27 



ii /./.' thus bein;; situated at an infinite distance. The two 



F! and F 2 (fig. 77) are then parallel; their distance 

 iv be </. The repeated reflection is now evidently equivalent 

 to a tninslntion = 2a. 



Such translations and parallel planes of reflection are often 

 ic t eristic of infinite figures or sys- 

 ;; for finite figures they have no im- 

 >rtance. It is moreover evident that in 

 last mentioned case the result will 

 main unaltered, if both planes are shifted 

 irallel to themselves, provided that their p, 

 iiitual distance be kept constant = a. 

 11. Now we shall consider the case 

 reflection occurs successively at four 

 jflecting planes which do not act inde- 

 mtly of each other, and which pass 

 through the same point 0. Then it can be 

 isily proved by the aid of the principle 

 the simultaneous rotation of two inter- 



p, 



Fig. 17. 



:ting planes just mentioned, that these successive reflections 

 four planes are equivalent to a reflection in two planes passing 

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

 i axis passing through 0. 



Let the four planes considered be S lf S 2 , S 3 , and 5 4 ; Si and 5 2 

 lay intersect along a straight line OL 1>2 , and S 3 and S 4 along OL 3 ,4. 

 Now we can first turn the two mirror-planes 5j and S z simul- 

 taneously round OL 1>2 , until S 2 passes through OL 3>4 ; the effect 

 )f the successive reflections in 5 X and S, will not be altered by this, 

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

 ic same. Now we will turn the planes 5 3 and S 4 together round their 

 itersection OL 3>4 , until 5 3 passes through OL 1>2 . There will be 

 10 change in the effect of the successive reflections in 5 3 and 5 4 

 >\ this. But now 5 2 , as well as 5 3 , coincides with the plane passing 

 through OL 1>2 ; and L 3 , 4> the reflections in the planes S' 2 and S' 3> 

 being the new positions of 5 2 and 5 3 , neutralising each 

 other. Thus there remain only the successive reflections in two 

 planes S\ and 5' 4 , these being the positions of 5 X and 5 4 finally 

 reached after completing the above mentioned turnings of the 

 four original planes. The reflections in S\ and 5' 4 are together 

 equivalent to a rotation round their intersection through a certain 



