This dyadic is homologous with snch as are obtained by varying the values of
and only such, unless
It is always possible to take three mutually perpendicular vectors for
and
or, if it be preferred, to take such values for these vectors as shall make the term containing
vanish.
158. The dyadics described in the two last paragraphs may be called shearing dyadics.
The criterion of a shearer is
![{\displaystyle \{\Phi -{\text{I}}\}^{3}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6826cd847f8897170e74bafdcb119c94693aac7e)
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The criterion of a simple shearer is
![{\displaystyle \{\Phi -{\text{I}}\}^{2}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f859b872220d526e9c9383124fdae71f44380555)
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The criterion of a complex shearer is
![{\displaystyle \{\Phi -{\text{I}}\}^{2}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f859b872220d526e9c9383124fdae71f44380555)
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Note.—If a dyadic
is a linear function of a vector
(the term linear being used in the same sense as in No. 105), we may represent the relation by an equation of the form
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or
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where the expression in the braces may be called a triadic polynomial, and a single term
a triad, or the indeterminate product of the three vectors
We are thus led successively to the consideration of higher orders of indeterminate products of vectors, triads, tetrads, etc., in general polyads, and of polynomials consisting of such terms, triadics, tetradics, etc., in general polyadics. But the development of the subject in this direction lies beyond our present purpose.
It may sometimes be convenient to use notations like
and
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to represent the conjugate dyadics which, the first as prefactor, and the second as postfactor, change
into
respectively. In the notations of the preceding chapter these would be written
and
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respectively,
denoting the reciprocals of
If
is a linear function of
the dyadics which as prefactor and postfactor change
into
may be written respectively
and
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If
is any function of
the dyadics which as prefactor and postfactor change
into
may be written respectively
and
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In the notation of the following chapter the second of these (when
denotes a position-vector) would be written
The triadic which as prefactor changes
into
may be written
and that which as postfactor changes
into
may be written
The latter would be written
in the notations of the following chapter.