crystal and the 
 light propagating medium which permeates it. 
 
 Rankine has shown mathematically that the axes of 
 elasticity in crystalline solids must correspond in their 
 arrangements with the normals to the edges and faces of 
 the primitive forms. 
 
 . By combining Rankine's result with Savart's we there- 
 fore see that there are important analogies between axes 
 of ordinary elasticity, axes of symmetry, and axes of elas- 
 ticity of the light propagating medium. 
 
 70. JTea/. The chief phenomena exhibited by crystals 
 in relation to heat are their dilatation and conductivity,
 
 CRYSTALLOGRAPHY. 63 
 
 and the degree of development of these phenomena will, 
 according to the assumed hypothesis, depend upon their 
 elasticity and molecular structure. 
 
 i. Dilatation. If in any crystal there is dilatation 
 taking place, this would seem to indicate an increase in the 
 inter-molecular space, or, in other words, a decrease in 
 density. And remembering the fact that specimens of the 
 same substance, differing only from each other in their 
 physical state, show differences in expansion, we have 
 additional reasons for believing that in any individual 
 crystal which under the action of heat shows different 
 variations in different directions, that in such directions 
 there is a corresponding difference in the molecular con- 
 stitution. 
 
 And if we imagine the medium through which light and 
 magnetic induction to act to have a relation to the material 
 they pervade, then the appearances which these two phe- 
 nomena present us with must likewise suffer differences in 
 change consequent on dilatation. 
 
 Further, as we find that bodies of the same substance 
 in different physical states when heated approximate to 
 one another in their physical states, (for example, both 
 fibrous and crystalline iron tend by heat to assume the 
 same conditions, and gaseous and liquid carbonic acid at 
 a high temperature under pressure are really the same 
 in every respect), we should expect that the indices of 
 refraction and the various coefficients of which we have 
 been speaking should tend to become equal in different 
 directions. 
 
 This is found to be true in the case of the magnetic 
 inductive capacity of a crystal. 
 
 From the observations of Mitscherlich it would seem
 
 64 CRYSTALLOGRAPHY. 
 
 that there is an intimate relation between the symmetry 
 which a crystal presents and its dilatation. The parallelism 
 of the optical phenomena and dilatation must also be 
 observed. 
 
 His results were as follows : 
 
 1. Isometric Crystals. These are single refracting. 
 
 They dilate uniformly, in all directions. 
 
 2. Tetragonal and Hexagonal Systems. The crystals in 
 
 these systems are double refracting but are uniaxial, 
 that is, they have one direction which corresponds 
 to the principal axis, in which direction they only 
 show single refraction. In these systems in the 
 direction of the principal axis there is a dilatation 
 which is different from the dilatations in the direc- 
 tions of the secondary axes, which latter are similar 
 to each other. 
 
 3. Orthorhombic, Monoclinic, and Triclinic Systems. 
 
 The dilatation is unequal in all directions. 
 
 In consequence of these differences in dilatations, which 
 have apparently a direct connexion with the differences 
 in symmetry, there is a change in the angular measure- 
 ments of crystals. Thus, for example, in the following 
 minerals when the temperature is raised 100C. the 
 changes which are noted are : 
 
 In Calc spar, Ca C, the obtuse angle of the primitive 
 Rhombohedron, which is usually 105 5', diminishes 8', 
 and the acute angle increases 8|'. 
 
 Dolomite Ca C, Mg C. The obtuse angle of the 
 primitive Rhombohedron, which is usually 106 15', di- 
 minishes 4' 6". 
 
 Chalybite Fe C. The obtuse angle of the primitive 
 Rhombohedron, which is usually 107, diminishes 2' 22".
 
 CRYSTALLOGRAPHY. 65 
 
 In Gypsum, Aragonite, and other minerals analogous 
 alterations have been observed. 
 
 Notwithstanding these various alterations, it is found 
 that the indices of the faces of a crystal retain their 
 rationality. From which it must mathematically follow, 
 that the dilatation in any direction depends upon that 
 direction, and is represented by the radius vector of an 
 ellipsoid, of which the principal axes are in the directions 
 of the best axes of reference in the crystal. 
 
 ii. Conduction of Heat. The conduction of heat in any 
 body means a transmission of heat from particle to particle 
 of that body, and, therefore, when the particles are closer 
 together or have more intimate connexion with one 
 another in one direction than in another, heat conductivity 
 will be less in the latter direction. The conductivity of 
 electricity is almost analogous. "We have few accurate 
 experiments to refer to in heat conductivity, but it is 
 known that it is different in the same substance in different 
 physical states, a fact which is known for electric con- 
 ductivity. The same conclusion may be come to by con- 
 sidering that increase of temperature which causes a 
 decrease of conductivity for both heat and electricity also 
 causes a separation of the particles of the body. 
 
 The most remarkable series of experiments which have 
 been made on the conductivity of crystals were those of 
 De Senarmont. Slices of crystals cut in certain directions 
 were taken, and these, after being coated with wax, were 
 heated at their central point, when they were perforated 
 by means of a fine wire kept in a heated condition by an 
 electric current. 
 
 The flow of heat was noted by the melted portion of the 
 wax, and at any instant on the surface of the slice an
 
 66 CRYSTALLOGRAPHY. 
 
 isothermal line was indicated, which was either a circle or 
 an ellipse, according to the nature of the crystal and 
 the direction in which the slice had been cut. 
 
 Thus, in a slice of Quartz or Calcite cut parallel to 
 the lateral axes, the conductivity was uniform, and the 
 isothermal curves circles. 
 
 In slices of the same mineral cut parallel to the prin- 
 cipal axis, the isothermal figures were elliptical. 
 
 In the case of Quartz the diameters of the ellipse were 
 as 100 : 131. In Calcite the diameters were as 100 : 111. 
 
 Such experiments would indicate that because in both 
 these cases the conductivity in the direction of the princi- 
 pal axis was greater than in the direction of the lateral 
 axis, therefore that along these two directions there was a 
 difference in molecular arrangement. In the direction of 
 the principal axis, there being probably a greater density 
 or a more intimate connexion among the molecules than 
 in the direction of the lateral axis. 
 
 From any one of these isothermal curves the conduc- 
 tivity in any direction may be calculated, or, rather, what 
 would be better called the diffusivity, which is the con- 
 ductivity divided by the specific heat per unit volume. 
 
 In this way, by observing the spread of temperature 
 upon various slices of any given crystal, by combining 
 together the figures which the various isothermal lines 
 would give us, the spread of temperature from any point 
 in a solid crystal can be easily conceived of.* 
 
 M. de Senarmont's results as to the conductivity of heat 
 within crystals were as follows : 
 
 1. Isometric System. In this system the conductivity 
 
 .* Since every slice gives an ellipse or a circle, the isothermal surface sur- 
 rounding a heated point in a crystal must be an ellipsoid.
 
 CRYSTALLOGRAPHY. 67 
 
 is equal in all directions, and the isothermal surface 
 is a sphere. 
 
 2. Tetragonal System. In this system the conductivity 
 
 is at a maximum or minimum in the direction of 
 the principal axis. In the directions of the lateral 
 axis it is equal, and the isothermal surfaces are 
 consequently surfaces of revolution round the prin- 
 cipal axis. 
 
 3. Rhombohedral and OrthorhomUc Systems. In these 
 
 systems the conductivity has a maximum, a mean, 
 and a minimum value in three rectangular direc- 
 tions, which correspond with the crystallographic 
 axis. The isothermal surfaces are therefore ellip- 
 soids, the axes of which correspond with the axes 
 of symmetry. 
 
 4. Monoclinic System. In this system the conductivity 
 
 has three different values at right angles. The 
 
 first of these is parallel to the crystallographic axis, 
 
 but the second and third directions do not depend 
 
 on the remaining axes, but on the nature of each 
 
 individual crystal. The isothermal surfaces are 
 
 therefore ellipsoids with unequal axes only, one of 
 
 which has its position previously determined. 
 
 71. Light. In a body where the molecular constitution 
 
 is different in different directions, the material elasticity 
 
 is different in different directions, and it is usually inferred 
 
 that the elasticity of the light propagating medium must 
 
 also be different in different directions ; and the directions 
 
 of greatest and least elasticity, and therefore the directions 
 
 of greatest and least velocity of light, which may be called 
 
 principal directions, should correspond with the principal 
 
 directions of material elasticity and other physical phe-
 
 68 CRYSTALLOGRAPHY. 
 
 That such analogies really exist will be seen from the 
 following epitome of the phenomena presented by crystals 
 of the various systems in regard to the manner in which 
 they transmit light : 
 
 1. Isometric System. Light passes through these crystals 
 
 in a similar manner in all directions, and there is 
 no double refraction. 
 
 2. Tetragonal and Hexagonal Systems. The direction of 
 
 the principal axis is the direction of no double 
 refraction, or of the optic axis. In all other direc- 
 tions double refraction is exhibited, and this is at a 
 maximum in the direction of the lateral axes, that 
 is, at right angles to the optic axis. Slices of these 
 crystals, which are said to be uniaxial, cut per- 
 pendicular to the principal axis, will, when viewed 
 by a polariscope, show a series of concentric cir- 
 cular rings, intersected by a cross, the axes of which 
 are at right angles. 
 
 3. Orthorhombic, Monoclinic, and Triclinic Systems. In 
 
 these systems there are two optic axes, or directions 
 of double refraction. 
 
 In the Orthorhombic system these axes lie in a 
 plane containing two of the crystallographic axes, 
 one of which bisects the two optic axes. 
 
 In the Monoclinic System they lie in a plane con- 
 taining two of the crystallographic axes, also in a 
 potential plane of symmetry at right angles to such 
 a plane of the axes to which they are equally in- 
 clined. 
 
 In the Triclinic System no relation between the 
 crystallographic and optic axes has yet been dis- 
 covered. Because these optic axes are two in
 
 CRYSTALLOGRAPHY. 69 
 
 number, these crystals are said to be biaxial. 
 Sections cut perpendicularly to these axes, when 
 viewed in a polariscope, show a series of rings 
 round each axis. Between the axes these are 
 drawn together and may meet to form a lemniscate. 
 
 72. Magnetism and Diamagnetism. From the researches 
 of Tyndall and Knoblauch, it would seem that the position 
 assumed by magnetic or diamagnetic bodies, when sus- 
 pended between the poles of a magnet, is due to the 
 constituent particles of the body being " packed " more 
 closely in one direction than in another 'the line of close 
 packing in magnetic bodies being the one which tends to 
 set axially or parallel to a line joining the poles, whilst in 
 diamagnetic bodies it sets equatorially or at right angles 
 to such a line. 
 
 In crystals, however, these directions appear to be inti- 
 mately connected with cleavage planes, the normals of 
 these setting equatorially or axially, according as the 
 crystal is magnetic or diamagnetic. Thus, a cube of 
 Topaz, which is diamagnetic, sets the principal axis in an 
 axial direction, so that the normals to the cleavage planes 
 are placed axially ; therefore Topaz is most closely packed 
 at right angles to the principal axis. 
 
 Continuing such experiments, it is found that some 
 crystals are therefore packed tighter along the principal 
 than along the lateral axis, and a relation between mole- 
 cular structure as exhibited by magnetic influence is found 
 to have a connexion with crystalline symmetry. 
 
 Just as heat, by its altering the molecular structure of a 
 body, affects the phenomena of light, so it affects magnetic 
 phenomena, the inductive capacities of a crystal which 
 may be in different directions unequal by the influence of 
 heat being rendered less unequal.
 
 70 CRYSTALLOGRAPHY. 
 
 73. Electric Conductivity. With, regard to the electric 
 conductivity of crystals, from the observations of M. 
 Wiedemann it appears to result that 
 
 Negative crystals have the best conductivity in the 
 
 direction of the principal axis. 
 Positive crystals have the best conductivity in the 
 
 direction of the lateral axis 
 
 a result which is in direct accordance with the phenomena 
 exhibited by light the direction in which light travels 
 fastest being that in which there is best conductivity. 
 
 74. Markings on Faces. In conclusion, it may be 
 observed that there are other appearances which crystals 
 present, such as the similar markings which are found 
 upon faces which are similarly placed about similar axes, 
 which also show a connexion between the symmetry of 
 crystals' and their molecular structure. 
 
 75. Conclusion. It must now be evident that all the 
 physical phenomena exhibited by crystals are to some 
 extent analogous with one another. 
 
 Crystals which have similar symmetry show similar 
 phenomena, and these usually in such a manner as to 
 characterise a group or system. 
 
 In the practical determination of a system, the cleavage 
 and optical properties will be the properties most con- 
 veniently experimented on, and although it is possible to 
 appeal to the other phenomena which have been referred 
 to, technical details of apparatus and manipulation are too 
 elaborate to be conveniently employed. 
 
 FINIS.
 
 ERRATA. 
 
 Page 7, 
 
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 18 (o*0 > (0- 
 
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 19 ;, (oko) (0/tO). 
 
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 4 from bottom, for read 0. 
 
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 15, 
 
 1 from top, 0. 
 
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 i 43, 
 
 7, for the letter D, read P.
 
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