Double Products and Strains in Hyperspace. 11 
more clearly the different classes of the antecedent and the con- 
sequent, the former will be denoted by a Greek small letter and 
the latter by a Greek small letter carrying a dash—thus «| 2. 
Dyads of the form «|i may be called secondary, and so on. 
The notation of the reciprocal set to ~ elements a, f, y, ... has 
been a‘, 6’, y’, ...3 and this will be adhered to. The dashes will 
not be introduced to call additional attention to the class of the 
reciprocals. The set of reciprocals of 2 independent elements of the 
(n—-1)st class a, 8, y, ... will be represented by «’, #’, y’,... and 
will be of the first class. 
The sum of any number of primary dyads 
(8) B= to| Go + Bo|Bo+70l¥o+--- 
is called a dyadic polynomial or simply a dyadic. As was stated 
in article 2, each of the dyads may be expanded as a block of x? 
terms by expressing the antecedents and the consequents in terms 
of a set of independent elements «, 8, y,..., » and «, B, y,...,%. 
If this be done the dyadic # takes the form 
(9) P = Cag c|a+ car alB+...+can e|v 
aE Charen hare. cs os Cone kD 
Gage | a he ee aa BD: 
If these terms be added according to rows or according to columns, 
® reduces to a sum of 7 terms: 
(9’) $=alat+BlAat...+tr|r, 
P= mlatAlB+r...+ |». 
In this reduction to a sum of w terms, either the antecedents or 
the consequents may be chosen arbitrarily, but not both. The most 
useful reduction to the form (9) is when the antecedents and con- 
sequents are reciprocal sets. 
Two dyadics # and # may be said to be equal if the x* coef- 
ficients cj are equal when the dyadics are both reduced to the 
form (9) in terms of the same antecedents and the same conse- 
quents. Another definition, which is preferable and obviously re- 
ducible to this, is contained in any one of the three statements: 
Two dyadics @ and ¥ are equal when and only when 
Pbo=VPo for all values of 9 
or oV=oP for all values of 6 
or ofo=o6 Fo for all values of @ and o. 
To insure the equality, it is not necessary to verify these equations 
for all values of 9 or of 6 or of g and o. If the equations hold 
respectively for 2 independent values of the elements in question, 
they will hold for all values. In fact, by the aid of reciprocals the 
