Mr. W. Barlow on Crystal Symmetry. 29 



axis at right angles to them and no other axis, as shown in 

 (PL II.) fig. IX. (Gadolin, fig. 47). 



As to the possibility of the existence of any case of this 

 polyhedral arrangement in which other axes are present in 

 addition to those found in the case just described : — . 



The existence of the trigonal axis perpendicular to the 

 plane of the three digonal axes requires that if an additional 

 axis is found, there shall be at least two other additional axes 

 identical with it. 



Now in the system described on p. 26 and (PI. II.) fig. VII. 

 there are six: digonal axes, and the twelve points in which thev 

 intersect the sphere of reference are, it is evident, the twelve 

 bisecting points of the edges of a cube so drawn that these 

 edges touch the sphere ; and these twelve points may be 

 regarded as lying on three parallel planes drawn perpendicular 

 to one of the trigonal axes of the system, six points being found 

 on the middle plane and three in each of the others. This is 

 the only possible way in which digonal axes can be added in 

 the case of the polygonal arrangement under consideration. 



For if the axes lying in the centre plane be the three 

 essential digonal axes, and so their distance apart, measured 

 by the side of the hexagon indicated by their points of inter- 

 section with the sphere, has the minimum value a, it is 

 evident that each of the three added digonal axes is at the 

 same minimum distance from the remaining two and also 

 from two of the three essential axes, and that any departure 

 from this relative situation for three additional axes would 

 bring one at least of the four distances referred to below the 

 minimum. Thus positions for the added axes further from 

 the centre plane would bring the distance apart of the three 

 added axes below the minimum, and positions nearer to the 

 centre plane would shorten some at least of their distances 

 from the essential axes. 



The question remains, can the system be enlarged in any 

 other way while a continues to be a minimum ? 



Now any position for an additional axis of a higher type 

 except the positions occupied by such axes in the enlarged 

 system referred to is precluded by proposition 22 ; the only 

 point left to consider is, therefore, the possibility of converting 

 the trigonal axis into a higher order of axis — an hexagonal 

 one. This is impossible because the resultant of the rotation 

 about such an axis and the rotation about one of the three 

 essential digonal axes, would be a digonal axis bisecting 

 one of the angles of G0° subtended by these axes; and this 

 would prevent a from being a minimum *. 



* See proof of general proposition 20, p. 24. 



