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



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

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

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

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

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