CHAPTER IV. 



Some general Remarks on Axes of the Second Order. Relations 

 between Groups of the Second order and those dealt with in the previous 

 Chapter. General Theorems concerning Groups of the Second Order. 

 Deduction of all possible Groups of this kind. Summary. 



1. We have now to consider such figures as have symmetry- 

 properties of the second order as well as those of the first. For we 

 have already seen that the existence of symmetry-properties of 

 the second order necessitates also the coexistence of those of the 

 first, unless the figure has only a single axis of the second order. 

 This follows from the simple fact that every combination of two 

 symmetrical operations of the second order is always equivalent 

 to some rotation. 



Besides of this conclusion, we shall later also make use of the 

 fact already stated, that in many cases a mirror-axis may be 

 replaced by a suitable combination of an axis of the first order 

 with an inversion-centre or a plane of reflection. 



Of course the most simple instances of groups of the second order 

 will be those which are completely analogous to the cyclic groups 

 in the previous chapter, but in this case having a single mirror- 

 axis of the period . We shall call them cyclic groups of the second 



order, and give them the symbol C n , the horizontal line above 

 the C indicating that the unique characteristic axis is now one 

 of the second order. Before describing their general character, it 

 will seem appropriate to extend our former considerations regarding 

 the axes of the second order (see p. 23 to 25) to every arbitrary 

 value of the period-number n. 



2. Let A n be an axis of the second order with a period - , 



n 



4 



