22 



8 



Fig. ii. 



is known when its characteristic angle of rotation * is given. That 

 angle is defined as the smallest angle, through whfch the symmetrical 

 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 ot, 

 the figure will also 

 be brought to self- 

 coincidence by rota- 

 tions through 2a, 3a 

 na, the number n being 

 an integer part of IK. 

 For if this were not 

 the case (fig. //), then, 

 after n times turning 

 round the axis A, a 

 straight line of the fi- 

 gure ALi situated in a plane perpendicular to A, would finally 

 reach the position AL S 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 (3, 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: 



2?r 



a. = - , where n = 1 , 2, 3, 4, 5, . . . . 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 , a, being 0. The 

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

 ternary, quaternary, quinary, senary, etc. axes. x ) 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. 



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

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

 tallography. 



