360 HITCHCOCK. 



and (18). Collecting the scalar coefficient of E^E* we find it to be 



— TZcikCLik summed on i, which is unity. Again, the coefficient of 

 Co 



E,Ea- when j and A- are unequal is —^CijOik summed on i, which is 



Co 



zero. Hence I = ZE^E* summed on k, and this latter expression for 



the idemfactor is symmetrical in the sense that it is unaltered by 



interchanging antecedents and consequents. It follows that the 



reciprocal relation between two sets of polyadics is a mutual one, and 



we have 



I = ZE,E, = 2A',A, = 2A,A' A - (20) 



and also 



M = / : M = M : I (21) 



or the idemfactor may be used either as prefactor or as postfactor. 



To illustrate, if we take K = 1 and A* = 3, our polyadics reduce 

 to vectors in ordinary space; and if our unit vectors be the usual 

 i, j, and k the idemfactor becomes the usual idemfactor of Gibbs, 

 namely ii + jj + kk. If K = 2 and N = 3 with the same notation, 

 our polyadics are ordinary dyadics; the idemfactor (20) is a double 

 dyadic, having nine terms obtained by doubling the nine fundamental 

 dyads, viz. 



I = iiii + jjjj + kkkk + ijij + jiji + jkjk + kjkj + kiki + ikik 



(22) 

 and it may easily be verified that the double dot product of any of the 

 nine fundamental dyads ii, ij, etc., either by or into this expression 

 gives the dyad unchanged; whence the same is true for any dyadic. 



4. Double Polyadics. 



Two polyadics A and M written together with no sign between them 

 will constitute a double poly ad AM. A sum of double poly ads is a 

 double polyadic, assuming always that the terms of the sum are of like 

 character. In the present investigation it will always be assumed that 

 the factors A and M of all double polyads are polyadics of the same 

 order K. \Yhen desirable, all the antecedents A may be expanded in 

 terms of the fundamental A'-ads E„ and the consequents collected so 

 that any double polyadic <p may be written in the form 



<p = EjM, + EoM 2 H h E„M„. (23) 



