( 725 ) 



angular to the right by the line of intersection of' the plane through 

 OW belonging to it with S, ± OW. That ^S^ is thus, entirely filled 

 with these representing lines which are in (l,l)-correspondence with 

 the represented systems of planes. 



We shall call that S^ J. W regarded as a complex of the rays 

 representing the equiangular to the right systems of planes, "the repre- 

 senting S^ to the right of S^' or shorter "the Sr of/S/'. In the same 

 way we form the "Si of S". Each pair of planes in S^ is then 

 unequivocally determined by its representants in S,- and 6'/ and rever- 

 sely the pair of planes determines unequivocally its representants. 



Theorem 4. An equiangular double rotation to the right of the S^ 

 about a certain equiangular system to the right which double rotation 

 leaves according to theorem 'J the position of Si unchanged, gives a 

 rotation of Sr about the representant of the system of the planes of 

 rotation over an angle equal to double the angle, over which the 

 equiangular double rotation to the right of S^ takes place. 



Proof. In the first place ensues from theorem 

 3 that the representants in S,- keep making mutu- 

 ally the same angles; so Sr has a "motion as a 

 solid". We suppose through OW io be laid its 

 plane of rotation a in S^ and its normal plane /?. 

 In fig. 8 we suppose the lines tending downward 

 (0 lie in a and those tending upward to lie in /? 

 whilst the indicated directions of rotation are 

 those of the equiangular double rotation to the 

 right which we consider. Angle IFOC is made equal 

 Fig. 8. to è ^- Then the Sr is the S^ through OC and 



/3. Let OP be an arbitrary vector in ^ and (p the angle, over which 

 the equiangular double rotation to the right takes place. The double 

 rotation leaves OC unchanged as representant of the equiangular 

 system to the right on («jS). Moreover it transfers OW'mio OIF' and 

 OP into OP'. If we then still make /^P"OP' equal to Z.P'OP 

 we have : 



(OP, OP \ 



I nwA ^qu^^ï^g^il^i' to the right, thus 



rOP\ OP \ 



\0W, OW') 



OP", OP' 



OW, OW' 



(OP', 0W\ 



\0P', OW') 



equiangular to the left, or also : 

 equiangular to the left, so at last 



equiangular to the right. 



48* 



