668 MOORE. 



Then the dyadic which represents the transformation r' = r-M is 

 (28) $ = mnihlci — hh) + msiikils — kzk^) + . . . 



+ ?W2p_],2p(^2pfc2p-l ~ ^'2p-2^'2p) 



If the transformation is appHed twice to the vector r = ± 5,A-, 

 we have 



r" = r-$ = (r-$)-$i = ;••($•$) 



= — [niii^ikiki + ^'2^-2) + viz^ihkz + A;4A-4) + . . . 



+ ?n2p_l,2p(A*2p-lA'2p-l + A'2pA'2p)j 



Then if $ and r are real and m^ = =t ?«34 = . . . = ± ?«2p-i,2P = 1 

 we return to r after repeating the transformation four times. Hence 

 if the complex co7isists of the sum of p mutually perpendicular %mit 

 planes, the transformation r' = r-M toiU he of order 4- 



5. The Hamilton-Cayley equation. From equation (12) we saw 

 that the transformation r' = r-M in 4-space depends on two para- 

 meters ?/?i and r»2 and to show the relation of these to the Hamilton- 

 Cayley identical equation which $ must satisfy we will change the 

 reference system to that of the invariant elements. We saw that the 

 invariant vectors were 



Now put 



ki ="= iki, ki ± iki. 



n = — ^ ((A'l + iki), r2 = —^ (A-i - ih) 

 n = —^ (A-3 + iki), ri = —^ (A-3 - iki), 



Then 



A-i = —^ (n + To), ko = - —^ {ri - ra), 



i i 



A'3 = -7^ (ra + ri), ki = — -^ (rg - ri). 



The dyadic $ then becomes 



(29) $ = ?[?«i(r2ri - ?-,/-o) -f »;2(r4r3 - r3r4)]. 



