418 MOOKE AND PHILLIPS. 



this is equivalent to 



[$z-$y] + [$y-$x] = 2[4>z-$F] = 0. 



This expresses that the complexes ^X and ^Y that correspond to 

 points harmonic with respect to the quadric Q are in involution. And 

 conversely, if the complexes are in involution, the points are harmonic 

 with respect to Q. 



With a two-three dyadic 



9/3 = 'Eqil3i 

 is associated a point 



This is a covariant of the dyadic. In order to obtain a geometrical 

 interpretation for it suppose /3i, (32, 183, ^i so chosen that the four 

 points [qi^il lie in a plane. Then P (being a linear function of the 

 points [qi^i]) lies in the same plane. That is, ij^ the vertices of a tetra- 

 hedron transform into four coviplexes such that the polar points of the 

 opposite planes are four points in a plane, that plane passes through the 

 point P. That such tetrahedra ^1, ^2, ^3, I3i exist can be shown as 

 follows. Let 



q^ = gijSi + q2^2 + 93)83 + 93184. 



If the points [qiiSi] satisfy a linear relation, the planes /?< have the 

 required property. If not, at least one of the products [qi^k], i 9^ k, 

 must be different from zero. Let [94183] be different from zero. The 

 dyadic can be written 



9^ = qA + 92)82 + (93 - X94)^3 + 94(^4 + X^3). 



Let a be the plane passing through [91)81], [92182] and P. Since [93183] 

 and [94/34] are not equal [94)83] cannot equal both of them. Suppose 

 [93184] and [94)83] are different points. Then X can be chosen such that 

 [91)81], [92182], [(93 — X94)53] and P lie in a plane. Since. 



P = [?i|8a] + [92^2] + [93 - X94)/33] + [94(/34 + XJ83)] 



this plane must pass through [94084 + X|33)]. Hence 181, J82, (83, /34 + 

 X/Ss have the required property. 



The discussion of the two-one dyadic can be taken by duality from 

 the discussion of the two-three. The dyadic can be written in the 

 form 



qB = qiBi + 92B2 + 93-83 + 94^4. 



