Introduction to Quaternions

56. The Ellipsoid. In discussing central surfaces of the second order, we shall speak as if our results were limited to the ellipsoid. That such limitation is not, in most cases, necessarily imposed on us, will be apparent to any one who has a slender acquaintance with ordinary Analytical Geometry. We adopt it in order that our language may have more precision, and that, in some instances, our analysis may have greater simplicity. If the center be made the origin it is clear that the scalar equation can contain no such term as ASαρ, for the definition of a central surface requires that the equation shall be satisfied both by +ρ and by -ρ.

If we turn to the equation of the ellipse (Art. 43), we shall see at once that the equation of the ellipsoid must have the form

  aρ˛ + bS˛αρ + 2cSαρSβρ + ... = 1. 

Now if, as in the Article referred to, we put

  φρ = aρ + bαSαρ + c(αSβρ + βSαρ) + ...

  Sρφρ = aρ˛ + bS˛αρ + 2cSαρSβρ + ... 

       = 1,

It will be seen that, as in Arts. 32, 33, one form of the equation of the straight line was found to coincide exactly with the equation of a plane, so a form of the equation of the ellipse coincides exactly with the equation of the ellipsoid.