Page:Scientific Papers of Josiah Willard Gibbs.djvu/244

Also^[1]

${\displaystyle F}$	${\displaystyle =\textstyle \sum '\displaystyle \left\{\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}\right)^{2}\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}\right)^{2}-\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)\right\}\quad }$
	${\displaystyle =\textstyle \sum '\sum \displaystyle \left\{\left({\frac {dx}{dx'}}\right)^{2}\textstyle \sum \displaystyle \left({\frac {dx}{dy'}}\right)^{2}-{\frac {dx}{dx'}}{\frac {dx}{dy'}}\textstyle \sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dx}{dy'}}\right)\right\}\quad }$
	${\displaystyle =\textstyle \sum '\sum \displaystyle \left\{\left({\frac {dx}{dx'}}\right)^{2}\left({\frac {dy}{dy'}}\right)^{2}+\left({\frac {dy}{dx'}}\right)^{2}\left({\frac {dz}{dy'}}\right)^{2}-{\frac {dx}{dx'}}{\frac {dx}{dy'}}{\frac {dy}{dx'}}{\frac {dy}{dy'}}-{\frac {dx}{dx'}}{\frac {dx}{dy'}}{\frac {dz}{dx'}}{\frac {dz}{dy'}}\right\}\quad }$
	${\displaystyle =\textstyle \sum '\sum \displaystyle \left\{\left({\frac {dx}{dx'}}\right)^{2}\left({\frac {dy}{dy'}}\right)^{2}+\left({\frac {dy}{dx'}}\right)^{2}\left({\frac {dx}{dy'}}\right)^{2}-2{\frac {dx}{dx'}}{\frac {dx}{dy'}}{\frac {dy}{dx'}}{\frac {dy}{dy'}}\right\}\quad }$
	${\displaystyle =\textstyle \sum '\sum \displaystyle \left({\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dy}{dx'}}{\frac {dx}{dy'}}\right)^{2}\cdot }$	(433)

This may also be written

${\displaystyle F=\textstyle \sum '\sum \displaystyle {\begin{vmatrix}{\frac {dx}{dx'}}&{\frac {dx}{dy'}}\\{\frac {dy}{dx'}}&{\frac {dy}{dy'}}\end{vmatrix}}^{2}\cdot }$

(434)

In the reduction of the value of ${\displaystyle G}$, it will be convenient to use the symbol ${\displaystyle \textstyle \sum _{3+3}}$ to denote the sum of the six terms formed by changing ${\displaystyle x,y,z,}$ into ${\displaystyle y,z,x;}$${\displaystyle z,x,y;}$${\displaystyle x,z,y;}$${\displaystyle y,x,z;}$ and ${\displaystyle z,y,x}$ and the symbol ${\displaystyle \textstyle \sum _{3-3}}$ in the same sense except that the last three terms are to be taken negatively; also to use ${\displaystyle \textstyle \sum _{3-3}'}$ in a similar sense with respect to ${\displaystyle x',y',z'}$; and to use ${\displaystyle {\text{x', y', z'}}}$ as equivalent to ${\displaystyle x',y',z'}$ except that they are not to be affected by the sign of summation. With this understanding we may write

${\displaystyle G=\textstyle \sum _{3-3}'\left\{\textstyle \sum \displaystyle \left({\frac {dx}{d{\text{x}}'}}{\frac {dx}{dx'}}\right)\textstyle \sum \displaystyle \left({\frac {dx}{d{\text{y}}'}}{\frac {dx}{dy'}}\right)\textstyle \sum \displaystyle \left({\frac {dx}{d{\text{z}}'}}{\frac {dx}{dz'}}\right)\right\}\quad \cdot }$

(435)

In expanding the product of the three sums, we may cancel on account of the sign ${\displaystyle \textstyle \sum _{3-3}'}$ the terms which do not contain all the three expressions ${\displaystyle dx,dy,}$ and ${\displaystyle dz}$. Hence we may write

${\displaystyle G}$	${\displaystyle =\textstyle \sum _{3-3}'\sum _{3+3}\displaystyle \left({\frac {dx}{d{\text{x}}'}}{\frac {dx}{dx'}}{\frac {dy}{d{\text{y}}'}}{\frac {dy}{dy'}}{\frac {dz}{d{\text{z}}'}}{\frac {dz}{dz'}}\right)}$
	${\displaystyle =\textstyle \sum _{3+3}\displaystyle \left\{{\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dz}{dz'}}\textstyle \sum _{3-3}'\displaystyle \left({\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)\right\}\quad }$
	${\displaystyle =\textstyle \sum _{3-3}\displaystyle \left({\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)\textstyle \sum _{3-3}'\displaystyle \left({\frac {dx}{dx'}}{\frac {dy}{dy'}}{\frac {dz}{dz'}}\right)\cdot }$	(436)