410 MOORE AND PHILLIPS. 



A similar result will be obtained if the planes jSj pass through a point. 

 Hence any singular dyadic can always be expressed as the sum of 

 three dyads. Conversely if the dyadic can be expressed in this form, 

 it is evidently singular. 



If X is the point of intersection of the planes 71, 72, 73 in (28) 



Bi^X) = 0. 



If a one-lhree dyadic is singular there is then a point X such that B ifiX) 

 is zero. Conversely, if there is such a point the dyadic is singular. 

 For, if 



Bim = B,{fi,X) + 52O32Z) + 53(^3^) + B,{^,X) = 



either (^iX) = ifi^X) = ifizX) = {fiiX) = and the four planes 

 pass through the point X, or the four points B i satisfy a linear relation 

 and so lie in a plane. 



If a one-three dyadic <l> is not singular it has an inverse $~^ such that 



$$-1 = / = <|.-i^ (29) 



is the one-three idemfactor. To show this, let 



$ = 5i/3i -f- 50/32 + 53/33 + 54^4. 



Since Bi, B2, B3, B^ do not lie in a plane, we can associate with them 

 four planes 71, 72, 73, 74 such that the points and planes form a 

 reciprocal system. Similarly let Ci, C2, C3, C4 form with /3i, ^2, 183, ^a 

 a reciprocal system. Then 



(7i5i) = (/3iC,) = 1, (30) 



{liBf) = {^iQ = 0, i9^ j. 



Using these equations it is easy to show that 



$-^ = Ci7i -f C272 + ^373 + ^474 



has the required property. Also 



$-1$ = / = $$-!. (31) 



The quantity 



(5/3) = (5i^i) + (52^2) + (53^3) + (54/34) 



is called the scalar of the dyadic. It is evidently independent of the 

 form in which the dyadic is written. For, if the 5's and /3's are 



