424 MOORE AND PHILLIPS. 



Each of the remaining terms contains p4, Ph, or pe. A sum of terms 



y^wiqilH — qilh) + X24(g2p4 — 94P2) + 



can be combined into a single term 



^\i{p\Pi — PiPi) 

 where 



Xi4Pi = X1471 + ^liq-i + 



Hence the dyadic can be reduced to the form 



$ = \u{p\Pi — PiPl) + X25OJ2P5 — phP-l) + \Z(,{PZP6 — P^Pz). 



The complexes pi, po. . .pe can be taken linearly independent. If, for 

 example, pe were a linear function of the others it could be replaced by 

 this linear function and the dyadic would then be expressed in terms 

 of 5 complexes, pi, p-i- ■ ■ -IH- The above process would then reduced 

 to two terms instead of three. This is a special case in which one of 

 the coefficients Xh, X25, X36 is zero. 



An anti-selj -conjugate dyadic $ transforms any complex p into a 

 complex in involution with p. For 



^p = p$c = — P^' 

 Hence 



[p^p] = - liJ^p] = 0. 



This expresses that ^p and p are in involution. 



A very important type of two-two dyadic is that which gives the 

 same transformation of lines and complexes as a point collineation or a 

 point-plane correlation. The peculiarity of such a transformation is 

 that it preserves contact, that is, it transforms intersecting lines into 

 intersecting lines and complexes in involution into complexes in 

 involution. If $ is such a dyadic 



^Pl-^P2 — Pl^c^P2 = 



whenever 



' P1P2 = 0. 



These are linear equations in the coordinates of pi and j)2 such that the 

 first is always satisfied when the second is. Hence there must be a 

 constant X such that 



Pl^c^P2 = Xpip2. 



