ROTATIONS IN SPACE OF EVEN DIMENSIONS. 



185 



If these equations are satisfied simultaneously 



cos {qi + ?2) — 1, — sin (gi + q-^ 

 sin iqi + 92), cos (gi + 9-2) — 1 



0, 



that is, 

 Hence 



cos (gi + go) = 1. 



gi = - g2 



Consequently M is anti-self-coniplementary. If then </> leaves any 

 self-complementary complex invariant other than A-i2 + /.•34, M must be 

 anti-self-complementary. In particular if M is seIf-comj)lemcntary, 

 the rotation cannot leave any self complementary complex invariant 

 except a multiple of M. Similar theorems hold for anti-self-comple- 

 mentary rotations. 



The self-complementary complexes are analogous to one system of 

 the generators on a quadric surface, the anti-self-complementary 

 complexes constituting the generators of the other system, each gener- 

 ator of one system cutting each generator of the other in the sense 

 that the star product is zero. The equiangular rotations, are the 

 rotations that leave this quadric invariant. Any rotation of the 

 self-complementary type leave all generators of the opposite type 

 invariant and one of the self-complementary type. Similarly for 

 the anti-self-complementary rotations. 



14. Analogy to Gibbs' vector semi-tangent.^^ Any self- 

 complementary complex can be written 



Hence 



M = g ihn + A-34). 



M-M = MX M = 2g2, 

 IM = g(A-2 ki - ki ko -\- ki ks - A's /m), 

 (I-M)-(I-M) = -qU = -\ {M-M) I. 



(49) 

 (50) 

 (51) 



16 Gibbs- Wilson Vector Analysis. 



