18 E. B. Wilson, 
8. Double powers.—A dyadic may be multiplied doubly into it- 
self. Thus if 
=alat+elPtyiyt... 
then 
(21) PY h=aBlabB+ayplayt... 
TPalpeatBy|By+... 
Frel|yeFty ply e+::. 
It will be noticed that the terms in #% occur in pairs. The ex- 
pression 
(22) 

2 
Dre DOD AIO 
which is one-half of Px, will be denoted, as indicated, by 2. It 
x 
may be called the double square of # or, more briefly and properly, 
the second of In like manner the expressions 
(22' 1 ie Lege 1 peas ee ae 
i) Pp— 3, PL PLO 4, FX, Pe= 4, PX, Sane Dp ae 
may be formed and will be called the third of #, the fourth of @, 
., the wth of @. Collectively the set ¢,, &;, ..., O, may be 
called the double powers of #, although it should be remembered 
: i ; 
that in % the factor Rl has been inserted. 
The double powers afford a ready means of formulating the con- 
ditions that a dyadic # possess a certain degree of nullity without 
the necessity of reducing # to the sum of the fewest possible 
dyads—a reduction which is by no means easily carried out on any 
assigned dyadic. If the dyadic has m—/ degrees of nullity, it may 
be written as 
b=alatBBtyiyt...+Ala (/ terms), 
where the antecedents and consequents are independent. In this 
case ; takes the form 
(23) Bi (CB eA (GIR en) 
and does not vanish. All the higher double powers will vanish be- 
cause one element will have to be repeated in each antecedent and 
consequent. The lower powers cannot vanish; for the double pro- 

second class, but as a vector, and vector multiplication is not associa- 
tive. Moreover, in the Vector Analysis, the scalar product of two vec- 
tors occurs, and hence there is the double scalar product of two dyadics. 
If Grassmann’s inner product were introduced into the system in addition 
to his outer product (the combinatorial product), there would be double 
inner products of dyadics. These were not taken up in Gibbs’s course, 
and they will be omitted at this point. 
