( 838 ) 



with each other. So to coiigTiieiit foiiibinatioiis in tlic group of 

 (lie eqiiiaiigiihir double rotations to the right (left) in S^ correspond 

 congruent combinations in the motion group of Sr {Si). As moreover 

 the 



'•2 3 "I ^14 

 ^•.. + K. 

 /l2 ~r ^3 4 



of a plane are the cosines of direction of the representant of the 

 system equiangular to the right with that plane with respect to the 

 system of axes OXr Vr ^r (defined Proceedings March! 904, p. 72ö), 

 and likewise 



the cosines of direction of the representant of the system equiangular 

 to the left with that plane with respect to the system of axes 

 OXi Yi Zi (defined in the same place) the S,. and >S'i introduced just 

 now prove to be identical with those introduced here formerly (see 

 Proceedings March, 1904, p. 725} so that they represent not oidy 

 l)y their motions the equiangular motion groups of S^ to the right 

 and to the left, but also by their vectors the systems of i>lanes equi- 

 angular to the right and to the left (with direction of rotation) of 

 S^ and so that the angle of Ihe representing vectors is tlie angle of 

 the systems of planes themselves. 



So also to congruent combinations in the twodimensicmal mani- 

 foldness formed by the equiangular systems of planes to the right 

 (left) correspond congruent combinations in S,- (<S,). This is an algebraic 

 proof for 2"f^. to its full extent. 



This deduction has at the same time made clear the meaning of 

 the four parameters of homogeneity for the general congruent three- 

 dimensional transformation about a fixed point, namely the cosines 

 of direction of the vector indicating the corresponding equiangular 

 double rotation to the right (left) of an S^ of which this S^ is 

 the Sr (Si) and the svstem of axes in S^ the system OXr Y, Zr 

 (OXiYiZ,). 



