Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/97

Now, by No. 174,

$\cos \Theta ={\tfrac {1}{2}}\{e^{\Xi .\Theta }+e^{-\Xi .\Theta }\}.$

Since

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

Therefore

$\cos \Theta =\alpha \alpha '+{\tfrac {1}{2}}(e^{c}+e^{-c})\cos b\{\beta \beta '+\gamma \gamma '\}-{\tfrac {1}{2}}(e^{c}-e^{-c})\sin b\{\gamma \beta '-\beta \gamma '\}.$

and

$\cos \Phi =\cos a\,\alpha \alpha '+{\tfrac {1}{2}}(e^{c}+e^{-c})\sin b\{\beta \beta '+\gamma \gamma '\}-{\tfrac {1}{2}}(e^{c}-e^{-c})\cos b\{\gamma \beta '-\beta \gamma '\}.$

In like manner we find

$\sin \Phi =\sin a\,\alpha \alpha '+{\tfrac {1}{2}}(e^{c}+e^{-c})\sin b\{\beta \beta '+\gamma \gamma '\}+{\tfrac {1}{2}}(e^{c}-e^{-c})\cos b\{\gamma \beta '-\beta \gamma '\}.$

180. If $\alpha ,\beta ,\gamma$ and $\alpha ',\beta ',\gamma '$ are reciprocals, and

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

and ${\text{N}}$ is any whole number,

$\Phi ^{\text{N}}=a^{\text{N}}\,\alpha \alpha '+b^{\text{N}}\{\beta \beta '+\gamma \gamma '\}+{\text{N}}b^{{\text{N}}-1}c\,\beta \gamma '.$

Therefore,

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

If $a$ and $b$ are unequal, and $c$ other than zero, we may add

$\log \Phi =\log a\,\alpha \alpha '+\log b\,\{\beta \beta '+\gamma \gamma '\}+cb^{-1}\beta \gamma .$

181. If $\alpha ,\beta ,\gamma ,$ and $\alpha ',\beta ',\gamma '$ are reciprocals, and

$\Phi =a{\text{I}}+b\{\alpha \beta '+\beta \gamma '\}+c\,\alpha \gamma ',$

and ${\text{N}}$ is a whole number,

$\Phi ^{\text{N}}=a^{\text{N}}{\text{I}}+{\text{N}}a^{{\text{N}}-1}b\{\alpha \beta '+\beta \gamma '\}+({\text{N}}a^{{\text{N}}-1}c+{\tfrac {1}{2}}{\text{N}}({\text{N}}-1)a^{{\text{N}}-2}b^{2})\alpha \gamma '.$

Therefore

${\begin{aligned}e^{\Phi }&=e^{a}{\text{I}}+e^{a}b\{\alpha \beta '+\beta \gamma '\}+e^{a}({\tfrac {1}{2}}b^{2}+c)\alpha \gamma ',\\\cos \Phi &=\cos a\,{\text{I}}-b\,\sin a\,\{\alpha \beta '+\beta \alpha '\}-({\tfrac {1}{2}}b^{2}\cos a+c\sin a)\alpha \gamma ',\\\sin \Phi &=\sin a\,{\text{I}}+b\,\cos a\,\{\alpha \beta '+\beta \alpha '\}-({\tfrac {1}{2}}b^{2}\sin a-c\cos a)\alpha \gamma '.\end{aligned}}$

Unless $b=0,$ we may add

$\log \Phi =\log a{\text{ I}}+ba^{-1}\{\alpha \beta '+\beta \alpha '\}+(ca^{-1}-{\tfrac {1}{2}}b^{2}a^{-2})\alpha \gamma '.$

182. If we suppose any dyadic $\Phi$ to vary, but with the limitation that all its values are homologous, we may obtain from the definitions of No. 171

$d\{e^{\Phi }\}=e^{\Phi }.d\Phi ,$

(1)

$d\sin \Phi =\cos \Phi .d\Phi ,$

(2)

$d\cos \Phi =-\sin \Phi .d\Phi ,$

(3)

$d\log \Phi =\Phi ^{-1}.d\Phi ,$

(4)