30 



for the ordinary rotation. Of course also in this case the figure 

 remains congruent to itself. 



12. We can now put the question: what will be the final result 

 of the successive reflections of a figure in three arbitrarily situated 

 mirror-planes? Here also only the result of their combined action is 

 investigated. Let the three planes be S 1} S 2 and S 3 . We will now 

 turn the planes S 1 and S 2 at the same time around their line of 

 intersection in the way mentioned before, until S 2 passes through Z, 

 being a perpendicular to S 3 . The successive reflections at S lt S 2 , and S 3 

 are now substituted by their equivalents in S\, S' 2 , and S 3 , the 

 plane S' 2 being thus perpendicular to 5 3 . Now in the same way we 

 can turn the planes S' 2 and 5 3 simultaneously round their inter- 

 section (their enclosed angle (= 90) of course being kept unaltered), 

 until at last S 3 passes through the perpendicular to S\ . The whole 

 series of original reflections in S lt S 2 , and S 3 , is thus substituted 

 by such in .S\, S" 2 , and S' 3 , of which S' 3 is perpendicular to S\ 

 as well as to S" 2 . 



But the reflections at S l and S" 2 being both perpendicular to 

 S' 3 , can be substituted by a rotation around their line of intersection 

 L, this of course being a perpendicular to S' 3 . The whole series of 

 operations thus appears to be equivalent to a rotation around an 

 axis L, combined with a reflection in a plane S' 3 perpendicular 

 to it; of course the figure F is transformed by this into its mirror- 

 image F'. 



We can therefore say in general l ) : The result of the successive 

 reflections of a figure F in three arbitrarily situated planes not acting 

 independently of e.ach other, is equivalent to a certain rotation round an 

 axis, combined with a reflection in a plane perpendicular to that axis, 

 their point of intersection being the common point of the three planes. 

 The figure F is changed thereby into its mirror-image F'. 



This resulting operation is evidently equivalent to what we have 

 previously called a rotation round an axis of the second order. 



13. It will be easily seen that the successive reflections at n 

 planes can always be reduced to one of the two preceding cases, ac- 

 cording as n is an even or an odd number. For if n is odd, it may be 

 reduced to the reflections in three planes; and if n is even, to such 

 in four planes. If n is odd, the figure F is finally changed into its 

 mirror-image F' , while if n is even, F always remains congruent with 



C. Viola, Zeits. f. Kryst. 26. 519. (1895). 



