408 MOORE AND PHILLIPS. 



It is clear that these symbohc forms will not be ambiguous if each 

 of the letters A, B, C, D does not occur more than once in a product. 

 If the same dyadic occurs more than once in a product, we represent 

 it in each of its positions by a different pair of letters. Thus to obtain 

 the product of AB with itself, we let AB = A'B' and so write the 

 result in the form 



A[BA']B' = i:Ai[BiAk\Bk 



If we write it in the form A[BA]B it is not clear whether this means 

 A[BA']B' or A[B'A']B. It is evident also that a product containing 

 one of the letters A or B without the other, such as 



[AC]D 

 has no significance. 



11. The one-three ond three-one dyadics. These dyadics 

 have been investigated quite extensively by Gibbs,^ Wilson^ and 

 Phillips.* We shall therefore state only a few facts about them. 



A one-three dyadic has the form 



B^ = 5A + £2|82 + ....+ 5„/?„ 



where the B's are points and the (3's planes. Since the planes can be 

 expressed as linear functions of any four not passing through a point, 

 the dyadic can be expressed in the simpler form, 



5/3 = 5i^i + Bo^, + Bs^s + 54/34, 



where j8i, 182, ^z, ^4 are any four planes not passing through a point. 

 In the same way instead of the four planes, the four points J5i, B2 

 Bz, Bi could be assigned arbitrarily. 



As an operator on points X, this dyadic gives a collineation 



Y = BifiX) = Bi(l3,X) + B,{fi2X) + Bzi^zX) + B,{fi,X). 



If X is the intersection of the planes 182, ^z, ^i 



X = ms^,], (/32Z). = (fizX) = {8,X) = 0, 

 and 



Y = ByWol33l3,). 



Hence the vertices of the tetrahedron (3i, ^2, jSs, ^i pass by the collinea- 

 tion into the points Bi, B2, Bz, Bi. 



