Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/44

It results from the principle stated in No. 85, that any vector equation of the first degree with respect to $\rho$ may be reduced to the form

	$\delta =$	$\alpha (\lambda .\rho )+\beta (\mu .\rho )+\gamma (\nu .\rho )+a\rho +\epsilon \times \rho .$
But⁠	$a\rho =$	$a\lambda '(\lambda .\rho )+a\mu '(\mu .\rho )+a\nu '(\nu .\rho ),$
and⁠	$\epsilon \times \rho =$	$\epsilon \times \lambda '(\lambda .\rho )+\epsilon \times \mu '(\mu .\rho )+\epsilon \times \nu '(\nu .\rho ),$

where $\lambda ',\mu ',\nu '$ represent, as before, the reciprocals of $\lambda ,\mu ,\nu$. By substitution of these values the equation is reduced to the form of equation (1), which may therefore be regarded as the most general form of a vector equation of the first degree with respect to $\rho$.

41. Relations between two normal systems of unit vectors.—If $i,j,k$, and $i',j',k'$ are two normal systems of unit vectors, we have

	$i'=(i.i')i+(j.i')j+(k.i')k,$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$⁠(1)
	$j'=(i.j')i+(j.j')j+(k.j')k,$
	$k'=(i.k')i+(j.k')j+(k.k')k,$
and⁠	$j=(i.i')i'+(i.j')j'+(i.k')k',$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$⁠(2)
	$j=(j.i')i'+(j.j')j+(j.j')k',$
	$k'=(k.i')i'+(k.j')j'+(k.k')k',$

(See equation (8) of No. 38.)

The nine coefficients in these equations are evidently the cosines of the nine angles made by a vector of one system with a vector of the other system. The principal relations of these cosines are easily deduced. By direct multiplication of each of the preceding equations with itself, we obtain siz equations of the type

$(i.i')^{2}+(j.i')^{2}+(k.i')^{2}=1.$

(3)

By direct multiplication of equations (1) with each other, and of equations (2) with each other, we obtain six of the type

$(i.i')(i.j')+(j.i')(j.j')+(k.i')(k.j')=0.$

(4)

By skew multiplication of equations (1) with each other, we obtain three of the type

${\begin{aligned}k'=\{(j.i')(k.j')-(k.i')(j.j')\}i+\{(k.i')(i.j')&-(i.i')(k.j')\}j\\&+\{(i.i')(j.j')-(j.i')(i.j')\}k.\end{aligned}}$

Comparing these three equations with the original three, we obtain nine of the type

$i.k'=(j.i')(k.j')-(k.i')(j.j').$

(5)

Finally, if we equate the scalar product of the three right hand members of (1) with that of the three left hand members, we obtain

${\begin{aligned}(i.i')(j&.j')(k.k')+(i.j')(j.k')(k.i')+(i.k')(j.i')(k.j')\\&-(k.i')(j.j')(i.k')-(k.j')(j.k')(i.i')-(k.k')(j.i')(i.j')=1.\end{aligned}}$

(6)

Equations (1) and (2) (if the expressions in the parentheses are supposed replaced by numerical values) represent the linear relations