Jan. 7, 1 875 J 



NA TURE 



'i . ' ; - represent the ratios of thejntcrcepts of the 



plane (// k I), and 1 : 1. -.t those of the plane (3 2 i). 



321 

 Where either of these values h, i, or / becomes 

 zero, this would represent an intercept indefinitely great 

 upon the axis to which it refers, since the algebraic 



value of a quantity of the form — is infinity. Referring 



o 

 again to the original zone on the crystal of barytes, 

 we see that the face, the trace of which on the zone 

 plane was taken for the axis of Z, will nowhere intersect 

 with that axis, so that its index for the axis of Z' becomes 

 o, and similarly for the plane parallel to the axis X. In 

 like manner if an axis 1' perpendicular to the zone plane 

 representing the profile of the zone of barytes had been 

 taken for a second axis, all the planes of that barytes zone 

 would have been parallel to that axis V, which is in fact 

 its :o;!c- axis, being parallel to the edges of the zone, and the 

 index with respect to that axis would for each plane of 

 the zone have been o. Thus, taking our indices in the 

 order corresponding to that of the axes .V Y Z, we can 

 now say that the plane, the trace of which gave us our 

 axis of Z, would have for its symbol (« o o), where n was 

 any whole number, or rather, since we may divide the 

 whole symbol by n without altering the ratio, (100). So, 

 the plane the trace of which gave us the direction for the 

 axis of A' would be (001) ; the standard plane that gave 

 the parameters a and c, having for its intercepts the values 



-;-, would be represented by the symbol (loi), while 



the other two planes would receive the symbols (201) and 

 (S02). .^ . ^ 



Since all planes on a crystal must intersect if continued 

 far enough with all three or with only two, or finally with 

 only one of the axes, they may be considered as falling 

 into one or other of three groups : such, namely, as have 

 three whole numbers in their symbol ; such as have one 

 zero in their symbol (the zero corresponding to the 

 axis with which they do not intersect) ; and such, thirdly, 

 as have two zeros with unity for their indices. 



Passing from a system with rectangular axes, the lec- 

 turer next considered the general case of an axial system in 

 w-hich the axes might be oblique to each other. In pointing 

 out that the three planes which contain these axes, namely, 

 the planes X V, Y Z, Z X, divided the space around the 

 point in which they and the axes intersected into eight 

 divisions or octants, he proceeded to designate the posi- 

 tion of a point situate anywhere in space by the Cartesian 

 method of co-ordinates. The point o of intersection of the 

 axes being called the origin, and positions to the right, 

 above, or in front of it, being considered as positive ; those 

 to the left, to the rear, and below it, as negative, it becomes 

 possible, by means of lines parallel to the axes projected 

 from the point, to determine its position in either octant. 

 Then taking two planes in a zone which intersected with 

 all three of the axes, such as two planes (iii) and (321), 

 the lecturer showed, by a representation in a model, how 

 the edge in which these two planes intersected could have 

 its direction determined by making it parallel to the 

 diagonal of a parallelepiped the sides of which would 

 represent the co-ordinates of any point in that line, in the 

 ratios of n a : 1' b : w c, where h,v, and 71' represented 

 values which the lecturer proceeded to educe from the 

 symbols of the faces. For this purpose he represented 

 the planes by two equations or expressions involving the 

 ratios of the co-ordinates of any point in the plane, in 

 terms of the parameters of the crystals and the indices of 

 the planes. 



Then, by a familiar algebraic method, he obtained an 

 expression for the relations between the co-ordinates 

 for any point in the Ime in which the planes intersected. 

 The expression thus obtained gave a symbol for the edge 

 in the form of the determinant of the indices of the two 



planes : thus a symbol [u v \v|, included in square 

 braces, representing the edge formed by the planes (f/;ij-) 

 and ill J; I), had for the values of its indices — 



V = oh -el 



w = e k - fh 

 and the lecturer proceeded to show that any third plane 

 with the indices pq r belonging to the zone [u v w] must 

 fulfil the condition — 



/ u -J- ? V + t/ w = o 

 and furthermore, that if two zones had a plane in common, 

 the symbol of that plane is found by taking the determi- 

 nant of the symbols of the zones. 



The next subject treated of had reference to the various 

 means which geometry offers for a more convenient treat- 

 ment and representation of the different zones of a crystal, 

 than that of making an elaborate drawing of its edges. 

 Of these, the method of referring the planes of a system to 

 a sphere by means of their normals was shown to possess 

 great simplicity. A sphere being conceived as described 

 around the point, or oyii:;in, in which the axes cross one 

 another as a centre, lines drawn from that point perpen- 

 dicular to each plane of the crystal — the normals to these 

 planes — are continued till they penetrate the surface of 

 the sphere in points that will be called the poles of the 

 planes, the symbol for a pole being identical with that for 

 the plane to which it belongs. The poles of a zone of 

 planes will thus be distributed along the arc of a great 

 circle of the sphere, its :;one ciicle. Hence the discussion 

 of the inclinations of the planes of a crystal, and so, many 

 of the chief problems of crystallography, becomes reduced 

 to their treatment by spherical trigonometry ; and what 

 has further rendered this mode of considering the rela- 

 tions of the planes of a crystal especially advantageous 

 has been the means which the principles of the projection 

 of the sphere afford us of graphically representing within 

 the circumference of a circle the poles corresponding to 

 all the faces, however numerous, that any single crystal 

 or that all the different crystals of a substance may pre- 

 sent, while the symmetry which they obey in their distri- 

 bution is seen at a glance. The stereographic projection 

 employed in Prof. JNIiller's system for this purpose affords 

 by its simplicity, its ready application, and the important 

 geometrical principles which it possesses, by far the most 

 practical, and with a httle experience in the student, much 

 the most intelligible representation of even the most com- 

 plex forms of crystallography. 



The characteristics of the stereographic projection were 

 exhibited in a small working model, in which it was 

 shown that the eye, supposed to be placed at a point on 

 the sphere of projection, would see the arcs of circles on 

 the opposite hemisphere as though projected on a plane 

 screen passing through the centre of the sphere and inter- 

 secting with its surface in a great circle, the circle 0/ pro- 

 jection, at the pole of which the eye was situate ; such 

 arcs of circles on the sphere were shown to be projected 

 as arcs that themselves were circular, and the method of 

 finding the centres for these projected arcs, and again the 

 mode of determining the value of an arc on the projected 

 circle by drawing lines from a projected pole of that circle 

 to the circle of projection, so as to intercept the required 

 arc upon the latter circle, were illustrated in the case of 

 arcs upon the model. 



The next subject taken up by the lecturer was in the 

 form of a digression in which he treated of the relations 

 of the parts into which a line was divided by four points, 

 two of which might be supposed to be stationary, while 

 the two others assume different positions on the line. 

 First the harmonic and then the anharmonic division of 

 such a line was discussed ; and from this, the lecturer 

 passed to the consideration of the harmonic and the 

 anharmonic division of an angle, contained by two and 

 divided by two other lines ; and he showed, firstly, that 

 when two lines out of four passing through the same 



