and consider the limits within which
varies, when we give
all possible values.
The products
and
are evidently planar dyadics.
124. Def.—A dyadic
is said to be an idemfactor, when
for all values of
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or when
for all values of
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If either of these conditions holds true,
must be reducible to the form
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Therefore, both conditions will hold, if either does. All such dyadics are equal, by No. 108. They will be represented by the letter
The direct product of an idemfactor with another dyadic is equal to that dyadic. That is,
![{\displaystyle I.\Phi =\Phi ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03ad2c41fab8ff9486c19a144b4fb79e47541616)
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where
is any dyadic.
A dyadic of the form
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in which
are the reciprocals of
is an idemfactor. (See No. 38.) A dyadic trinomial cannot be an idemfactor, unless its antecedents and consequents are reciprocals.
125. If one of the direct products of two dyadics is an idemfactor, the other is also. For, if
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for all values of
and
is complete;
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for all values of
therefore for all values of
and therefore
Def. — In this case, either dyadic is called the reciprocal of the other.
It is evident that an incomplete dyadic cannot have any (finite) reciprocal.
Reciprocals of the same dyadic are equal. For if
and
are both reciprocals of
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If two dyadics are reciprocals, the operators formed by using these dyadics as prefactors are inverse, also the operators formed by using them as postfactors.
126. The reciprocal of any complete dyadic
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is
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where
are the reciprocals of
and
are the reciprocals of
(See No. 38.)