By Carl Graham

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Our problem is to simplify the general equation of a quadric by transition to an appropriate rectangular coordinate system. We shall find it convenient to write the equation of our surface as follows: + lai^x + 2a24y + 2^342 + Ö44 = 0. z are num bered 1, 2 , 3 and the indices /,y in aij indicate that this co efficient is followed by the ith variable and the jth variable. A coefficient 0 , 4 , 1 = 1, 2 , 3, is followed by the ith variable only. , Oij = αμ. The reason for writing the factor 2 next to the mixed terms is apparent from the identity αιιΛΓ^ -f -f 2023;^^ + 2α24>' + 2 Λ 3 4 Ζ = ( « 1 1 ^ + ai2y + Ö13Z + ai^)x + {a2lX + a22y + ^23^ + a2A)y 022;^^ + « 3 3 ^ ^ + 'i^llXy + 2JI3XZ + 2^14:^ 34.

Then ^ 1 , ^ 2 , A3 have the same sign. Assume for definiteness that > 0, λι > Ο, A3 > 0. If > 0, then it is apparent from (6) that our surface represents an ellipsoid with semiaxes '=V£' ''-^i- If / i = 0, (5) is satisfied by the single real point x" = /' = z" = 0. In this case we say that (5) defines an imaginary cone. An imaginary cone may be regarded as a degenerate ellipsoid [in the sense that (5) with H-Q may be viewed as the limit as / / -»^ 0 of a sequence of ellipsoids]. If Η (5) defines no real points.

After an appropriate rotation, (9) is reduced to the canonical form (10) II. General Theory of Quadric Surfaces 60 56. If we compare the latter equation with Eq. (5), §10, we see that the free term, H, in (5), §10, can be computed directly from (1) without effecting any coordinate transfor mations, namely, -Δ/δ. We established in §10 that for / / = 0, (5) defines a degenerate surface (cone). Hence a central quadric is degenerate if and only ifA = 0. We wish to add (without proof) that Δ = 0 also characterizes degenerate paraboloids (cylinders).