10 E. B. Wilson, 
follows the theorem: If any set of ~ quantities of the (7—1)st class, 
B1, Bx, ..-, Bn, satisfy the equations (6), the elements —; and a’; 
are identical. For the ~’s may be expressed linearly in terms of 
the a’s as 
Bi = aye ay + ay Og +... + ani O'n 
and then equations (6) give the equations aj; = 1, a =0. Thus 
the uniqueness of the reciprocal set is established. Furthermore 
equations (6), (6’), ... may be written in the form 
Ge — ead a ay =0 
(6) aa ay aj; = +1 a ay a’; a’; = 0, 
where the negative signs hold when and only when z is even, and 
then only in every alternate equation. From the uniqueness of sets 
of reciprocals the theorem may therefore be stated that: The reci- 
procals of a given set of elements are equal to the given set except 
when their class-number is odd and »# is even, in which case they 
are the negative of the given set. 
One of the prime uses of the reciprocals is to express the idem- 
factors. To avoid the introduction of subscripts, let «. ~B, y,... be 
a set of m independent primary elements and a’, #', y‘, ... their 
reciprocals. The dyadic expression 
(7) U 02 id esp || op P| ieee oh nm terms, 

is called an idemfactor for primary elements. It has the property 
that when multiplied combinatorially, see (4’), into a primary ele- 
ment, it reproduces that element: that is, 
Te= (ale + BIR Tyl7/ +...) e=a(e'8) + BBQ) Tye) 

Saree eg Ne 
This may be seen by considering 9 as expressed in terms of a@, ~. 
y,.... In lke manner the dyadic expression 
(7?) I= aBla' B, n(n—1)/2 terms, 
is an idemfactor for elements of the second class, that is, 0 6=@ 6. 
And 
(7%) h=DaPsyl\a' py n(n—1) (n—2)/6 terms, 
is an idemfactor for elements of the third class. And so on. 

Elementary Properties of Dyadics. 
4. Various representations of dyadics.—For the present purposes, 
the primary dyad will be defined as one whose first factor is a 
primary element and whose second factor is an element of the 
(n—1)st class. For brevity, these factors will be designated re- 
spectively as the antecendent and the consequent. To bring out 
