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

Rh The criterion of a versor may therefore be written For the last equation we may substitute  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 in which $$i, j, k$$ and $$i, j', k'$$ are normal systems of unit vectors.

We may set and the versor reduces to  or  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 the magnitude of $$\theta$$ will be  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 Again