Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/82

66 as will be evident on considering separately in the expression $$\Phi. \rho$$ the components perpendicular and parallel to $$\theta,$$ or on substituting in for $$\cos q$$ and $$\sin q$$ their values in terms of $$\tan \tfrac{1}{2}q.$$

If we set, in either of these equations, we obtain, on reduction, the formula  in which the versor is expressed in terms of the rectangular components of the vector semitangent of version.

146. If $$\alpha, \beta, \gamma$$ are unit vectors, expressions of the form are biquadrantal versors. A product like is a versor of which the axis is perpendicular to $$\alpha$$ and $$\beta,$$ and the amount of rotation twice that which would carry $$\alpha$$ to $$\beta.$$ It is evident that any versor may be thus expressed, and that either $$\alpha$$ or $$\beta$$ may be given any direction perpendicular to the axis of rotation. If we have for the resultant of the successive rotations  This may be applied to the composition of any two successive rotations, $$\beta$$ being taken perpendicular to the two axes of rotation, and affords the means of determining the resultant rotation by construction on the surface of a sphere. It also furnishes a simple method of finding the relations of the vector semitangents of version for the versors $$\Phi, \Psi,$$ and $$\Psi. \Phi.$$ Let Then, since  which is moreover geometrically evident. In like manner,