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

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^[1] 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

${\mathfrak {R}}={\mathfrak {A}}+t{\mathfrak {B}}+t^{2}{\mathfrak {C}}+t^{3}{\mathfrak {D}}+t^{4}{\mathfrak {E}},$

having five vector constants. The actual motion rigorously satisfies six conditions, viz., if we write $\tau _{e}$ for the interval of time between the

1. 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 ${\mathfrak {B}},$

   ${\mathfrak {A}}.{\mathfrak {B}}=xx'+yy'+zz'.$

   ${\mathfrak {A.A}}$ 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

   $yz'-zy',$⁠$zx'-xz',$⁠$xy'-yx'.$

   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 $x'',y'',z'',$

   $({\mathfrak {ABC}})={\begin{vmatrix}x&y&z\\x'&y'&z'\\x''&y''&z''\\\end{vmatrix}}.$

   It follows, from the above definitions, that for any vectors ${\mathfrak {A}},{\mathfrak {B}},$ and ${\mathfrak {C}}$

   ${\mathfrak {A}}.{\mathfrak {B}}={\mathfrak {B}}.{\mathfrak {A}},$⁠${\mathfrak {A}}\times {\mathfrak {B}}=-{\mathfrak {B}}\times {\mathfrak {A}},$

   $({\mathfrak {ABC}})=({\mathfrak {BCA}})=({\mathfrak {CAB}})=-({\mathfrak {ACB}})=-({\mathfrak {CBA}})=-({\mathfrak {BAC}}),$

   and

   $({\mathfrak {ABC}})={\mathfrak {A}}.({\mathfrak {B}}\times {\mathfrak {C}})={\mathfrak {B}}.({\mathfrak {C}}\times {\mathfrak {A}})={\mathfrak {C}}.({\mathfrak {A}}\times {\mathfrak {B}});$

   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 ${\mathfrak {A}}={\mathfrak {L}}+{\mathfrak {M}},$

   ${\mathfrak {A}}.{\mathfrak {B}}={\mathfrak {L}}.{\mathfrak {B}}+{\mathfrak {M}}.{\mathfrak {B}},$⁠${\mathfrak {A}}\times {\mathfrak {B}}={\mathfrak {L}}\times {\mathfrak {B}}+{\mathfrak {M}}\times {\mathfrak {B}},$⁠$({\mathfrak {ABC}})=({\mathfrak {LBC}})+({\mathfrak {MBC}}),$

   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.