436 MOOKE AND PHILLIPS. 



of 5/3 with itself represents the same transformation as an operator 

 on planes. For, if B^ transforms three points X, Y, Z into X', Y', Z', 

 then 



[BB'B"] m'^"] = B^ : [B'B"] [^'^"] 



transforms the plane [XFZ] into a linear function of the planes 

 [X'Y'Z'l [Y'Z'X'], etc., that is into the plane [X'Y'Z']. 



In the same way it is easily shown that the double and the triple 

 product of a one-one dyadic with itself determine the same trans- 

 formation in lines and in points. 



The double product of a one-two dyadic with itself is zero. For, 

 if Ap = A'p', 



[AA'tpv') = [A'AWp) = - [AA'Mpp'). 



Since the double product is equal to its negative, it is zero. The same 

 is true of the double product of a two-three dyadic wit^ itself. 



20. Hamilton- Cayley equations. It has been shown by vari- 

 ous writers that a one-three or a three-one dyadic in three dimen- 

 sions satisfies an algebraic equation of the fourth degree called the 

 Hamilton-Cayley equation ^^ of the dyadic. 



That the two-two dyadic satisfies an equation of the sixth degree 

 might be inferred from the fact that the transformation set up by a 

 two-two dyadic in three dimensions can be interpreted as a transforma- 

 tion of points in a space of five dimensions. In general, there will 

 then be six linearly independent complexes left invariant by the 

 dyadic. Taking these as prefactors, the dyadic can be written 



$ = rs = \\piqi + \2p2q2 + . . . + Xepe^e- 

 Since 



r{spi) = yupi, 

 where /x is constant, it follows that 



{Piqd = (Pm) = . . . = (piqe) = 0. 



Thus each p is in involution with all the q's except the one associated 

 with it. If the p's and ^''s are lines, this is the configuration called a 

 double six. In general we may call it a double six of complexes. 



17 Whitehead's Universal Algebra, page 261. Bocher's Introduction to 

 Higher Algebra, Chapter XXII. 



