12 E. B. Wilson, 
dyadic which converts ” given independent elements «a, 8, ..., v 
into 2 elements «, (1, ..-, %1, not necessarily independent, may 
be written down as 
(10) &d—a|e'+B|8'...- tml, Ba = aq, etc. 
If the antecedents of this dyadic are not linearly independent, 
the expression may be reduced to a sum of / dyads where /<n. 
In general if the antecedents or consequents or both, which occur 
in the reduction of a dyadic to a sum of w terms, be not independent, 
the dyadic may be reduced to a sum of dyads less than z in number. 
If 7 be the least number of dyads which may be obtained in the 
reduction 
(11) @—alatel@t...+ala (¢ terms), 
where the antecendents and consequents are now linearly independent, 
the dyadic is said to have nullity of degree n—/. If the elements 
be interpreted as vectors issuing from an assumed origin in space 
of x dimensions, the nullity may be stated geometrically by saying 
that by the operation of the dyadic the -dimensional space has 
been converted into a flat subspace of / dimensions passing through 
the assumed origin. 
In like manner the dyadic which, used as a postfactor, converts 
the independent elements @, 8, ..., » into @%, B,, ..., ¥ is 
(10) (-1)1—e@'|a+@lii+...+7 lm, «ef= aq, etc, 
where the consequents need not be independent. If the dyadic 
reduces to a sum of / terms, of which the antecedents and con- 
sequents are then linearly independent, the degree of nullity is 
again w—/. The application of the dyadic has converted space 
into a subspace of dimensions /in hyperplanes. This subspace may 
or may not be identical with that previously obtained by using the 
dyadic as a prefactor to elements of the first class. In general the 
two subspaces will not be identical. 
5. Combinatory products of dyadics.—As the individual dyads 
satisfy the distributive law, the definition of the product of two 
dyadics follows immediately from the definition of the product of 
the dyads as given in article 2. It also follows that the product 
of dyadics is itself a distributive operation. An examination of the 
definition shows, however, that the product is not in general com- 
mutative but that it is associative, that is, 
(12) (PQ) = ($B) Q=F #2. 
The associative property is not lost if elements of the proper class 
are multiplied into the products at either end or at both ends, that is, 
(12') (6 ®) (Po) = 6( V0) = (6 Bo = 68 Vo. 
