Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/83

Therefore,

${\begin{aligned}\theta _{1}\times \theta _{2}&={\frac {[\alpha \times \beta ]\times [\beta \times \gamma ]}{\alpha .\beta \,\beta .\gamma }}={\frac {\alpha \times \beta .\gamma \,\beta }{\alpha .\beta \,\beta .\gamma }}\\&={\frac {\beta .\alpha \,\beta \times \gamma +\beta .\beta \,\gamma \times \alpha +\beta .\gamma \,\alpha \times \beta }{\alpha .\beta \,\beta .\gamma }}\cdot \end{aligned}}$

(See No. 38.) That is,

$\theta _{1}\times \theta _{2}=\theta _{2}-{\frac {\alpha .\gamma }{\alpha .\beta \,\beta .\gamma }}\theta _{3}+\theta _{1}.$

Also,

$\theta _{1}.\theta _{2}={\frac {\alpha \times \beta .\beta \times \gamma }{\alpha .\beta \,\beta .\gamma }}=1-{\frac {\alpha .\gamma }{\alpha .\beta \,\beta .\gamma }}\cdot$

Hence,

${\begin{aligned}\theta _{1}\times \theta _{2}&=\theta _{2}-(1-\theta _{1}.\theta _{2})\theta _{3}+\theta _{1},\\\theta _{3}&={\frac {\theta _{1}+\theta _{2}+\theta _{2}\times \theta _{1}}{1-\theta _{1}.\theta _{3}}},\end{aligned}}$

which is the formula for the composition of sucoessiye finite rotations by means of their vector semitangents of version.

147. The versors just described constitute a particular class under the more general form

$\alpha \alpha '+\cos q\{\beta \beta '+\gamma \gamma '\}+\sin q\{\gamma \beta '-\beta \gamma '\},$

in which $\alpha ,\beta ,\gamma$ are any non-complanar vectors, and $\alpha ',\beta ',\gamma '$ their reciprocals. A dyadic of this form as a prefactor does not affect any vector parallel to $\alpha .$ Its effect on a vector in the $\beta {\text{-}}\gamma$ plane will be best understood if we imagine an ellipse to be described of which $\beta$ and $\gamma$ are conjugate semi-diameters. If the vector to be operated on be a radius of this ellipse, we may evidently regard the ellipse with $\beta ,\gamma ,$ and the other vector, as the projections of a circle with two perpendicular radii and one other radius. A little consideration will show that if the third radius of the circle is advanced an angle $q,$ its projection in the ellipse will be advanced as required by the dyadic prefactor. The effect, therefore, of such a prefactor on a vector in the $\beta {\text{-}}\gamma$ plane may be obtained as follows: Describe an ellipse of which $\beta$ and $\gamma$ are conjugate semi-diameters. Then describe a similar and similarly placed ellipse of which the vector to be operated on is a radius. The effect of the operator is to advance the radius in this ellipse, in the angular direction from $\beta$ toward $\gamma ,$ over a segment which is to the total area of the ellipse as $q$ is to $2\pi .$ When used as a postfactor, the properties of the dyadic are similar, but the axis of no motion and the planes of rotation are in general different.

Def.—Such dyadics we shall call cyclic.

The Nth power (N being any whole number) of such a dyadic is obtained by multiplying $q$ by N. If $q$ is of the form $2\pi$ N/M (N and M being any whole numbers) the Mth power of the dyadic will be an idemfactor. A cyclic dyadic, therefore, may be regarded as a root of