24. E. B. Wilson, 
dyadic differs trom the corresponding invariants of the transformed 
dyadic. 
By virtue of the identity = %; J; it appears that 
(37) Dee (Pp) — On ee Oe 
This may be taken as the definition of #5 in place of (32) and 
analogous equations, and it offers a ready interpretation of the 
scalar invariants of @ according to the matricular form (9) in which 
the antecedents and consequents are reciprocal sets. In this case 
@, is merely the sum of all the coefficients in the main diagonal ; 
@,, is the sum of all two-rowed minors of the matrix which have 
two terms of the main diagonal as their main diagonal; #3; is the 
sum of all three-rowed minors which have the three terms of their 
main diagonal selected from the terms of the main diagonal of the 
matrix; and so on until finally #, is the determinant of the matrix. 
The values of these sums would be unchanged if the matrix under- 
went a transformation of coordinates. The importance of these 
invariants to the theory of matrices and to the mathematical 
theories of elasticity is well known. It would be possible indefin- 
itely to multiply the interpretation of the theory of dyadics in the 
theory of matrices by reference to the expression of the dyadics in 
the form (9) where the antecedents and consequents are reciprocal 
sets: but this would not be worth while. 
11. The identical equation.—It was shown in article 6 that any 
dyadic satisfies an equation of degree not greater than ?, and 
from this fact was deduced the existence of an equation of least 
degree. Consider the relation (28’) of article 9 as applied to the 
dyadic —g /, where @ is any dyadic and g is any scalar. 
(22 Tn LP 8d Oe in 
The left-hand side may be expanded by the binomial theorem (25) 
and simplified by the relations (37); the right-hand side may also 
be expanded by the binomial theorem and then multiplied out. 
The result is 
n—1 n 
[Pnr—ge Pn—1,s ane de Dips ore ale (=1) Brews ais (1) en i IT(g), 
where JI (g) is a polynomial of degree z in g with dyadic coef- 
ficients. The relation is an identity in g. By the same reasoning 
which shows that two identical algebraic polynomials with scalar 

*) The relation of a dyadic to its family of transformed dyadics # & P—1, 
where # is any complete dyadic, was apparently left unmentioned by 
Gibbs. Perhaps this was due merely to the brevity of his course on 
multiple algebra. 
