Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/223

CHAPTER X.

CONFOCAL QUADRIC SURFACES^[1].

147.] Let the general equation of a confocal system be

${\frac {x^{2}}{\lambda ^{2}-a^{2}}}+{\frac {y^{2}}{\lambda ^{2}-b^{2}}}+{\frac {z^{2}}{\lambda ^{2}-c^{2}}}=1,$

(1)

where $\lambda$ is a variable parameter, which we shall distinguish by the suffix $\lambda _{1}$ for the hyperboloids of two sheets, $\lambda _{2}$ for the hyperboloids of one sheet, and $\lambda _{3}$ for the ellipsoids. The quantities

$a,\ \lambda _{1},\ b,\ \lambda _{2},\ c,\ \lambda _{3}$

are in ascending order of magnitude. The quantity $a$ is introduced for the sake of symmetry, but in our results we shall always suppose $a=0$.

If we consider the three surfaces whose parameters are $\lambda _{1},\ \lambda _{2},\ \lambda _{3}$, we find, by elimination between their equations, that the value of $x^{2}$ at their point of intersection satisfies the equation

$x^{2}\left(b^{2}-a^{2}\right)\left(c^{2}-a^{2}\right)=\left(\lambda _{1}^{2}-a^{2}\right)\left(\lambda _{2}^{2}-a^{2}\right)\left(\lambda _{3}^{2}-a^{2}\right).$

(2)

The values of $y^{2}$ and $z^{2}$ may be found by transposing a, b, c symmetrically.

Differentiating this equation with respect to $\lambda _{1}$, we find

${\frac {dx}{d\lambda _{1}}}={\frac {\lambda _{1}}{\lambda _{1}^{2}-a^{2}}}x.$

(3)

If $ds_{1}$ is the length of the intercept of the curve of intersection of $\lambda _{2}$ and $\lambda _{3}$ cut off between the surfaces $\lambda _{1}$ and $\lambda _{1}+d\lambda _{1}$ then

$\left({\frac {ds_{1}}{d\lambda _{1}}}\right)^{2}=\left({\frac {dx}{d\lambda _{1}}}\right)^{2}+\left({\frac {dy}{d\lambda _{1}}}\right)^{2}+\left({\frac {dz}{d\lambda _{1}}}\right)^{2}={\frac {\lambda _{1}^{2}\left(\lambda _{2}^{2}-\lambda _{1}^{2}\right)\left(\lambda _{3}^{2}-\lambda _{1}^{2}\right)}{\left(\lambda _{1}^{2}-a^{2}\right)\left(\lambda _{1}^{2}-b^{2}\right)\left(\lambda _{1}^{2}-c^{2}\right)}}$

(4)

1. This investigation is chiefly borrowed from a very interesting work, Leçons sur les Fonctions Inverses des Transcendantes et les Surfaces Isothermes. Par G. Lamé. Paris, 1857.