ROTATIONS IX SPACE OF EVEX DIMENSIONS. 163 



the summation being for all values of i and _;' in (8), represents a 

 motion in which each plane A',-/ is rotated through an angle qij. If the 

 rotation in kij is proper the upper signs in formula (9) are used and if 

 improper the lower signs are used. Any rotation ^ in 2?;, dimensions 

 can be expressed in this form. To do so let kij be the ?t invariant 

 planes of ^ and qij the angles of rotation in those planes. Then 

 4r^ ^ is a dyadic leaving all the vectors ki invariant. Hence 



(^-1^ = 7 

 and so = ^. 



If the rotations in two of the planes knave improper, these can be 

 replaced by two new planes in which the rotations are proper. Thus 

 suppose ki2 and ku are the planes in question. By proper choice of 

 ki, A-2, ks, ki the part of (9) belonging to these planes can be reduced, 

 by (7), to the form 



A'l A'l — /I'o A'2 ~i A'3 A'3 — A'4 A'4. 

 This is equivalent to 



A'l A'l + A'3 A'3 — (A'2 A'2 -\- A'4 A'4) 



which represents a proper motion leaving all the vectors in the plane 

 A'la invariant and rotating the vectors in the plane A'24 through 180°. 

 The corresponding rotations will be proper. By a continuation of 

 the process we can ultimately reduce (9) to a form in which not more 

 than one of the rotations in the invariant planes will be improper. 

 If the motion in only one of the planes is improper the 2n dimensional 

 rotation represented by ^ is improper. Hence any proper rotation 

 can be represented by a dyadic of the form 



= S {ki ki + A'j A'y) cos qij + (kj ki — ki kj) sin qij. (10) 



We shall now obtain the characteristic equation ^^ satisfied by cj). 

 In case of a rotation '^ in a plane A- 12 we have 



"if = (A'l A'l + A-2 A-2) cos qi2 + (A'2 A'l — A'l ko) sin qu, 

 '^^= (A:i A'l + A-2 A-2) (cos ^qn — sin -q^) + (A-2 A'l — A-i A-2) 2^sin qvi cos ^12, 

 I = k\ A'l -\- k<y k-i- 



Hence 



^2 _ 2 >ir cos 912 + / = 0. 



14 See E. B. Wilson, note 8. 



