16 E.B. Wilson, 
but the existence of this equation is not necessary for many of the 
theorems concerning polynomials. 
As @ satisfies an equation of degree v*, it may be inferred that 
f must satisfy an equation of least degree. Let this equation be 
of degree f, so that 
(16”) A(@) = GP + a, GP-14+....+ a 16+ a, 1=0. 
The equation of least degree is unique. For if there were two 
different equations of least degree, their difference would be an 
equation of less degree—which is absurd. It follows that any 
equation E(@)—0 in ® must be the product of the equation of 
least degree and a polynomial in #. For it is possible to write, 
in accordance with (17), 
(17°) O= E(®)= B(@) A(#) + P(#). 
Hence P(#) vanishes and the statement is proved. The equation 
of least degree is therefore a necessary factor of any equation in #. 
Double Multiplication. 
7. Introduction to double products.—The developments of the two 
preceding sections do not differ materially from the ordinary 
treatments of the generalized linear vector function (Hamilton) or 
the simplest type of Liickenausdriicke and quotients (Grassmann) 
or the theory of matrices (Cayley, Sylvester, Frobenius, and others). 
They have been passed hastily in review, partly for the purpose of 
outlining Gibbs’s course on multiple algebra, partly for the purpose 
of establishing the notations, methods, and fundamental theorems 
which will be useful in the future. With his usual reticence, Gibbs 
apparently did not think that this part of his work on multiple al- 
gebra was of sufficient importance and originality to warrant his 
printing it. With regard to double multiplication it was different. 
He seemed to feel that here he had introduced a new idea and 
a new set of methods, which might be of considerable importance 
in a complete treatment of multiple algebra. In fact I remember 
that he once told me that he had in mind several points in mul- 
tiple algebra which he hoped to find time to publish after he had 
completed the revision of his published papers on thermodynamics. 
Very likely he was thinking of his theory of double multiplication. 
Unfortunately, however, the revision of his thermodynamic papers 
was cut short, almost before it had begun, by his sudden death; 
and the only portions of his work on double multiplication which 
were published during his life consist of the few words on the 
subject in his address On Multiple Algebra and of the discussion 
