PEINCIPLES OF CRYSTALLOGRAPHY. 265 



systems, all sections perpeudiculiir to it similar, on account of the sym- 

 metry accoi(lin£»- to the six tautozonal planes of symmetry; on this 

 account, therefore, having the same optical relations. The base 1 1 1 is 

 perpendicular to the optical axis. 



7. Tesseral (isometric) system. — Nine planes of symmetry, three 

 perpendicular to each other and similar, the other six intercalated tau- 

 tozoually at 45° between each two of the first. 



If we take the first three planes of symmetry parallel to the three 

 principal sections, it results immediately, from the existence of the other 

 planes of symmetry, that the ellipsoid of polarization must be a sphere 

 whose radius is difterent for different colors. A sphere has only circu- 

 lar sections ; therefore, simple refraction is produced in all directions. 



We have above considered only the cases where the similar principal 

 sections of all colors very nearly coincided ; the exceptions to this law 

 are really very rare, and present no difficulties. Observation by means 

 of monochromatic glasses or sources of light always allows a very quick 

 orientation. 



We have also in the above description left out crystals with one axis, 

 which polarize circularly, because they, in spite of the greatest theoret- 

 ical differences, can practically be regarded exactly as the other mono- 

 axial crystals, with the exception of the image of the axis, which inside 

 of the rings shows that the black cross is replaced by a uniform color, 

 which is dependent on the thickness of the plate. 



It is now no longer necessary to describe the special behavior of sec- 

 tions of different crystals with respect to the orientation of their direc- 

 tion of vibration. The orientation of the ellipsoid, with regard to the 

 axes of the crystal and their respective planes of symmetry, is given 

 above ; if, therefore, the crystallographic orientation of a plate is known, 

 the kind of section in the ellipsoid and the directions of vibration can 

 be at once determined. Inversely, the experimentally easily-determined 

 position of the direction of vibration of a section of known crystal- 

 lographic orientation gives a starting-point for the determination of 

 the system. 



Eeviewing the method of development of the foregoing sketch, we 

 see, as the starting-point, the law of experience, that by the selection 

 of a certain method of representation, the symbols of all faces and zones 

 consist of whole numbers, whose relations with one another are therefore 

 rational numbers. 



From the rationality of these numbers follows, in a way which we 

 could only briefly dwell upon, that only such groupings of faces are 

 possible which belong to one of the seven different kinds of symmetry, 

 the seven systems of crystallization. From the general law of the 



