26 



independently 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. 16). A 

 point of the figure P is reflected in 

 Vi, and its mirror-image is P' ; then P' 

 is reflected in F 2 , and arrives in P z . 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 2 a 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 easily 

 be demonstrated too, that every rotation 

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

 sive reflections in two mirror-planes, intersecting along LL' under 



x 

 an angle = . 



If both mirrors are simultaneously turned around LL' over an 

 angle /3, while keeping the enclosed angle between them unaltered (= #), 

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

 ber of very interesting theorems in the doctrine of symmetry. A 

 special case is that in which the angle a is infinitely small, the inter- 



Fig. 1 6. 



1) Boldyrew, loco cit. 



