THE DYADICS IN THREE DIMENSIONS. 437 



If / is the two-two idemfactor, it is easily seen that 



Similarly, 



{$ — X2(p25'2)/} •;> = 0, 



etc. Consider the product 



^ = {$ - \i(pjq{)I\ {^ - XoXp-zqo)!] ...{$- XeCpeg-e)/}. 



It is clear that 



^i^6 = 0. 



Since ^ and I are commutative, any factor could be put last. Hence 



•^i^i = "^pi = . . . = "^Pg = 0. 



Since pi. . .peon assumed linearly indepedent, 



^ = 0. 

 When expanded this has the form 



Ao^' + ^1*^ -\-...-\- As-^-h AJ = 0. 



This is the Hamilton-Ca^dey equation satisfied by the dyadic. 



That a polynomial equation is satisfied by any dyadic whose pre- 

 factors and postfactors on dual may be shown in the following way. 

 Let Ri and S^ be dual. Let i?,-. . .Rn be linearly independent and let 

 Si. . .Sn be linearly independent. Suppose 



^^ZAaRiSk, 

 ^ = ^bikRiSk 



are commutative. Then 



$^ _ ^4> = ^bami)Sk - Ai,{^Ri)Sk = 0. 



Since the S'* are linearly independent, the coefficient of Su in this 

 equation is zero. That is, 



blkmi) - ai,($i?i) + . . . + bnk{<^Rn) - Anki'^Rn) 

 = (6u$ - A,k^)-Ri + . . . + {bnk^ - Ank-^)-Rn = 0. 



