20 



characteristic rotations round the axes of symmetry A and B, inter- 

 secting in the fixed point of the figure, this last must necessarily 



have a third characteristic rotation 

 round another symmetry-axis pas- 

 sing through 0. The last rotation is 

 equivalent to the former two, ex- 

 ecuted in succession. Thus from 

 the existence of two axes of sym- 

 metry the existence of a third 

 necessarily follows. 



But if the figure considered has 

 two axes of the second order, in- 

 tersecting in 0, the resulting mani- 

 pulation must be an ordinary rota- 

 tion about an axis of symmetry 

 of the first order, passing through 

 0. For by the first operation of 

 the second-order the figure F is 

 transformed into its mirror-image 

 F', and this is in its turn changed 

 by the operations corresponding to 

 the second mirror-axis into a figure 

 congruent with the initial one. 

 Executed successively they are 

 therefore together equivalent to 

 an ordinary rotation; and this is 



evidently the same thing as the assertion made already (page 14 

 and 15), that a figure cannot possess several symmetry-properties 

 of the second order alone, without the simultaneous possession 

 of those of the first order too. 



From both these examples it will be clearly understood that there 

 are essential mathematical relations connecting the different sym- 

 metry-properties of one and the same figure simultaneously present, 

 Thus the simultaneous existence of two such properties always 

 requires the existence of a third one equivalent to the other two 

 combined. It is precisely the existence of such relations, that makes 

 the whole doctrine of symmetry a possibility, and permits us to 

 deduce the general classes of symmetrical types by mere mathe- 

 matical reasoning. 



8. However in this connection the question might rise: what 



A 



