26 Mr. W. Barlow on Crystal Symmetry. 



axes of rotation than one are present, this kind of polyhedral 

 arrangement must be the one displayed. 



With regard to trigonal axes, the only ease in which a set 

 of these axes will mark out the angular points of a polyhedron 

 is the one in which the polyhedron is a cube (Proposition 21). 



Enumeration of the Cases of Plurality of Axes in which 

 there is no Mirror-linage Repetition. 



Since one at least of the polyhedra named in proposition 21 

 must be traceable in every centred system having a plurality 

 of axes, all such systems will be exhaustively traced if every 

 possible case in which one of these polyhedra is formed is 

 discovered. 



In making this exhaustive examination all cases which are 

 without mirror-image repetition are described first. 



Taking the cases in the order given in proposition 21: — 



1. Where the polyhedron traced by a set of axes of the same 



rotation-angle is a cube: n = 4; 0=120°; cf> = 2 .— =180°. 



The axes necessarily involved in the existence of this form 

 of polyhedral arrangement of some of them, are therefore 

 four trigonal axes in the four cube diagonals and three digonal 

 axes through the centres of opposite faces of the cube. And 

 as all the axes of each of these sets are brought to coincidence 

 b}^ coincidence-rotations of the system, the axes of each set 

 are identical. 



The type in which no other coincidence-axis than these 

 exists is given in (PI. I.) fig. VI. (Gadolin, fig. 29). 



As to whether any type in which any other coincidence-axis 

 together with these occurs is possible : — 



{Since the cube-edge a is the minimum distance between 

 two trigonal axes, and there is no point on the sphere of 

 reference whose distance from one or other of the cube- 

 corners is not less than this, it is evident that no additional 

 trigonal axis can exist. 



And it follows that the axis of any additional coincidence- 

 rotation must be so placed that its movements bring the 

 cube referred to to coincidence with itself. Consequently 

 any such additional axis must either pass through the centre 

 point of a face or the centre point of an edge of the cube. 



If one of the digonal axes already occupying the face-centres 

 is converted to a tetragonal axis, or if a digonal axis is added 

 which passes through the points of bisection of two opposite 

 cube-edges, the requisite condition referred to is fulfilled and 

 it can be fulfilled in no other way. 



