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

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This may also be written 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 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