Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/136

120 first and second positions, and $$\tau_{1}$$ for that between the second and third, and set $$t = 0$$ for the second position,

for $$t = -\tau_{3},$$ for $$t = 0,$$  for $$t = \tau_{1},$$  We may therefore write with a high degree of approximation  From these six equations the five constants $$\mathfrak{A, B, C, D, E}$$ may be eliminated, leaving a single equation of the form  where    This we shall call our fundamental equation. In order to discuss its geometrical signification, let us set so that the equation will read  This expresses that the vector $$n_{2}\mathfrak{R}_{2}$$ is the diagonal of a parallelogram of which $$n_{1}\mathfrak{R}_{1}$$ and $$n_{3}\mathfrak{R}_{3}$$ are sides. If we multiply by $$\mathfrak{R}_{3}$$ and by $$\mathfrak{R}_{1},$$ in skew multiplication, we get whence