34 



r\ 



a period: a = , it possesses evidently n such non-equivalent ro- 

 tations. For the whole series of rotations: A (#), A(2a), A (3#). . . . 



A(no) corresponds to a series of successive positions Sj, S 2 , S n 



which are all different from each other, S n being finally identical 

 with the initial position S . 



Now these n non-equivalent rotations are connected with each 

 other in a particular way. For if we take any two of them at random, 

 and if we perform these rotations successively, the result will be 

 a third rotation which, however, appears to be equivalent to one or 

 other contained in the series of n terms. Thus for instance the combi- 

 nation of A(ja) and A[(n 5)^], will equal the rotations A[(n 2)0,] 

 or A[(n 8)oi\, which are included in the series mentioned above; 

 if we take A(6x) and A[(n i)oi\, the resulting rotations will be 

 A[(n /)#] or A[(n + 5)^], i. e. = A(^), both of these being also 

 included in the same series, etc. Looked at from this standpoint, 

 we can say that this series of n non-equivalent rotations represents 

 in a certain sense a closed system of rotations. 



Such a closed system of quantities which are different from each 

 other, but of which each two are together equivalent to some other 

 term of the system, is called a group of quantities. The n non-equi- 

 valent rotations corresponding to an axis of symmetry with a period 



r\ 



ex, = , therefore, form an instance of this kind: they represent 



a finite group of rotations. 



That the symmetry-properties of each figure now really represent 

 a system of things forming a closed group of this kind, will be at once 

 clear when the two facts mentioned in the last chapter are remem- 

 bered : viz : i) that every two symmetry-properties of a figure when 

 combined, always necessitate the existence of a third one equivalent 

 to the two combined ; and 2) that the symmetry of a figure is abso- 

 lutely known when all its non-equivalent symmetry-properties are 

 known. Finite figures are characterised by a limited number of non- 

 equivalent symmetry-properties ; and the problem of finding all pos- 

 sible combinations of symmetry-properties of finite figures, is that of 

 finding out all finite groups of these properties. The latter being deter- 

 mined by the corresponding rotations round axes of the first or of the 

 second order, the problem to be solved can be described as that of deter- 

 mining all finite groups of rotations of the first and the second order. 



2. In this chapter we shall only deal with the relatively simple 



