430] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 477 



the result obtained is a displacement of the most general type called a 

 motor. We now prove Clifford s great theorem that a right vector and a 

 left vector can be determined so as to form any motor, i.e. to accomplish 

 any required homographic transformation that conserves intervene. 



For, if we identify the several coefficients at the foot of p. 474 with those 

 of 420, we obtain equations of the type 



(11)= ace - ftp - 77 - 88 , 

 (21) = -a/9 -/9a + 78 -87 , 

 (31) = - ay - 8 - ya + 8/3 , 

 (41) = - 08 + /3y - 7/3 - 8a . 



These can be simply reduced to a linear form ; for multiply the first by a , 

 and the second, third, and fourth by ft, 7 , 8 , respectively, and add, 

 we obtain 



for a 2 + /3 /2 + 7 2 -f 8 2 =1. 



In a similar manner we obtain a number of analogous equations, which 

 are here all brought together for convenience 



(11) a! - (21) ft - (31) 7 - (41) 8 = a, 



- (21) a - (11) ft + (41) 7 - (31) 8 = & 



- (31) a - (41) ft - (11) 7 + (21) 8 = 7, 



- (41) a + (31) ft - (21) 7 - (11) 8 = 8. 

 + (22) a! + (12) ft - (42) 7 + (32) 8 = a, 

 + (12) a - (22) ft - (32) 7 - (42) 8 - ft, 



- (32) a! - (42) ft - (12) 7 + (22) 8 = 8. 

 + (33) a + (43) ft + (13) 7 - (23) 8 = a, 

 + (43) a + (33) ft - (23)7 - (13)8 = & 



- (13) a - (23) ft - (33) 7 - (43) 8 = 7, 

 + (23) a! + (13) ft - (43) 7 + (33) 8 = 8. 

 + (44) a - (34) /3 + (24) 7 + (14)8 = a, 

 + (34) a! + (44) ft + ( 14) 7 - (24) 8 = & 



- (24) a - (14) ft + (44) 7 - (34) 8 = 7, 

 + (14) a! - (24) ft - (34) 7 - (44) 8 = 8. 



These will enable a, ft, 7, 8 and a , ft, 7 , 8 to be uniquely determined. 



