425] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 471 



From (iv), 



From (v), 



by multiplication, 



but, from (vi), 

 whence, we deduce, 



= -(24) (34). 

 = + (23) (34); 



= -(23)(24)(34) 2 ; 

 7 S = -(23)(24); 



ft 2 - (34) 2 . 



The significance of the double sign in the value of ft will be afterwards 

 apparent ; for the present we take 



From (ii) B = + (23), 



From (iii) 7 = - (24), 



while the group (vii) will be satisfied if 



The scheme of orthogonal transformation for the Right Vector (for so we 

 designate the case of ft = + (34),) is as follows : 



+ a + ft +7 + B 



-/3 + a. + B - 7 



-7 - B + a +/3 



- B +7 -ft + a 

 If we append the condition 



then we have completely defined the Right Vector. 

 We now take the other alternative, 



= -(34); 



then, from (ii), B = (23), 



then, from (iii), 7 = + (24). 



We thus have for the Left Vector, the form, 



+ a + ft +7 + B 

 ft + a. 8 +7 

 -7 +8 + a -ft 



- B -7 +/3 + a 



