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

Also^[1]

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

This may also be written

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

(434)

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

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

(435)

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

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

1. The values of $F$ and $G$ given in equations (434) and (438), which are here deduced at length, may be derived from inspection of equation (430) by means of the usual theorems relating to the multiplication of determinants. See Salmon's Lessons Introductory to the Modern Higher Algebra, 2d ed., Lesson III; or Baltzer's Theorie und Anwendung der Determinanten, §5.