ROTATIONS IN HYPERSPACE. 669 



Tlie multiplication table for the r's is as follows 



ri,-r = ro-i-i = is- is = rr r^ = 0, 

 ri • rs = /-i • 7-4 = ro • /'s = ro- r^ = 0, 

 n ■ ro = rs Ti = 1 . 



The idemfactor Ii becomes 



(30) /i = /-irs + roz-i + '•3^4 + rir3. 

 The Hamilton-Cayley equation then becomes 



(31) (<l> — imJi)(^ + imi/i)($ — inioliji^ + im^Ii) = 



(*2 + neh){^'~ + m,Ur) = 0. 

 If nil = ± »22 



(29') $ = iwi[(r2ri — nro) ± (r4r3 — r3r4)] 



from which we see at once that the vectors 



ari ± brs, ar^ ± 6r4 



are left invariant (or multiplied by constants only) for all values of 

 a and h. The Hamilton-Cayley equation in this case becomes 



($2 _|_ „^^27j) = 



In terms of this new reference system the dyadic $ expressed in 

 plane coordinates becomes 



(32) ^ = ^x^ = — [mihnrn + mi-rsiVsi + Wi??i2(ri3r24 



+ r42ri3 — r23ri4 — rur23)] 



The multiplication table for the coordinate planes is 



rn'Tvi = '■34 •'■34 = — 1, 

 ris-roi = ri4-ro3 = 1, 



and all the other products are zero. The idemfactor is 



h = 2^1xA = — rurn — /•34/-34 + '•l3'"24 + '•24ri3 + ri4r23 + r23ri4 



and the Hamilton-Cayley equation is 



(^ - 7?il2/2)(^ - »?2-/2)(^- - mMi'h) = 0, 



and if nii = =•= mo the equation is 



(^ - »?i-/2)(^ + mi-h) = 0. 



