29 



12. \\V ran now put the question: what will be the final result 

 the sneerssive reflections of a figure in three arbitrarily situated 

 -plunest Here also only tin- iv-ult ot their combined action i- 

 tvestigated. Let the three planer he .S',, S., and .S'.,. \\Vwillnow 

 nn the planes S l and S., together around their line of inter- 

 tinn Hinultaneously, in the way mentioned before, until 5 2 

 through /. being a perpendicular to 5 3 . The successive reflec- 

 >ns at >,. S a , and .S 3 , are now substituted by their equivalents in 

 ',, >',, and 5 3 , S' 2 being thus perpendicular to S 3 . Now in the 

 ie way we can turn the planes S' 2 and S 3 simultaneously round 

 ieii intersection (their enclosed angle ( 90) of course being 

 ;ept nnaltered), until at last S 3 passes through the perpendicular 

 > ",. The whole series of original reflections in 5,, S 2 , and 5 3 , 

 thus substituted by such in S\, S" 2 , and S' 3 , of which S' 3 is 

 irpendicular to S' lt as well as to .S'",. 



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

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

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

 rations thus appears to be equivalent to a rotation around an 

 ds L, combined with a reflection in a plane S' s perpendicular 

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

 ige F'. 



We can therefore say in general *) : The result of th? successive 

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

 lependently of each other, is equivalent to a certain rotation round an 

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

 rir point of intersection being the common point of the three planes, 

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

 This resulting operation is evidently equivalent to what we have 

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

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

 planes can be always reduced to one of the two proceeding 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 

 itself at the end. The reduction to the two cases described in the 

 above, takes place by turning every two new planes simultaneously, 



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



