Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/106

lie in any same real plane.) If ${\mathfrak {a}}\times {\mathfrak {b}}.{\mathfrak {c}}$ is not equal to zero, the equation shows that any fourth bivector may be expressed as a sum of ${\mathfrak {a}},{\mathfrak {b}}$ and ${\mathfrak {c}}$ with biscalar coefficients, and indicates how these coefficients may be determined.

11. The equation

$({\mathfrak {r}}.{\mathfrak {a}}){\mathfrak {b}}\times {\mathfrak {c}}+({\mathfrak {r}}.{\mathfrak {b}}){\mathfrak {c}}\times {\mathfrak {a}}+({\mathfrak {r}}.{\mathfrak {c}}){\mathfrak {a}}\times {\mathfrak {b}}=({\mathfrak {a}}\times {\mathfrak {b}}.{\mathfrak {c}}){\mathfrak {r}}$

is also identical, as may easily be verified. If we set

${\mathfrak {c}}={\mathfrak {a}}\times {\mathfrak {b}},$

and suppose that

${\mathfrak {r}}.{\mathfrak {a}}=0,$⁠${\mathfrak {r}}.{\mathfrak {b}}=0,$

the equation becomes

$({\mathfrak {r}}.{\mathfrak {a}}\times {\mathfrak {b}}){\mathfrak {a}}\times {\mathfrak {b}}=({\mathfrak {a}}\times {\mathfrak {b}}.{\mathfrak {a}}\times {\mathfrak {b}}){\mathfrak {r}}.$

This shows that if a bivector ${\mathfrak {r}}$ is perpendicular to two bivectors ${\mathfrak {a}}$ and ${\mathfrak {b}},$ which are not parallel, ${\mathfrak {r}}$ will be parallel to ${\mathfrak {a}}\times {\mathfrak {b}}.$ Therefore all bivectors which are perpendicular to two given bivectors are parallel to each other, unless the given two are parallel.

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[Note by Editors.—The notation $\left\vert \Phi \right\vert \Phi _{\text{C}}^{-1}$ used on page 64, was later improved by the author by the introduction of his Double Multiplication, aooording to which the above expression is represented by $\Phi _{2},$ and $\left\vert \Phi \right\vert$ by $\Phi _{3}.$ See this volume, pages 112, 160, and 181. For an extended treatment of Professor Gibbs's researches on Double Multiplication in their application to Vector Analysis see pp. 906–321, and 333 of "Vector Analysis," by E. B. Wilson, Chas. Scribner's Sons, New York, 1901.]