Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/81

The criterion of a versor may therefore be written

$\Phi .\Phi _{\text{C}}={\text{I}},$⁠and⁠$\left\vert \Phi \right\vert =1.$

For the last equation we may substitute

$\left\vert \Phi \right\vert >0,$⁠or⁠$\left\vert \Phi \right\vert \gtrless -1.$

It is evident that the resultant of successive finite rotations is obtained by multiplication of the versors.

143. If we take the axis of the rotation for the direction of $i,i'$ will have the same direction, and the versor reduces to the form

$ii+j'j+k'k,$

in which $i,j,k$ and $i,j',k'$ are normal systems of unit vectors.

We may set

${\begin{aligned}j'&=\cos q\,j+\sin q\,k,\\k'&=\cos q\,k-\sin q\,j,\end{aligned}}$

and the versor reduces to

$ii+\cos q\,\{jj+kk\}+\sin q\,\{kj-jk\},$

or

$ii+\cos q\,\{{\text{I}}-ii\}+\sin q\,{\text{I}}\times i,$

where $q$ is the angle of rotation, measured from $j$ toward $k,$ if the versor is used as a prefactor.

144. When any versor $\Phi$ is used as a pref actor, the vector $-\Phi _{\text{X}}$ will be parallel to the axis of rotation, and equal in magnitude to twice the sine of the angle of rotation measured counter-clockwise as seen from the direction in which the vector points. (This will appear if we suppose $\Phi$ to be represented in the form given in the last paragraph.) The scalar $\Phi _{\text{S}}$ will be equal to unity increased by twice the cosine of the same angle. Together, $-\Phi _{\text{X}}$ and $\Phi _{\text{S}}$ determine the versor without ambiguity. If we set

$\theta ={\frac {-\Phi _{\text{X}}}{1+\Phi _{\text{S}}}},$

the magnitude of $\theta$ will be

${\frac {2\sin q}{2+2\cos q}}$⁠or⁠$\tan {\tfrac {1}{2}}q,$

where $q$ is measured counter-clockwise as seen from the direction in which $\theta$ points. This vector $\theta ,$ which we may call the vector semitangent of version, determines the versor without ambiguity.

145. The versor $\Phi$ may be expressed in terms of $\theta$ in various ways. Since $\Phi$ (as pref actor) changes $\alpha -\theta \times \alpha$ into $\alpha +\theta \times \alpha$ ($\alpha$ being any vector), we have

$\Phi =\{{\text{I}}+{\text{I}}\times \theta \}.\{{\text{I}}-{\text{I}}\times \theta \}^{-1}.$

Again

$\Phi ={\frac {\theta \theta +\{{\text{I}}+{\text{I}}\times \theta \}^{2}}{1+\theta .\theta }}={\frac {(1-\theta .\theta ){\text{I}}+2\theta \theta +2{\text{I}}\times \theta }{1+\theta .\theta }},$