372 



HITCHCOCK. 



where the last n — p rows are alike, a circumstance which need cause 

 no surprise, since, as already pointed out, the rows are non-signant in 

 the very nature of the definition of these polyadic determinants. 8 



If we still further specialize (56) by letting the remaining p double 

 polyadics all become equal to <p we shall have, by (48), 



Ei, Eo, 



= pi (?i — p)! rtij 



(57) 



where, on the left, p rows are like the first row, and (n — p) rows are 

 like the last row. 



9. An Invariant Property of the Scalars (36). 



From the form of the definition of the scalars of the present discus- 

 sion, they are independent of development in terms of a particular 

 set of polyadics. Throughout the argument the fundamental polyads 

 Er -E„ may be replaced by any other set of n polyadics forming a 

 normal, orthogonal system with respect to Zv-tuple dot product, for 

 no other properties of these polyads have been so far used. These 

 scalars have another kind of invariance of a somewhat wider sort, with 

 respect to a reference system which need no longer be normal nor 

 orthogonal but only linearly independent. The properties in question 

 are essentially included in the following theorem : — 



Theorem. The scalar ((<pi, ^>, • ••, <p n ))s formed from a set of n 

 arbitrary double polyadics is unaltered when each of these double 

 polyadics is multiplied by (or into) the same double polyadic 6, (by 

 iv-tuple dot product), except for a factor which is the determinant of 6. 



This theorem is equivalent to the identity 



((iPi, &,-■-, <p n ))s i(G n ))s = n! ((^: 6, ^:d, ■ ■ -,<p n : d)) s (58) 



for, by (37), ((0 n ))s equals ».' times the coefficient of 0" in the Hamilton- 

 Cayley equation for 6, which is the determinant of 0; and 6:<pi = 

 <P\':0, while by (55) the left side of (58) is unchanged if we change 

 every <p into <£>'; hence it is sufficient to prove (58) as written. 



