42 



In these cases too it is obvious that all objects and figures, 

 having this particular kind of symmetry, may take a second form 

 which is the mirror-image of the other. In the case of our stirrer, 



the one would correspond to a 

 right-handed, the other to a left- 

 handed screw. 



6. III. So far we have considered 

 those figures which have one axis 



r\ 



of the period , or such as possess 



Fig. 41. 



Propeller. 



two or more binary axes. The 

 only case yet remaining is there- 

 fore that, where the figure has 

 more than one axis with a period- 

 number higher than 2. If this 

 case is treated in the most general 

 way, we can be sure that no other 

 types of symmetry-groups only 



having rotations round axes of the first order, are omitted, and 

 that, therefore, the question of the possible groups of this kind has 

 been finally and exhaustively settled. 



Let us suppose that a figure possesses rotations round an axis A 



of the period , and also such round an axis B of the period . Re- 



membering our previous conclusion, that by the characteristic motions 



of the figure, it itself as well as its R , 



whole system of axes must be made 5 



to coincide with itself, it follows 



necessarily from this, that round A 



there must be a number of n axes 



B equivalent to each other, and in 



the same way round B a number of 



p axes A , all again of the same kind. 



If a sphere with radius r be con- 



structed round the fixed geometrical 



centre of the figure, the points of 



intersection of all these axes B will 



be situated in the corners of a regular polygon with n sides, 



and those of the axes A in the corners of a regular polygon 



with p sides. As the whole system of axes must include a finite 



-pig. 42. 



