33 



2r 



. 1: at = '-, it possesses evidently n such non-equivalent ro- 

 tations. For the whole series of rotations: A(a), A(2), /1(3).... 



.\(nx) corresponds to a series of successive positions S lt S 2 , S, 



\\hii li air all different from each other, 5 n being finally identical 

 ith the initial position 5 . 



ow these n non-equivalent rotations are connected with each 

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

 if we perform these rotations successively, the result will be 

 ird rotation which however appears to be equivalent to one or other 

 ined in the series of n terms. Thus the combination of A(j<z) and 

 )x] e.g., will equal the rotations A [(n 2)0] or A [(n <?)#], which 

 included in the series mentioned above ; if we take A (6<x) and 

 'n /)], the resulting rotations will be A[(n /)<*], or ^4[(+j)#], 

 . = A(5&), both of these being als included in the same series, etc. 

 ked at from this standpoint, we can say that this series of n non- 

 ivalent rotations represents in a certain sense a closed system 

 rotations. 



uch a closed system of quantities which are different from each 

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

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

 ent rotations corresponding to an axis of symmetry with a period 



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



finite group of rotations. 



lat the symmetry-properties of each figure now really represent 

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

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

 red : viz : /) that every two symmetry-properties of a figure when 

 ibined, always necessitate the existence of a third one, equivalent 

 them together ; and 2) that the symmetry of a figure is absolutely 

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

 ''iin'te figures are characterised by a limited number of non-equivalent 

 symmetry-properties; and the problem of finding all possible combi- 

 nations of symmetry-properties of finite figures, is that of finding out 

 all finite groups of these properties. The latter being determined 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 determi- 

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

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



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