()70 MOORE. 



From (32) we see that the complex 2-veetors 



aru + broi, aru + br23 



are left invariant for all values of a and b. The only planes belonging 

 to this system are ra, r^, ru, r^z. If ?»i = =t=-?H2 we see at once that 

 the complex 



arn + br^ + cvi^ + dr^i, 

 or ari2 + 6r34 + f'"i4 + f/''23 



is left invariant and the planes belonging to this system are the 

 invariant planes discussed before. 



Choosing the invariant vectors for the coordinate system in space 

 of 2p dimensions and proceed as above we at once arrive at the Hamil- 

 ton-Cayley equation 



and if Wi = =^vh = =^^3. . . = ='=?«p this equation becomes 



$2 + mi^h = 0. 

 The equation which ^ = ^^x*^ satisfies follows in a similar manner. 



III. 



6. Rotations in 4-space. We saw that an infinitesimal rotation 

 could be represented by 



(5) / = /• + M-rdt. 



If M is a simple plane vector Mi say, and if r is perpendicular to Mi, 

 then since Mi-r = 0, the ^"ector /• is left absolutely fixed, and therefore 

 the plane completely perpendicular to Mi is left absolutely fixed. 

 Also if r lies in Mi it is evident that r' will likewise lie in Mi and there- 

 fore Ml is left invariant but not point for point. If in this rotation 

 we take Mi as a unit plane and write (5) in the form 



?•' = /• + viiMi-r dt 



the constant mi measures the rate of rotation in the plane Mi. For 

 if r lies in Mi 



r' - r dr 



= — = miMi-Ti 



dt dt 



