14 EE. B. Wilson, 
The existence of a reciprocal for any complete dyadic establishes 
the principle of cancelation for such dyadics. Thus 
if dP — O then 1 GV = 162 and F=2Q 
or if do = &o then -! ogo = 'Po and o=— 6. 
In the first of these equations the second dyadic. @ or 2 need not 
be complete. Although a complete dyadic may be canceled from 
an equation, an incomplete generally cannot be canceled. 
A dyadic may be multiplied into itself; the product #@ will be 
denoted by #?. In like manner all the successive powers may be 
formed. From the theorem on nullity, it follows that if a dyadic 
is complete, all its powers will be complete. As the reciprocal of 
any power of a dyadic is the same as that power of the reciprocal, 
it is seen that negative as well as positive integral exponents are 
applicable to complete dyadics. Incomplete dyadics will be con- 
sidered to have only positive powers. It may happen that the 
successive powers have an increasing degree of nullity, so that there 
is a certain least power fp such that ? = 0. In this case @ is said 
to be a nilpotent dyadic. It is not necessary, however, that the 
nullity of the successive powers should increase to the value z. 
This may be seen by a simple example. Consider the dyadic 
P=) Bien ates ate A A 1) wee eens) 
This has nullity of degree —/, and as all its powers are identical 
with it, they also have nullity of degree m—/. In general, however, 
the powers of an incomplete dyadic have increasing nullities up to 
a certain power, from which on they all have the same _ nullity. 
And by reasoning like that employed in proving Sylvester’s theorem 
on nullities, it is seen at once that if m—/, n—/+ 4, n—/+44+4,... 
are the respective nullities of ¢, #7, , ..., then nJZ2LZ=h 
=.... Gibbs apparently did not state this last fact in his lectures. 
6. Homologous dyadics.—It has been stated that in general, dyadics 
are not commutative in their multiplication. If two dyadics # and 
‘P are such that 6 #— WP, they will be said to be homologous. 
Any dyadic is homologous with the idemfactor /, and all powers 
of a dyadic are homologous with one another. Moreover, if two 
or more dyadics are homologous, any dyadics which may be ob- 
tained from them by the algebraic processes of addition, multiph- 
cation, and so forth, are also homologous with the original dyadics 
and with each other. Thus the algebra of homologous dyadics 
does not differ essentially from the algebra of ordinary real and 
complex numbers except as regards the extraction of roots. It will 
be seen later, in article 14, that even in such simple cases as the 
