13 



.mil ivtrrrnce, are numbered / to 8. Let us moreover take 

 arbitraiy point P in space, outside the cube A, and draw a 

 i-ht lint /./.' through it parallel to one of the four upright ed 

 tin- rnhr. lt'now.1 be revolved round the axis L' through angles 

 90, 2 x 90, 3 x 

 the cube A 

 successively 

 o the positions 

 C, f>, which posi- 

 ns differ from A, 

 tin- numbers at 

 clearly 



le corners 

 -how. Because the 

 now has a spe- 

 ll symmetry of its 

 however, the 

 ires B, C, and Z>, 

 be made to coin- 

 le eventually with 



by merely shifting them parallel to themselves along the plane 

 revolution. Thus, by definition, the rotations through angles of 90, 

 *: 90, 3 x 90 round an axis parallel to one of the edges of the 

 ibe, are characteristic rotations for the symmetry of this figure. 

 the positions B, C, and I) differ from A, these three characteristic 

 itions are non-equivalent. A rotation through an angle of 360 how- 

 , r er would be equivalent to that through 90, etc. 

 In the same way we should find, that if through some point in 

 ice Q a straight line were drawn parallel to one of the four longest 

 liagonals of the cube, rotations through angles of 120 and 2 x 120 

 nmd this line as an axis, would appear to be "characteristic motions" 

 It is easily seen that for the plane drawing of fig. 3, rotations 

 rough 72, 2 x 72, 3 x 72, and 4 x 72 round an arbitrary axis 

 ssing through a point R in space and perpendicular to the plane of 

 tin drawing, are also characteristic motions, and non-equivalent ones. 

 In the same way it must be evident that if the rotation of the 

 cube A in fig. 6 around LL' had been through an angle differing from 

 those mentioned, coincidence could not have been realised by shifting 

 alone ; and the same would have been the case, if the rotations around 

 /-// through 90, etc., were applied to a different figure instead of 

 to a cube, e. g. to an unsymmetrical figure. 



