17 



dicular to this axis LL', and meeting it at P. From fig. n it will be 

 seen at once, that a point 5 in the figure F, by the rotation through 

 180 round any axis LL' through P will arrive at s, and by 

 a further reflection in the plane perpendicular to LL' will be 

 brought to S' . The transformation of S into 5' is, however, evidently 

 equivalent to an inversion with respect to P, and it can easily 

 be seen that the result is quite independent as well from the 

 special choice of LL', as from the sequence of reflection and 

 rotation, so long as the reflecting plane W be kept perpendicular 

 to the axis LL'. 



4. From this it will be easily understood that every transformation 

 of a figure F from a definite original position S^ into its mirror-image 

 F' with a position S 2 ', can always be executed by a combination of some 

 rotation round an axis LL' and a successive reflection in a plane V 

 perpendicular to that axis. 



For by the inversion of F with respect to an arbitrarily chosen 

 point in space P, it moves from the position 5 X into a position S/, 

 in which it is changed then into its mirror-image F'. Since the 

 figure in this new position S/ and that in the desired final position 

 S 2 ' are now congruent, (for they are both mirror-images of the 

 same figure F) the transition of S/ to S 2 ' can be made by a single 

 rotation through an angle a round an axis LL' passing through P, 

 if only the point P be suitably chosen, so as to coincide with the 

 geometrical centre of F : otherwise a translation must also be finally 

 made to complete the transitio nof S/ into 5 2 '. This, however, does 

 not affect the general validity of the demonstration. Now the 

 inversion can be substituted by a rotation through 180 round an 

 axis which for this purpose can be made coincident with the above 

 mentioned axis LL', the whole angle of rotation now becoming 

 (a + ir), and by a reflection in a plane V perpendicular to LL'. The 

 total transition from 5 T to S 2 ' i s thus performed by a single rotation 

 through an angle (& -f TT) round an axis LL', and a reflection 

 combined with it, in a plane V perpendicular to this axis LL'. 



Another demonstration of this important theorem will be given at 

 the end of this chapter as a consequence of our considerations of 

 repeated reflections in several planes. 



5. A few remarks must now be made on the difference of finite 

 (limited) and infinite (unlimited) figures in general. Instances of 

 such finite figures are polyhedra, and all objects with a limited form. 



On the other hand unlimited systems of points distributed in 



