634 REPORT— 1894. 



encouraging the adoption of the true Millerian symhol in the still outstanding case 

 of the Ehomhohedral System. 



Rationality of Indices and the Laio of Zones. — It may here be pointed out 

 that, although the importance of zones for the simplification of crystaUographic 

 calculation had been recognised by Weiss, it was only later that Neumann proved 

 that the fact that all possible crystal faces can be derived by means of the inter- 

 section of zones is a necessary consequence of the rationality of the indices ; that, 

 indeed, the law of zones is mathematically identical with the law of rationality. 

 To the same able physicist and mathematician we owe the development of the 

 method of stereographic projection now in common use by crj'stallographers for 

 the representation of the poles of crystal faces. 



Symmetry. — We have said that the recognition of six systems of crystallisation 

 was a result of consideration of the lengths and mutual inclinations of certain 

 lines called axes. Now, it had long ago been remarked that any one face of a 

 crystal is accompanied by certain others similarly related to the geometrically 

 similar parts of what may be regarded as a fundamental figure : such a group of 

 concurrent faces is called a simple form. It came to be recognised, too, that all 

 the faces of such a form can be geometrically derived from any one of them by 

 repetition, according to certain laws of symmetry, and that the same laws of 

 symmetrj^ are binding for every simple form or combination of forms exhibited by 

 crystals of the same substance. Hence it came to be perceived, though very 

 slowly, that the essential diflerences of the systems of crystallisation are not mere 

 differences of lengths and mutual inclinations of lines of reference, but are really 

 differences of symmetry. Ever since his appointment to the professorship of 

 Mineralogy in this University, now tliirty-eight j-ears ago, Mr. Maslcelyne has 

 been persistent in directing attention to the importance of symmetry', and such 

 importance now receives universal recognition. 



Thirty-two Types of Symmetry in Crystals. — But in each system of crystallisation 

 it becomes necessary to recognise both completely and partially symmetrical types. 

 In the latter, the symmetry is in abeyance relative to various planes or lines which 

 In other crystals of the same system are active as planes or axes of symmetry. But 

 this abeyance of symmetry is itself found to be subject to a law, for all planes or 

 axes of symmetry which are geometrically similar are either simultaneously active 

 or simultaneously in abeyance. By means of this law relating to partial symmetry, 

 it has been inferred that altogether thirty-two types of symmetry are possible in 

 the six crystalline systems. 



The possible existence of these thirty-two types of symmetry of crystals is thus 

 an induction from observation : the question naturally arises as to why only these 

 thirty-two exist or are inferred by analogy to be possible. Axes of symmetry are 

 observed round which faces of crystals are symmetrically repeated by twos or threes 

 or fours or sixes ; why is it that in crystals no axis of symmetry is ever met with 

 round which the faces are symmetrically repeated by fives or sevens ? A few words 

 as to how this most important problem has been attacked and solved may be of 

 interest. 



We know that the characters of a crj'stal relative to any line in it vary with 

 the direction of the line, but are the same for all lines parallel to each other. Such 

 a property will result if we imagine with Bravais that in a crystal elementary 

 particles are arranged at equal distances from each other along every line, and are 

 similarly arranged in all those lines which are parallel to each other ; the distances 

 separating particles being, however, in general different for lines which are inclined 

 to each other. Such an arrangement of particles is termed parallelepipedal : space 

 may be imagined to be completely filled with equal and similarly disposed parallele- 

 pipeds, and an elementary particle to be placed at every corner or quoin of each. 

 Further, each particle is regarded, not as being spherical, but as having diflerent 

 characters on its different sides ; and the particles must be similarly orientated — that 

 is, have similar sides in similar positions. 



Now, it will be seen on examination of a model or figure that with such an 

 arrangement any plane containing three particles will contain an infinite number, 

 all arranged at the corners of parallelograms. Further, any such plane will clearly 



