176 ' PHILLIPS AND MOOKE. 



In four dimensions all vectors X through the origin making a given 

 angle with il/o belong to the quadrie cone 



{MrX)-{MrX) = 0. 



This cone has two sets of generating planes. Any plane M of the one 

 set is such that M * Mi is self-complementary and any plane of the 

 other set such that M * il/2 is anti-self-complementary. 



II. Rotations in four dimensions. 



9. Identities. In terms of unit two-vectors I'ij the laws of star 

 multiplication are 



A'i2 * /i'l2 = A'l2 * n 34 = 0, /ooN 



A.'i2*/t'l3 = kiZ- 



The case of four dimensions is the one in which we shall be most inter- 

 ested. Let 



be any two-vector in four dimensions. Then 



A'i2 M = Xi3 A'23 ~1~ Xi4 A" 24 — A23 A"l3 — A24 "'l4, 

 ^"34 * il/ = — Xi3 /r4i 4" X4 ksi — X23 A'42 ~\~ X24 A'32. 



(34) 



Denoting the complement by the subscript c we then have 



{kn * M)o = (/.•34 * M) = {kn)c * M. 

 Since any unit plane could be taken as kn 



(Ml * il/2)c = (il/i)c * M2 = M, * {Mo},. (35) 



The operations in (35) being distributive the equations are valid 

 whether Mi and M2 are planes or complex two-vectors in four dimen- 

 sions. 



Since the complement o,f (Mi * M2)c is Mi * M2 we also have 



Mi*M2= (M{)c*{M2)c. (36) 



