Analytical Geometry of Three Dimensions

William H. McCrea

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486154886
• Topic: Mathematics
• Subtopic: Analytic Geometry
• Index: Mathematics

Brief but rigorous, this text is geared toward advanced undergraduates and graduate students. It covers the coordinate system, planes and lines, spheres, homogeneous coordinates, general equations of the second degree, quadric in Cartesian coordinates, and intersection of quadrics.
Mathematician, physicist, and astronomer, William H. McCrea conducted research in many areas and is best known for his work on relativity and cosmology. McCrea studied and taught at universities around the world, and this book is based on a series of his lectures.

CHAPTER VI

QUADRIC IN CARTESIAN COORDINATES; STANDARD FORMS

39. Algebraic Lemmas

The results of this section, in the form required, are not readily accessible in textbooks of elementary algebra; their derivation here may therefore assist the student.

I. Discriminating cubic.* Let us consider the matrices

where a, b, c, ƒ, g, h are given real numbers, and write

This is in accord with the practice we shall adopt of denoting the cofactors of a, h, …, d in Δ by A, H, …, D, where

We shall denote the cofactors of a, …, h in D by

and the cofactors of α − λ, …,h in D_λ by

In the application required, the equation D_λ = 0, which is in fact ^†

will be called the discriminating cubic of the quadratic form having matrix D.

Suppose first that f, g, h ≠ 0. Let λ = γ ₁, γ₂ be the roots of

Then γ₁, γ ₂ are real, and γ₁ < a, b < γ₂.

By Jacobi’s theorem, or by direct verification,

Also

and ƒ ≠ 0; therefore

vanishes for one and only one value of λ.

Case (i).

. Put λ = γ₁ in (4); since a− γ₁ > 0,

we obtain D_γ1 < 0. Analogously, D_γ3 > 0. Therefore, when λ = −∞, γ₁, γ₂, ∞, the sign of D_λ 0. Therefore, when λ = −∞, γ₁, γ₂, ∞, the sign of D_λ is +, −, +, −. Thus (2) has three real and distinct roots λ₁, λ₂, λ₃ such that λ₁ < γ₁ < λ₃ < γ₃ < λ₃.

Case (ii).

Put λ = γ₁ in (4); since α−γ₁ > 0,

we obtain D_γ1 = 0. As in (i), D_y3 > 0. Therefore D_λ has the same sign when λ = −∞, γ₂, and has one zero γ₁ between these values; consequently it has a second zero between them. As in (i), there is a third zero between λ = γ₂, ∞. Thus (2) has three real roots, λ₁, λ₂, λ₃, two at least being distinct, such that λ₁ = γ₁ < γ₂; λ₂ < γ₂ < λ₃.

Now suppose (2) has a double root; from what has just been proved, this is necessarily λ₁(= γ₁) Then in (4) λ₁ is a double zero of D_λ and of

and so is a double zero of

Therefore, since λ₁( = γ₁) is only a simple zero of

it must be a zero also of

Using these properties in further relations like (4), we can show that λ₁, is a zero also of

Thus a double zero of D_λ is a zero of every cofactor of D_λ.

It is ea...