184 PHILLIPS AND MOORE. 



where tti and 7r2 are completely perpendicular unit planes. Now tti 

 and X2 being fixed planes of 4>, are left invariant. Hence leaves M' 

 invariant. Similarly the rotation 



A — J-M' 

 (p — C 



leaves any self-complementary two-vector M invariant. 

 Let 



The rotation evidently leaves the self-complementary-complex 



Icu + hi 



invariant. Any self-complementary complex can be written 



X (kn + ^"34) + M (A-13 — hi) + v (Xi4 + hs). 



If leaves this invariant, it must leave 



H (hs — hi) + V {ku + i-23) 



invariant. Since ki transforms into 



ki cos qi -\- ki sin qi, etCf 

 we must then have 



li{kn — ku) + vikii + kis) = 

 = /i [(ki cos qi + A'2 sin qi) X (^3 cos 92 + ^4 sin 92) — fe cos ^i 



— ki sin gi) X (^4 cos 92 — /<^3 sin qz)] 

 + v [(A'l cos qi -f- A-2 sin g'l) X (ki cos 92 — ks sin ^2) + fe cos qi 



— ki sin 9i) X (^3 cos 92 + ^4 sin q^). 



Equating coefficients of ^-13, we get 



IX cos iqi + q2) — V sin {qi -\- qo) = /x. 

 Similarly from the coefficients of ^14 we get 



ju sin (gi + ^2) + i' cos (gi -f- 92) = i*. 



