372 C. K. SWARTZ PROPOSED CLASSIFICATION" OF CRYSTALS 



Theorems. — Axes or planes of symmetry must necessarily intersect 

 symmetrically, since all parts are symmetrically disposed with reference 

 to them. AVe have the following theorems concerning their combinations : 



1. The least number of axes of symmetry that can enter into combination is 



three. Two only can not combine. 2 



2. If a singular axis be intersected by other axes, then the latter are (a) in 



number equal to period of singular axis; (6) in period, two-fold; 

 (c) in position, perpendicular to the singular axis. (See figure 1, 

 where a four-fold axis is intersected by 4 two-fold axes.) Proof: 



(a) If lateral axes are present, their number equals period of singular 

 axis, since all parts are repeated about any axis as often as its 

 period. For example, three axes occur about a three-fold axis, 

 four about a four-fold axis, etcetera. 



(&) If the lateral axes were not two-fold, the singular axis would be re- 

 peated in a new position — that is, it would not be singular. Thus 

 it would be repeated three times about a three-fold axis, etcetera. 



Figure 5. — Alternating 



Figure 1. — Singu- Fig ike 2. — Repetition Figure 3. — Trigonal axes Figure 4. — Axis of sym- axis developed by alter - 

 lar axis with i lat- of singular axis about developed by equal rectan- metry developed at inter- nating axes and symnie- 

 eral axes. oblique axis. gular axes. section of symmetry planes, try planes. 



(c) If the accessory axes were oblique to the singular axis, the latter 

 would be repeated in a new position — that is, it would not be sin- 

 gular. Thus, in figure 2, the singular axis SS would be rotated to 

 position S'S' about an oblique axis a a and no longer be singular. 



3. Intersection of equal axes. — If equal axes intersect, then 



(a) The least number of such axes is three. (See theorem 1.) 

 (&) If the three axes are equal, they are rectangular, since each axis is 

 equally distant from the others if all are equal — that is, perpen- 

 dicular to them. 



(c) If rectangular, their periods can be two or four only. Three and six- 



fold axes can not produce 90 degree positions, since they must 

 intersect at 120 or 60 degrees. 



(d) Four trigonal axes are present, equally distant from the three equal 



axes and intersecting in the center of figure. They are the signs 

 of the equality of the three axes. (See projection, figure 3.) 



4. Intersection of axes and planes of symmetry. 



(a) The intersection of several axes of symmetry lying in one plane pro- 

 duces an axis of symmetry at their point of intersection whose 

 period equals their number. (See figure 1.) This is the converse 

 of theorem 2a. 



2 See proof of this very elementary proposition in Groth's Phvsikalische Krystallo- 

 graphic, fourth edition, 1895, pp. 315-316. 



