Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/135

Rh If we adopt such a unit of time that the acceleration due to the sun's action is unity at a unit's distance, and denote the vectors Vectors, or directed quantities, will be represented in this paper by German capitals. The following notations wiU be used in connection with them: The sign $$=$$ denotes identity in direction as well as length. The sign $$+$$ denotes geometrical addition, or what is called composition in mechanics. The sign $$-$$ denotes reversal of direction, or composition after reversal. The notation $$\mathfrak{A}. \mathfrak{B}$$ denotes the product of the lengths of the vectors and the cosine of the angle which they include. It will be called the direct product of $$\mathfrak{A}$$ and $$\mathfrak{B}.$$ If $$x, y, z$$ are the rectangular components of $$\mathfrak{A}$$ and $$x', y', z'$$ those of $$\mathfrak{B},$$ $$\mathfrak{A. A}$$ may be written $$\mathfrak{A}^2$$ and called the square of $$\mathfrak{A}.$$ The notation $$\mathfrak{A}\times \mathfrak{B}$$ will be used to denote a vector of which the length is the product of the lengths of $$\mathfrak{A}$$ and $$\mathfrak{B}$$ and the sine of the angle which they include. Its direction is perpendicular to $$\mathfrak{A}$$ and $$\mathfrak{B}$$ and on that side on which a rotation from $$\mathfrak{A}$$ to $$\mathfrak{B}$$ appears counter-clockwise. It will be called the skew product of $$\mathfrak{A}$$ and $$\mathfrak{B}.$$ If the rectangular components of $$\mathfrak{A}$$ and $$\mathfrak{B}$$ are $$x, y, z,$$ and $$x', y', z',$$ those of $$\mathfrak{A} \times \mathfrak{B}$$ will be The notation $$(\mathfrak{ABC})$$ denotes the volume of the parallelepiped of which three edges are obtained by laying off the vectors $$\mathfrak{A}, \mathfrak{B},$$ and $$\mathfrak{C}$$ from any same point, which volume is to be taken positively or negatively, according as the vector $$\mathfrak{C}$$ falls on the side of the plane containing $$\mathfrak{A}$$ and $$\mathfrak{B},$$ on which a rotation from $$\mathfrak{A}$$ to $$\mathfrak{B}$$ appears counter-clockwise, or on the other side. If the rectangular components of $$\mathfrak{A}, \mathfrak{B},$$ and $$\mathfrak{C}$$ are $$x, y, z; x', y', z';$$ and $$x, y, z'',$$ It follows, from the above definitions, that for any vectors $$\mathfrak{A}, \mathfrak{B},$$ and $$\mathfrak{C}$$   and  also that $$\mathfrak{A}. \mathfrak{B}, \mathfrak{A} \times \mathfrak{B}$$ are distributive functions of $$\mathfrak{A}$$ and $$\mathfrak{B},$$ and $$(\mathfrak{ABC})$$ a distributive function of $$\mathfrak{A}, \mathfrak{B},$$ and $$\mathfrak{C},$$ for example, that if $$\mathfrak{A} = \mathfrak{L} + \mathfrak{M},$$ and so for $$\mathfrak{B}$$ and $$\mathfrak{C}.$$ The notation $$(\mathfrak{ABC})$$ is identical with that of Lagrange in the Mécanique Analytique, except that there its use is limited to unit vectors. The signification of $$\mathfrak{A} \times \mathfrak{B}$$ is closely related to, but not identical with, that of the notation $$[r_{1}r_{2}]$$ commonly used to denote the double area of a triangle determined by two positions in an orbit. drawn from the sun to the body in its three positions by $$\mathfrak{R}_{1}, \mathfrak{R}_{2}, \mathfrak{R}_{3},$$ and the lengths of these vectors (the heliocentric distances) by $$r_{1}, r_{2}, r_{3},$$ the accelerations corresponding to the three positions will be represented by $$-\frac{\mathfrak{R}_{1}}{r_{1}^3}, -\frac{\mathfrak{R}_{2}}{r_{2}^3}, -\frac{\mathfrak{R}_{3}}{r_{3}^3}\cdot$$ Now the motion between the positions considered may be expressed with a high degree of accuracy by an equation of the form having five vector constants. The actual motion rigorously satisfies six conditions, viz., if we write $$\tau_{e}$$ for the interval of time between the