THE DYADICS IN THREE DIMENSIONS. 403 



Similarly there are two products C$ and $C of C and a dyadic $. 

 These are obtained by multiplying C and each dyad of $ as above and 

 adding the results. 



If A' is complementary to the consequents 



$Z = A,{B,X) + A,{B,X) +....+ An{BnX), 



is an extensive quantity of the same kind (dimensions) as the ante- 

 cedents. Hence 



Y = $Z 



is a linear transforviation in which to each element X corresponds an 

 element Y of the same kind as the antecedents. It can be shown that the 

 most general linear transformation of these elements can be expressed 

 in this way. Similarly if X is complementary to the antecedents 



. F = Z<l> 



is a transformation of elements X into elements Y of the same kind 

 as the consequents. 



In this paper we shall consider the following types of dyadics. 



(a) The one-one dyadic, in which the antecedents and conse- 

 quents are both points. 



(b) The three-three dyadic, in which the antecedents and conse- 

 quents are both planes. 



(c) The one-three dyadics, in which the antecedents are points and 

 the consequents are planes. 



(d) The two-two dyadics, in which the antecedents and conse- 

 quents are both lines or complexes. 



(e) The one-two and two-one dyadics. 



(f) The two-three and three-two dyadics. 



7. Idemf actors. A dyadic I such that 



X = IX (24) 



for all elements X of a given kind, is called an idem factor. In this 

 case the antecedents and consequents must be complementary in 

 kind. Therefore there are three idemfactors, a one-three, a three-one 

 and a two-two. 



There is only one idemfactor of each of these types, For, if Ii and I2 

 are dyadics such that 



X = hX, X = hX, 

 then 



(/i - h)X = 0. 



