THE DYADICS IN THREE DIMENSIONS. 411 



expressed as linear functions of new elements B'^ and /S',- and so the 

 dyadic is expressed in a new form, by the distributive law of outer 

 multiplication, the scalar of the dyadic will at the same time be 

 transformed into the scalar of the dyadic in the new form. The same 

 conclusion follows also from the fact that the laws of the indetermi- 

 nate multiplication are included among those of any distributive 

 multiplication whose operations are commutative with multiplication 

 by a scalar. Hence if an equation is satisfied by dyads, the equation 

 will still be satisfied, if the indeterminate products are replaced by 

 any such distributive products. 



A function of a dyadic independent of the form in which the dyadic 

 is written will be called an invariant if it is a scalar, a covariant if it is 

 an extensive quantity or dyadic. In a similar way we define invari- 

 ants and covariants of two or more dyadics. 



If the scalar of the dyadic is zero it was shown by Pasch ^^ that there 

 exist certain tetrahedra such that each vertex passes, by the colline- 

 ation, set up by the dyadic, into a point of the opposite plane and, 

 conversely, if any such tetrahedron exists the scalar of the dyadic is 

 zero. 



The discussion of the three-one dyadic runs exactly similar to the 

 one-three of which it is the conjugate. As an operator it gives also a 

 collineation but in this case it is a collineation in planes instead of in 

 points. If the antecedents aj and the consequents A^ form a recipro- 

 cal system, the dyadic 



aA = - [aiAi + 03^2 + 03^3 + 04^4] (32) 



is the three-one idemfactor. 



The scalar of a three-one dyadic is the negative of the scalar of its 

 conjugate (which is one-three) and consequently the vanishing of 

 this scalar has the same signification. 



12. One-one and three-three dyadics. A one-one dyadic has 

 the form 



Since the points Bi, B2,. . . .Bn can be expressed as linear functions 

 of four, the dyadic can always be reduced to the form 



BC = B,Ci + B2C2 + B^Cz + B,Ci. (33) 



13 VoUkommene Invariante, Math. Ann. Vol. 52, page 128. 



