ROTATIONS IN HYPERSPACE. 675 



planes changed by the rotation into multiples of themselves. In 

 this case 



which leads to the relations 



— WiC.23 — tn^cu = XCl3, 



lUiCn — m-ic-iz = Xf24, 



— W1C24 + m2Ciz = Xcu. 



Four values of X render this system consistent and the corresponding 

 invariant planes are 



(A-13 + A-42)=^ i{kli + A'23), A-13 — A- 42=^ ?(^'14 " hi) ■ 



No real plane satisfies this condition. 



7. Rotations in any even space. Equation (5) is a rotation in 

 a space of 2p dimensions if we consider M as a complex lying in that 

 space. As before the dyadic representing the rotation, if we write M 

 in terms of p mutually completely perpendicular unit planes which 

 for convenience we will take as coordinate planes, is 



^ = /i+ [miihh — A-iA-2) + vi2{kih — h^i) +. • ■+ mp{k2php-\ — 



hp-ihp)]dt. 

 = h+ ^dt. 



The same transformation expressed in plane coordinates is 



= 72+ hx^dt 

 where /] = '^kiki 



T^ — yb -h ■■ 



The same argument used in the preceding section will show that the 

 p mutually perpendicular planes into which M is resolved are all left 

 invariant. If the m's are all distinct these are all the invariant planes, 

 but if 71 of them are equal there are oo2("-i^ invariant planes and 

 the rotation can be resolved in an infinite number of ways into rota- 

 tions parallel to j^ mutually perpendicular planes. 



