${\displaystyle {\frac {dx}{dx'}},...{\frac {dz}{dz'}}}$ under the most general supposition with respect to the position of the co-ordinate axes.

For any principal axis of strain we have

${\displaystyle {\frac {d(r^{2})}{d\alpha '}}d\alpha '+{\frac {d(r^{2})}{d\beta '}}d\beta '+{\frac {d(r^{2})}{d\gamma '}}d\gamma '=0,}$

when⁠${\displaystyle \alpha 'd\alpha '+\beta '\beta '+\gamma 'd\gamma '=0,}$

the differential coefficients in the first of these equations being determined from (420) as before. Therefore,

${\displaystyle {\frac {1}{\alpha '}}{\frac {d(r^{2})}{d\alpha '}}={\frac {1}{\beta '}}{\frac {d(r^{2})}{d\beta '}}={\frac {1}{\gamma '}}{\frac {d(r^{2})}{d\gamma '}}\cdot }$

(426)

From (420) we obtain directly

${\displaystyle {\frac {\alpha '}{2}}{\frac {d(r^{2})}{d\alpha '}}+{\frac {\beta '}{2}}{\frac {d(r^{2})}{d\beta '}}+{\frac {\gamma '}{2}}{\frac {d(r^{2})}{d\gamma '}}=r^{2}.}$

(427)

From the two last equations, in virtue of the necessary relation ${\displaystyle \alpha ^{'2}+\beta ^{'2}+\gamma ^{'2}=1}$, we obtain

${\displaystyle {\tfrac {1}{2}}{\frac {d(r^{2})}{d\alpha '}}=\alpha 'r^{2},}$⁠${\displaystyle {\tfrac {1}{2}}{\frac {d(r^{2})}{d\beta '}}=\beta 'r^{2},}$⁠${\displaystyle {\tfrac {1}{2}}{\frac {d(r^{2})}{d\gamma '}}=\gamma 'r^{2},}$

(428)

or, if we substitute the values of the differential coefficients taken from (420),

${\displaystyle \alpha '\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}\right)^{2}+\beta '\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)+\gamma '\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dz'}}\right)=\alpha 'r^{2},}$	${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}}$ (429)
${\displaystyle \alpha '\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}{\frac {dx}{dx'}}\right)+\beta '\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}\right)^{2}+\gamma '\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dz'}}\right)^{2}=\beta 'r^{2},}$
${\displaystyle \alpha '\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}{\frac {dx}{dx'}}\right)+\beta '\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}{\frac {dx}{dy'}}\right)+\gamma '\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}\right)^{2}=\gamma 'r^{2}.}$

If we eliminate ${\displaystyle \alpha ',\beta ',\gamma '}$ from these equations, we may write the result in the form,

${\displaystyle {\begin{vmatrix}\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}\right)^{2}-r^{2}&\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)&\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dz'}}\right)\\\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}{\frac {dx}{dx'}}\right)&\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}\right)^{2}-r^{2}&\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}{\frac {dx}{dz'}}\right)\\\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}{\frac {dx}{dx'}}\right)&\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}{\frac {dx}{dy'}}\right)&\textstyle \sum \displaystyle \left({\frac {dx}{dz'}}\right)^{2}-r^{2}\end{vmatrix}}=0.}$

(430)

We may write

${\displaystyle -r^{5}+Er^{4}-Fr^{2}+G=0.}$

(431)

${\displaystyle E=\textstyle \sum '\sum \displaystyle \left({\frac {dx}{dx'}}\right)^{2}\cdot }$

(432)