23 



figure must be rotated round the axis considered, to make it coincide 

 with itself. Of course the axis is supposed to pass through the geo- 

 metrical centre of the figure 0. 



If this angle is , the figure will also be brought to self -coincidence 

 by rotations through 



2a, 3a na, the 



number n being an in- 

 teger part of 27T. For 

 if this were not the 

 case (fig. ij), then, after 

 n times turning round 

 the axis A, a straight 

 line of the figure AL 

 situated in a plane per- 

 pendicular to A , would 

 finally reach the posi- 

 tion AL 8 , and because 

 the figure is still con- 

 gruent with itself, AL 

 could be brought into 

 the position AL 8 by 

 turning the figure round A in an opposite direction over the 

 angle 0, this being smaller than a. It then follows that /3 would 

 also be a characteristic angle of the figure when rotated round its 

 axis of symmetry A, which is impossible because a is supposed 

 to be the smallest angle of this kind. Obviously /3 must thus be 

 equal to zero, and therefore: 



r\ 



x = , where n = 1 , 2, 3, 4, 5, 



8 



etc. to oo 



n 



The number n determines the period of the axis A, and can have 

 all integer values between 1 and oo , for n = oo , # being 0. The 

 axes are, according to the values of n, regarded as binary (n = 2) 

 ternary, quaternary, quinary, senary, etc. axes. l ) For n = oo , the 

 axis is called an axis of isotropy, and it has a special importance with 

 respect to many physical phenomena, as we shall see afterwards. 



b. Axes of the second order. 



In the same way we may distinguish the axes of the second order 



1 ) They are also called axes of threefold, fourfold, etc. symmetry, or trigonal, 

 tetragonal, etc. axes. These last names are however exclusively used in crys- 

 tallography. 



