Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/96

80 Now $$\text{I} + q\text{N}^{-1} \{ kj - jk \}$$ evidently represents a versor having the axis $$i$$ and the infinitesimal angle of version $$q\text{N}^{-1}.$$ Hence the above exponential represents a versor having the same axis and the angle of version $$q.$$ If we set $$qi = \omega,$$ the exponential may be written  Such an expression therefore represents a versor. The axis and direction of rotation are determined by the direction of $$\omega,$$ and the angle of rotation is equal to the magnitude of $$\omega.$$ The value of the versor will not be affected by increasing or diminishing the magnitude of $$\omega$$ by $$2\pi.$$

178. If, as in No. 151, the definitions of No. 171 give  If $$a, b, c$$ are positive and unequal, we may add, by No. 172,  179. If, as in No. 163, we have by No. 173  But  Therefore,  Hence, if $$a$$ is positive,  Since the value of $$\Phi$$ is not affected by increasing or diminishing $$q$$ by $$2\pi,$$ the function $$\log \Phi$$ is many-valued.

To find the value of $$\cos \Phi$$ and $$\sin \Phi,$$ let us set Then, by No, 175,  But  Therefore,