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This theorem shows that the algebraic nature of the equation of a sur face and the degree of that equation are characteristics of the surface and do not depend on the choice of rectangular coordinate system. 31. The above theorem implies that the curve of intersection of a quadric with a plane is an algebraic curve whose order does not exceed two. Yrwif Let 5 be a quadric and let α be a plane. Select a rectangular coordinate system with axes Ox and Oy in a. According to para. 30, 5 is described in our rectangular coordinate system (as well as in any other rectangular coordinate system) by an equation of the second degree: Ax^ + jBy2 + Cz2 + IDxy + lExz + 2Fvz + 2Gjc + 2H>^ + 2 A z 4 - L = 0.
Lemma 2. If in addition to the assumption of Lemma 1 we have ^11 + ^ 2 + ^33=0 (24) then all the ba in (21) vanish. Proof, In view of Lemma 1 the rows of the determinant are propor tional to each other so that all its 2 χ 2 subdeterminants vanish. In par ticular, bn bii bi2 b22 = 0, bn ^31 ^33 = 0, 022 ^23 = 0. ^32 ^33 Hence ^11^22 = ^12» ^11^33=^13» ^22^33 = ^ 2 3 · (25) But then 622 > 0, bn ^33 > 0, ^22^33 > 0, which means that ^ n , ^22, ^33 cannot have different signs. But then, in view of (24), we must have ^11 = bii = ^33 = 0.
The characteristic equation of Φ is - 18λ2 + 99λ - 162 = 0. The number of changes of sign in Η h •- is three. Hence all the 10. Reduction to Canonical Form, General Equation 51 characteristic numbers of the form are positive. The form is elliptic and positive definite. We note that in this case the roots are easy to find, namely, λι = 3, λ 2 = 6, λ 3 = 9. Hence, as M(JC, z) varies over the unit sphere x^-\-y'^-\Γ2 = 1, 3 < 7χ2 + 6^2 4- 5^2 - 4xy - Ay ζ < 9. Example. Of what type is the quadratic form Φ = χ2 2>'2 + ^2 + Axy-%xz-Ayz, Solution.