ROTATIONS IN SPACE OF EVEN DIMENSIONS. 181 



in TTi or TTo. J^cdors in one of the invariant planes are rotated through the 

 maximum angle and vectors in the other are rotated through the minimuTri 

 angle and those in neither plane are rotated through an intermediate angle. 

 If gi= ± qo 



cos 6 = cos q\ = COS q-2 



and it would appear that all vectors are rotated through the same 

 angle. In this case M is self-complementary or anti-self-comple- 

 mentary. 



12. Equiangular rotations. If J/ is any two-vector and A:i a unit 

 one-vector, M can be expressed in terms of four perpendicular vectors 

 including ^'i. The terms of M containing ^'i will have for sum a plane 

 since the sum of three plane vectors with a vector in common, is a 

 plane vector. Let ^2 be perpendicular to ^'i in that plane and let 

 ks, ki, form with ki and k2 a system of orthogonal unit vectors. Then 



M = Xi2 ki2 -\- X23 A:23 + ^24 A*24 + X34 ^"34 



and 



Mc = X12 A:34 "f" X23 ku + X24 ""31 "I" X34 A^io. 



If M = Mc 



X12 = X34, X23 = X24 = 



and 



M = X12 {kn + A-34). 



If M = - Mc 



X12 = — X34, X23 = X24 — 

 and 



M = X12 (^-12 — A-34). 



In either case M is a linear function of two completely perpendicular 

 planes one of which may be taken through the arbitrary vector ki. 

 By properly choosing k2 we can always make X12 positive. Then X12 

 will have the same value for all choices of ki. For if k'n ar ^ '-'4 

 are two other completely perpendicular planes and 



M = X {kn ± kzi) = M {k'n ± ^34), 

 M-M = 2x2 = 2/x2. 



