THE DYADICS IN THREE DIMENSIONS. 415 



and the transform of ^ is on tj, (A) and (C) are satisfied and so (B) 

 must be satisfied. 



If p= [BC] = Z[B,C,] = 0, (D) 



the transform of r] will always lie on ^ if that of ^ lies on 77. This 

 indicates a polarity. In fact, if p = 0, (p^) = and so 



\{BC - CB)^ = 



where ^ is any plane, since the dyadic | (BC — CB) gives the same 

 transformation as the null system set up by p. Then it is easy to see 



that BC - CB = or 



BC = CB. 



The dyadic being self conjugate represents a polarity. In this case 

 equation (D) shows that the four lines [Bid] belong to the same 

 system of generators on a quadric. For it is a linear relation between 

 four lines. A polarity thus transforms the planes of any tetrahedron 

 into the vertices of a secojid tetrahedron such thai the lines joining corre- 

 sponding vertices of the two tetrahedrons belong to the same set of genera- 

 tors on a quadric. 



In general two sets of four points Bi, and d can be found such that 

 two one-one dyadics <l> and ^ can be written in the form 



$ = 5iCi + 52C2 + 53C3 + B,d, ,„„v 



■^ = \iBid + \oBoC. + X3B3C3 + X454C4, ^ ' 



where the X's are numbers. For, in general, there exists a set of four 

 independent planes ai which transform by both dyadics into the same 

 set of four independent points 5,. If we take the C's as the vertices 

 of the tetrahedron formed by the a's and properly choose the magni- 

 tudes of the B's,, the dyadics will take the above forms. 



If the planes of a tetrahedron Ci, C2, C-a, C4 transform into points 

 B\, B2, -B3, Bi such that the lines [BiCi] belong to one system of generators 

 of a quadric, the other system of generators of that quadric belong to p. 

 For, if / is a generator of that second system, {I Bid) = 0. Hence 



(Ip) = MBid) = 0, 



and consequently I belongs to the complex p. We shall now show 

 that such points Bi,d exist. We have just seen that two dyadics 

 $, ^ can be reduced to the form (36). This form shows that $ 



