27 



in different mirror-planes, simultaneously present. It is supposed 

 in this and all following cases, that the reflecting planes do not act 

 /wdependently of each other, but that only the result of their combined 

 action is always considered. 



The case of a single reflecting plane has already been dealt with, 

 and its general character is now assumed to be understood. 

 We will therefore proceed to the case when two planes of symmetry 



intersect in a line LL' (fig. 18). A 

 point of the figure P is reflected in 

 F 1? and its mirror-image is P'; then P' 

 is reflected in V 2 , and arrives in P 2 . The 

 figure F after these two consecutive 

 reflections will be congruent with itself, 

 and therefore the final position could 

 also be obtained from the initial one 

 by rotating every point of it P through 

 a characteristic angle 2a round LL' , 

 the axis of intersection of the two reflec- 

 ting planes, containing between them 

 an angle, the value of which is #. The 

 repeated, reflection in two planes inter- 

 secting under an angle a, thus appears 

 to be equivalent to a rotation about the 

 line of intersection through an angle 2&. 

 Of course it is clear, and it can also 

 be easily demonstrated, that every rota- 

 tion about an axis LL' through an angle a may be replaced by two 

 successive reflections in two mirror-planes, intersecting along LL' 



under an angle . 



If both mirrors are simultaneously turned around LL' through an 

 angle /3, while keeping the enclosed angle between them unaltered 

 (= a), P! will reach the same final position P 2 , and the same is true 

 for every point of the figure P. Of course the succession of both the 

 reflections considered must remain the same as before. 



The change of position of F thus appears to be quite independent 

 of such a simultaneous motion of both mirror-planes. 



This is a very important principle, and it can be used, as e. g. 

 Boldyrew l ) showed in many cases, for the demonstration of a num- 



L' 



Fig. 18. 



Boldyrew, loco cit. 



