ROTATIONS IX HYPERSAPCE. 667 



If the 2-vectors -1, B, C. . .P are all multiples of Mi by a proper 

 choice of axes the complex can be written in the simple form 



M = mi(ki2 ± ^'34 ^ /i-56 ± . . . ± A-Op_i, 2p). 



We will confine our attention to the single case 



(23) M = /«l(A-12 + A-34 + ...k2p-U2p) 



and the other combinations of sign can be disposed of in a similar 

 manner. The dyadic which represents the transformation r' = r-M 

 is 



(24) <!> = miihh — kiko + kih — hki + • • ■ + hphp-i — hp-ihp) 

 and the same transformation expressed in plane coordinates is 



which can be proved by the same method used for the case of 4-space. 

 Similarly it can be shown that the complex 



(25) C = ai2^'i2 + (isil'si + ase^'se + ■ • • CE2p-i)2pA'2p-i,2p 



+ (iisihs — ^'42) + aii(ku — ^3) + ai5(^i5 —[^62) + • • • 



+ 035(^-35 — A-46) +/'36(A-36 — A'45) + 



is left invariant by the transformation C-^ where the coefficients in C 

 are entirely arbitrary. The complex will be a simple plane if C^C = 0. 

 This gives for the invariant planes, putting Ojo = 1, for convenience. 



(26) A = kn-\- {an- + fli4-)A'34 + (015" + «i6-)A-56 + • • • 



+ (ab2p-i^ + ai,2p^)A*2p-i,2p 

 + ai3(A-i3 — A- 42) + 0i4(A-i4 — A-oo) + f/i5(A-i5 — A-oe) + . ■ ■ 



+ (ai4«15 — «13«16)(A-3o — A-46) + («13«15 " «14ai6)(A-36 " A-45) + • • • 



+ (ai30i7 + anOis) (A'ss — A-47) + • • • 



Hence oo-p-2 simple planes are left invariant. Then in resolving 

 this complex into the sum of p similarly perpendicular simple planes 

 the first plane can be chosen in co -p-- different ways, the second in 

 oo2p-4 ^ayg and so on. Hence the resolution can be effected in 

 oo2(p-i)[ different ways. 



By proper choice of axes the general complex can be put in the 

 form 



(27) .1/ = mvihn + W34A-34 + . . . + »«2p-i,2pA:2p-i,2p- 



