ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 367 



7. SCALARS FORMED IN COMBINATION WITH THE IdEMFACTOR. 



It may happen that one or more of the double polyadics of the 

 scalar (36) is the idemfactor. The scalar formed from any p double 

 polyadics together with the idemfactor taken q times is the same as 

 the scalar formed from the p double polyadics alone, aside from a 

 numerical factor/ that is 



((<Pu <p2, ••■, <p P , 1,1, ■■■))s = f ((<Ph <&,"-, <p p ))s (42) 



where, however, p -f- q is not to be greater than n. It is sufficient to. 

 prove the proposition for one idemfactor and any p double polyadics, 

 assuming p + 1 not greater than n; the general form of (42) then 

 follows by letting some of the polyadics equal the idemfactor. I shall 

 in fact prove the identity for one idemfactor with p = 2; the proof 

 for other values of p is quite similar. The scalar ((<p\, <pz, I))g is by 

 Art 6 the sum of all cubic determinants of the third order of the form 



(43) 



which on development yields the sum of three cubic determinants of 

 the second order having the same form as (41), and corresponding to 

 the three pairs of subscripts ij, jk, hi. Consider some particular one. 

 of these cubic determinants of the second order, say that one which 

 involves the subscripts ij. It will evidently occur once in the expan- 

 sion of every cubic determinant of the form (43), provided the sub- 

 scripts are i, j and any other subscript not greater than n. That is,, 

 the cubic determinant (41) will occur n — 2 times in the development 

 of ((ipi, (p 2 , 1))s- But i and j are any pair of different subscripts. 

 Hence 



((<£>i, tp2, 1))s = (n - 2) ((<pi, (p 2 )) s 



The same reasoning applies for any value of p. 



((<Ph V2,' m 't <Pp, I))s = (W — p) ((<Pl, (P2,- 



(44) 

 Hence we have 



■,<P P ))s (44a) 



where p has any value from 1 to n — 1 and <pi ■ • ■ <p p are any double 

 iv-adics whatever. By letting several of these /v-adics successively 

 become equal to the idemfactor we obtain (42) and see that the 

 numerical factor/ is given by 



/ = (n-p-q+1) (n - p - q .+ *) • • • (n - p) 



(45) 



