JoLY — The Associative Algebra applicable to Hyperspace. 105 



Operating then on «i, suppose, it is easily shown that 



liiHIvi^ = (cos i-Wig + «i4 sin ^Ui^) ii (cos ^Ui^ - %«2 sin ^Ui^) 

 = i (cos i-Wi2 - «i4 sin ^^12)^ = «i (cos W12 - «i4 sin W12) 

 = ii cos W12 + 4 sin W12, 



or «i is tiirned through the angle iii^ towards i^. 



Hence, if 5-34 = cos ^u^i + 44 sin ^n^^, q^^ ( ) ^34"' 

 turns a line in the plane of 44 through the angle W34 ; and 



qxiizi ( ) 2'3rVi2"' 



turns the component of a vector in the plane of 44 through an angle M12, 

 and the component in the plane of 44 through an angle ^<34. Also, 

 §■12 and ^34 are commiitative in multiplication, or the rotations in the 

 hyper-per]3endicular planes may be effected in any order. 



Further, the general operator Q ( ) Q"^ of Art. 37 affecting m units 

 may be reduced to the type 



^n^zi • • • (I21-11 21 { ) ^21-1) 21 ' • • isi ^12 > 



where 21 = ni or m- 1, since these two operators produce the same 

 effects on all vectors. 



42. Having seen in Art. 37 that m vectors may be changed into m 

 others, equally inclined but otherwise perfectly general, by an operator 

 of the type Q{) Q"S it appears that the general displacement about a 

 fixed point of a rigid body of m dimensions in a space of m dimensions 

 may be represented by operating by Q ( ) Q~^ on the vectors from 

 the fixed point to the various points of the body. The most general 

 displacement of a body is produced by adding to this an arbitrary 

 disj)lacement of translation 8. 



The displacement, then, of the point originally at the extremity of 

 p is h-^-QpOr^ - p. If the rotation is made about the extremity of c 

 instead of about the origin, the same motion is produced, provided the 

 new displacement of translation S' satisfies 



S + (3p<3-^ - P = 8' + (3 (p - e) Q-^ - (p - €). 



Hence, 8' = 8 + (3e(2-' - e = 8 + (<^ - 1) e. 



In spaces of even order it is generally possible to determine c, so that 



8' = 0, or 8 + (<^-l)e = 0. 



This is not generally possible in spaces of odd order, for in such spaces 



