Double Products and Strains in Hyperspace. 24 
seen from the work on double multiplication, satisfy the obvious 
equations 
(27) [= In-1, ¢) I, =Jy-2) Choatenerenstcs Ln=2 — = Tee, Ln = I, 
The conjugate of the conjugate of any dyadic is the given dyadic, 
that is (% )e-=@. The process of taking the conjugate is involutory. 
As to the rules of operation with conjugates, a number of theorems 
may be stated. The conjugate of the sum of any dyadics is the 
sum of the conjugates. The conjugate of the product of two 
dyadics is the product of the conjugates taken in inverse order. 
For let 
@=alatB@tylyt+..., 
p— q' lA+ Ble |e 7’ |2 (Sai oe 
By merely forming the expression for (® #), and #, @, the truth of 
the theorem is evident in this case. The proof for dyadics other 
than primary would be similar, and the theorem may evidently be 
extended to any number of factors by induction. Hence the con- 
jugate of any power of a dyadic is that same power of the con- 
jugate of the dyadic, and the result may be extended to negative 
powers if a reciprocal exists. It may also be seen that the double 
product of the conjugates of two or more dyadics is the conjugate 
of the double product of the dyadics. Here the order of the factors 
is immaterial. As a special case, the conjugate of a double power 
is that double power of the conjugate. 
As @ and ®,_; are both of the (#—1)st class, it is natural to 
seek a relation between them. Let 
@=alat+B\Bt+ylyt+...+v|v, (# terms) 
Now #,_; is the sum of all combinations (not permutations, for the 
factor 4 “1! has been thrown out) of z—1 antecedents and their 
eee adie consequents. These combinations may be represented 
in terms of the reciprocal sets provided that the dyadic # is com- 
plete so that the antecedents and consequents are linearly inde- 
pendent. Thus in this case 
On1=(aBy...v)(aBy...v) (ale tele ty|yt+. 
+ v'|9*) 
and #,—=(—1)r—1 (aja+,| Boylyt...tripy) 
The negative sign occurs in precisely fhece cases where the theory 
of reciprocals in article 3 requires it. Hence 
(28) PD, Py—1 = Py [n-1. 
If @ has one or more degrees of nullity, it may be written as the 
sum of —1 dyads, which need not have independent antecedents 
and consequents, unless the degree of nullity is one; and hence 
