where
are the reciprocals of
and
and
are scalars, of which
is positive, will be most evident if we resolve it into the factors
![{\displaystyle a\alpha \alpha '+\beta \beta '+\gamma \gamma ',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8186e890d6f5e16d676c10a6a8cbb29ef201480f)
![{\displaystyle \alpha \alpha '+p\beta \beta '+p\gamma \gamma ',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c96c335994feabc17314faa09a04eeb193569375)
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of which the order is immaterial, and if we suppose the vector on which we operate to be resolved into two factors, one parallel to
and the other in the
plane. The effect of the first factor is to multiply by
the component parallel to
without affecting the other. The effect of the second is to multiply by
the component in the
plane without affecting the other. The effect of the third is to give the component in the
plane the kind of elliptic rotation described in No. 147.
The effect of the same dyadic as a postfactor is of the same nature.
The value of the dyadic is not affected by the substitution for a of another vector having the same direction, nor by the substitution for
and
of two other conjugate semi-diameters of the same or a similar and similarly situated ellipse, and which follow one another in the same angular direction.
Def. — Such dyadics we shall call cyclotonic.
154. Cyclotonics which are reducible to the same form except with respect to the values of
and
are homologous. They are multiplied by multiplying the values of
and also those of
and adding those of
Thus, the product of
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and
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is
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A dyadic of this form, in which the value of
is not zero, or the product of
and a positive or negative integer, is homologous only with such dyadics as are obtained by varying the values of
and
156. In general, any dyadic may be reduced to the form either of a tonic or of a cyclotonic. (The exceptions are such as are made by the limiting cases.) We may show this, and also indicate how the reduction may be made, as follows. Let
be any dyadic. We have first to show that there is at least one direction of
for which
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This equation is equivalent to
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or,
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