368 HITCHCOCK. 



We might also have obtained (44) by a count of the number of terms 

 obtained by developing either side. If n C p+ \ denote the number of 

 combinations of n things p + 1 at a time, this will be the number of 

 cubic determinants of order p + 1 obtained on developing the left side 

 of (44). Each of these cubic determinants yields p + 1 cubic de- 

 terminants of order p, because one layer is taken from the idemfactor, 

 hence contains p + 1 non-vanishing elements. Thus the left side of 

 (44) may be written as the sum of (p + l) n C p+ i cubic determinants 

 of order p. On the right the scalar {{<p\, <p2,- • •, <p P ))s is the sum of 

 n C p cubic determinants of order p. But we have identically 



(p+ l)»C P n= (n - p)nC p (46) 



agreeing with (44). 



In particular it follows from (42) that any scalar (36) can be written 

 as a single cubic determinant of order n by adjoining the idemfactor 

 n — p times, so that p -4- q = n and the numerical factor becomes 

 (n — p)!. Thus 



{{(pi, w • •, <p P , I, I,- ■ -))s = (n - p)'. {{(pi, <P2- ■ -,<Pp))s (4") 



in the case where the idemfactor occurs just n — p times. 



It also follows that any coefficient in the Hamilton-Cayley equation 

 for <p can be written as a single cubic determinant of order n. For by 

 letting the p arbitrary double polyadics all equal <p, we have in virtue 

 of (37) 



{{<p, <p,--,<p,I,I,--- 1)) a = P- (« - pV m P (4S) 



pro\dded <p enters p times and / enters n — p times. 



S. Scalars Expressed as Square Determinants whose Ele- 

 ments are K-adics. 



By virtue of the result of Art. 7, it becomes possible to express any 

 scalar of the form (36) without the use of cubic determinants, if we 

 are willing instead to use determinants whose elements are not scalars 

 but polyadics, and to make certain slight but needful modifications in 

 the usual rules for determinate expansions. 



Suppose, as before, that there are » arbitrary double polyadics 

 <pv • -<pn and let each of them be expanded in the manner of (27), thus 



fi 



= B n E! + BfoEs + • • • + B W! E n (49) 



