28 



angle, this axis of course also passing through 0. The figure F 

 remains obviously, congruent with itself in these successive ope- 

 rations because their number is even. 



It will be immediately clear from this that the theorem of 

 Euler previously mentioned can now be demonstrated with out the 

 slightest difficulty. For in this theorem it was stated that the suc- 

 cessive rotations over angles a and /3 round two axes A and B, 

 intersecting in 0, are always equivalent to a rotation over a certain 

 angle 7 round a third axis C, also passing through 0. Now for every 

 rotation of both A and B, there may be substituted a successive reflec- 

 tion in two mirror-planes passing through A and B, and intersecting 



at angles -=- and -^ respectively. 



*. 



But then we have the above 



mentioned case of four planes intersecting in a single point 0. Thus 

 the result is equal to a single rotation round C, also passing through 

 0, through a definite angle y .which can be easily found by 

 geometrical construction from the angles a and /3 1 ). This funda- 

 mental theorem is now sufficiently demonstrated, and the base is 

 established for the mathematical treatment of the symmetry-problem 

 in those cases in which only axial symmetry is considered. 



If the four planes do not pass through the same point 0, an 

 additional translation will happen to add to the resulting rotation, 

 a helicoidal motion thus being substituted for the single final rotatioi 

 mentioned above. But in this case also the general conclusions will 

 still be valid, it being only necessary (for wfinite figures) to sub- 

 stitute the helicoidal for the ordinary rotation. Of course in this 

 case too, the figure remains congruent to itself. 



Take O as centre of a sphere (fig. 18), a being the point of intersection 

 of its surface with A, b being the same for 

 B. Let ab be joined by a great circle. Now 

 if the great circle aa' be so constructed, that 



the angle a'ab is and in a sense opposite to 



the direction of the rotation round A ; and if 

 bb' be a great circle, constructed in the true 

 sense of the rotation round B, b'ba being 



2 



equal to , the intersection c is the point, 



where the new axis OC pierces the surface 

 of the sphere, and its characteristic angle is 

 equal to 2 = y, the sense of rotation being 

 readily found. 



Fig. 1 8. 



