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

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

$e^{{\text{I}}\times \omega }.$

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,

$\Phi =a\alpha \alpha '+b\beta \beta '+c\gamma \gamma ',$

the definitions of No. 171 give

${\begin{aligned}e^{\Phi }&=e^{a}\alpha \alpha '+e^{b}\beta \beta '+e^{c}\gamma \gamma ',\\\cos \Phi &=\cos a\,\alpha \alpha '+\cos b\,\beta \beta '+\cos c\,\gamma \gamma ',\\\sin \Phi &=\sin a\,\alpha \alpha '+\sin b\,\beta \beta '+\sin c\,\gamma \gamma ',\end{aligned}}$

If $a,b,c$ are positive and unequal, we may add, by No. 172,

$\log \Phi =\log a\alpha \alpha '+\log b\beta \beta '+\log c\gamma \gamma '.$

179. If, as in No. 163,

$\Phi =a\alpha \alpha '+b\{\beta \beta '+\gamma \gamma '\}+c\{\gamma \beta '-\beta \gamma '\}=a\alpha \alpha '+p\,\cos b\,\beta \beta '\{\beta \beta '+\gamma \gamma '\}+p\,\sin \,q\{\gamma \beta '-\beta \gamma '\},$

we have by No. 173

$e^{\Phi }=e^{a\alpha \alpha '}.e^{b\{\beta \beta '+\gamma \gamma '\}}.e^{c\{\gamma \beta '-\beta \gamma '\}}.$

But

${\begin{aligned}e^{a\alpha \alpha '}&=e^{a}\alpha \alpha '+\beta \beta '+\gamma \gamma ',\\e^{b\{\beta \beta '+\gamma \gamma '\}}&=\alpha \alpha '+e^{b}\{\beta \beta '+\gamma \gamma '\},\\e^{c\{\gamma \beta '-\beta \gamma '\}}&=\alpha \alpha '+\cos \,c\{\beta \beta '+\gamma \gamma '\}+e^{b}\sin \,c\{\gamma \beta '-\beta \gamma '\}.\end{aligned}}$

Therefore,

$e^{\Phi }=e^{a}\alpha \alpha '+e^{b}\cos \,c\{\beta \beta '+\gamma \gamma '\}+e^{b}\sin \,c\{\gamma \beta '-\beta \gamma '\}.$

Hence, if $a$ is positive,

$\log \Phi =\log a\,\alpha \alpha '+\log p\,\{\beta \beta '+\gamma \gamma '\}+q\{\gamma \beta '-\beta \gamma '\}.$

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

${\begin{aligned}\Theta &=b\{\beta \beta '+\gamma \gamma '\}+c\{\gamma \beta '-\beta \gamma '\},\\\Xi &=\gamma \beta '-\beta \gamma '.\end{aligned}}$

Then, by No, 175,

$\cos \Phi =\cos\{a\alpha \alpha '\}+\cos \Theta -{\text{I}}.$

But

$\cos\{a\alpha \alpha '\}-{\text{I}}=\cos a\alpha \alpha '-\alpha \alpha '.$

Therefore,

$\cos \Phi =\cos a\alpha \alpha '-\alpha \alpha '+\cos \Theta .$