Double Products and Strains in Hyperspace. 7 
definition, a mere formal juxtaposition of two letters subject to the 
distributive law; that is, 
a(@+y)=abtay and (et+s)y=ay+éy. 
Hence if « and # are each expressed in terms of ” given independent 
elements (which need not be the same set of ~ for both vectors), 
the product «8 may be expanded into a block of 7? terms or dyads. 
It is through this fact that connection is made with the theory of 
matrices. There is no necessity that the two elements in a dyadic 
product should belong to the same class, whether primary or other- 
wise. If I’ and 4 belong respectively to the &th and /th classes, 
the dyad [74 may be defined in a manner similar to « as a formal 
juxtaposition of two elements subject to the distributive law. As 
the Ath and /th classes contain respectively 
n(n—1)...(n—k+1 n(n—1)...(~—/l+1 
( ) = pe sees 10.2 ) a arly) 
linearly independent elements, the dyad [4 may be expanded into 
a block of w~...(m~—-+1).n...(n—H-1)/A!/! terms. These 
terms will not form a square matrix unless k=/ or k+/=n; in 
other cases the matrix will be rectangular. 
Gibbs applied also the name indeterminate product to the com- 
bination «@ @ or I'd, and he was very particular to state that he 
considered it the most general and most essential product with 
which multiple algebra has to dealt. Other products may be re- 
garded as functions of the dyadic product. This product determines 
its constituent elements « and ~, or I’ and 4, except that a scalar 
factor may be transferred from one to the other. The proof of this 
is not essentially different from that given for the simple case of 
vectors in the Vector Analysis, page 272. In what follows, the only 
dyadic products which will be considered are those in which the 
sum of the class-numbers & and / is equal to ”. In this case the 
combinatorial product of two like dyads [4 and I” 4 is detined by 
the simple equation 
(4) (A) >< (FA) = P(A) SHAK Ps 
where 4 =< I” is necessarily a scalar. The product therefore reduces 
to a similar dyad I'4’ modified by a scalar factor. In like manner 
the product of a dyad into an element of the same class as the 
first member of the dyad is defined by the equation 



1 See his address On Multiple Algebra, pp. 283-25; The Scientific Papers, 
volume 2, pp. 109—111: also Vector Analysis, article 102, pp. 271-275. 
The question is also treated in my communication On Products in Ad- 
ditive Fields: Verhandlungen des dritten internationalen Mathematiker- 
Kongresses. Teubner, 1905, pp. 202-215. 
