127. Def.—We shall write
for the reciprocal of any (complete) dyadic
also
for
etc., and
for
etc. It is evident that
is the reciprocal of
,
128. In the reduction of equations, if we have
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we may cancel the
(which is equivalent to multiplying by
) if
is a complete dyadic, but not otherwise The case is the same with such equations as
![{\displaystyle \Phi .\sigma =\Phi .\rho ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afc21ea823941c99ffb45f37a5074c9e4c2c4c91) ![{\displaystyle \Psi .\Phi =\Omega .\Phi ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e118b10edcc5d70eb56be461847ac8f4c5b53e65)
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To cancel an incomplete dyadic in such cases would be analogous to cancelling a zero factor in algebra.
129. Def.—If in any dyadic we transpose the factors in each term, the dyadic thus formed is said to be conjugate to the first. Thus
and
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are conjugate to each other. A dyadic of which the value is not altered by such transposition is said to be self-conjugate. The conjugate of any dyadic
may be written
It is evident that
and
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and
are conjugate functions of
(See No. 106.) Since
we may write
etc., without ambiguity.
130. The reciprocal of the product of any number of dyadics is equal to the product of their reciprocals taken in inverse order. Thus
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The conjugate of the product of any number of dyadics is equal to the product of their conjugates taken in inverse order. Thus
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Hence, since
![{\displaystyle \Phi _{\text{C}}.\{\Phi ^{-1}\}_{\text{C}}=\{\Phi ^{-1}.\Phi \}_{\text{C}}=I,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1750480b2e8ee835033000287bc2b38b32dc4f66)
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and we may write
without ambiguity.
131. It is sometimes convenient to be able to express by a dyadic taken in direct multiplication the same operation which would be effected by a given vector (
) in skew multiplication. The dyadic
will answer this purpose. For, by No. 117,
![{\displaystyle \{I\times \alpha \}.\rho =\alpha \times \rho ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f38b4e3b60bdd3fe914f1b25188381eadf6fd246) |
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![{\displaystyle \{I\times \alpha \}.\Phi =\alpha \times \Phi ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47bfce1db5260a67259bd6e49f52ecde5df5a862) |
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The same is true of the dyadic
which is indeed identical with
as appears from the equation