ROTATIONS IN HYPERSPACE. 073 



The dyadic ^ can be expressed in terms of plane coordinates by- 

 means of the double product. 



(36) ^2 = §^^^ = |(/i + $f/Ox(/i + ^dt) = |7i^7x + I^^dt 



= h + Il^dt 



and the rotation is then expressed by the formula 



C = €■{!, + hl^dt). 



If we write .V = mik]^. + »;2^"'34 



$ = miihki — kik-y) + m-2(kik3 — k^ki) 

 Ix^ = niiik^skn — A-isA-og + A-24/i'i4 — kukoi) + m2(kukis — hzku 



~\~ "'24^*23 — ko'ikii) 



If the complex 2-vector 



is left invariant, that is if C-^ = C 



C-(/ix$) = miicizk'iz — coskis + Ci4A-24 — Coiku) + m^icnku 



— Cuhz + C23A'24 — C24^23) = 0. 



From which we get 



(37) ?»iCi3 = — '"2C24, '»if23 = m^cu, miCu = viic^, — miC^i = moCn. 

 If in\ 9^ ± mi these equations are satisfied only when 



Cl3 = C'24 = Cl4 = ("23 = 0. 



Thus any complex of the linear pencil 



Cnkn + C34A-34 



is left invariant. The only simple planes belonging to this pencil are 

 ^12 and ^•34. Hence these are the only planes whose magnitude and 

 position are left invariant by the rotation. 



If m\ = ±???2 equations (37) can evidently be satisfied if C\z = =^C24> 

 Ci4 = =^C23. In this case the complex 



C = C]2A'i2 + ciz(kn ± kio) + Cu(ku + ^'23) + 034^:34 



is left invariant for all values of the coefficients. The planes of this 

 system of complexes are determined by the relation 



CxC = C12C34 =^ (ci3- + C14-) 



