28 



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

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

 section LL' thus being situated at an infinite distance. The two 

 planes V l and V 2 (fig. zp) are then parallel; their distance apart 

 may be a. The repeated reflection is now evidently equivalent 

 to a translation = 2a. . 



Such translations and parallel planes of reflection are often 

 characteristic of infinite figures or sys- 

 tems; for finite figures they have no im- 

 portance. It is moreover evident that in 

 the last mentioned case the result will 

 remain unaltered, if both planes are shifted 

 parallel to themselves, provided that their 

 mutual distance be kept constant = a. 



11. We shall now consider the case 

 when reflection occurs successively at four 

 reflecting planes which do not act inde- 

 pently of each other, and which pass 

 through the same point 0. Then it can be 

 easily proved by the aid of the principle 

 of the simultaneous rotation of two inter- 



P, 



Fig. 19. 



secting planes just mentioned, that these successive reflections 

 in four planes are equivalent to a reflection in two planes passing 

 through 0; or, which is the same thing, to a single rotation around 

 an axis passing through 0. 



Let the four planes considered be 5j, S 2 , S 3 , and S 4 ; S l and S 2 

 may intersect along a straight line OL lj2 , and S 3 and S 4 along OL 3>4 . 



Now we can first turn the two mirror-planes S l and S 2 simul- 

 taneously round OL 1>2 , until 5 2 passes through OL 3>4 ', the effect 

 of the successive reflections in 5 3 and S 2 will not be altered by this, 

 provided that the angle of intersection ot, between S and S 2 remain 

 the same. Now we will turn the planes S 3 and S 4 together round their 

 intersection OL 3}4 , until S 3 passes through OL 1>2 . There will be 

 no change in the effect of the successive reflections in S 3 and 5 4 

 by this. But now 5 2 , as well as S 3 , coincides with the plane passing 

 through OL 1)2 and OL 3>4 , the reflections in the planes S' 2 and S' 8 , 

 being the new positions of S 2 and S 3 , - - neutralising each 

 other. Thus there remain only the successive reflections in two 

 planes S\ and S' 4> these being the positions of 5j and S 4 finally 

 reached after completing the above mentioned turnings of the 



