Introduction to Quaternions

51. As already stated, most of the properties of the hyperbola are the same as the corresponding properties of the ellipse, and proved by the same process, e being greater than 1. There are, however, some properties both of it and of the parabola which may be conveniently developed by a process more analogous to that of the Cartesian geometry. This process we shall develop presently. In the meantime we proceed to give a brief outline of the application to the parabola of the method employed in the preceding Chapter for the ellipse.

52. If S be the focus of a parabola, DQ the directrix, we have SP = PQ, SA = AD = a.

If SP = ρ, SD = α, we have (EX. 5, Art. 35)

  α²ρ² = (α² - Sαρ)²              (1). 

If

        ρ - α-1Sαρ
  φρ = ————————————               (2),
            α2

  Sρ(φρ + 2α-1) = 1               (3). 

If ρ be another point in the parabola, ρ' - ρ = β, the limit to which β approaches is a vector along the tangent; so that if xβ = π - ρ, π is the vector to a point in the tangent; this gives

  S(π - ρ)(φρ + α-1) = 0          (4);