ROTATIONS IN HYPERSPACE. 671 



Since r does not change in magnitude as it rotates, the magnitude of 

 -r divided by Vr-r will be the rate at which r is turning in the plane 

 Mi. That is 



J 



(Ij = VmriMrry = mWr-r 



Ml being a unit-plane J/i • r has the same magnitude as r if r lies in Mi. 

 Therefore 



m-i 



JC-fJ 



V 



/•• r 



which shows that ?»i measures the rate of rotation in Mi. 



If in (5) M is a complex 2-vector it can be resolved into the sum of 

 two perpendicular planes and can then be wTitten in the form 



(33) r' = r+ {viiMi + m.Mo) -rdt 



where Mi and M^ are unit planes. In this case the motion consists 

 of a double rotation, one parallel to the simple plane Mi and the other 

 parallel to the simple plane M-i. The same argument as used above 

 will show that vii measures the rate of rotation in Mi and iiiz the rate 

 of rotation in il/o. If r lies in Mi, then 



r' = r -\- Ml -r dt 



and the same argument used above shows that Mi is left invariant. 

 The same reasoning also shows that Mo is left invariant. 



In order to exhibit the whole list of invariants we will represent 

 the transformation (33) as a dyadic, choosing the reference system so 

 that 



M = viikYi + vhk'si. 



The same argument as used in §2 shows that the dyadic sought is 



(34) ^ = /i + [mi(k2ki - A-iA-2) -\-m2{kih - hki)dt = /i + $ dt 

 and (33) then becomes 



