16 



dicular to this axis LL' , and meeting it at P. From fig. 9 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 5 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 VV be only kept perpendi- 

 cular to the axis LL'. 



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

 of a figure F from a definite original position Si into Us 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 Sj into a position Sj', 

 in which it is changed simultaneously into its mirror-image F'. 

 Since the figure in this new position Si', and that in the desired final 

 position 82' now are 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 finally also be made to complete the transition of Si' into Sg'. 



This however does not affect the general validity of the demon- 

 stration. 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 (ot, -f TT), and a reflection in a plane V perpendicular 

 to LL. The total transition from Sj to S 2 ' is thus performed 

 by a single rotation through an angle (# + IT) 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. 





