QUADRIC SURFACES
§ 91. The conics, the cones of the second order, and the ruled quadric surfaces complete the figures which can be generated by projective rows or flat and axial pencils, that is, by those aggregates of elements which are of one dimension (§§ 5, 6). We shall now consider the simpler figures which are generated by aggregates of two dimensions. The space at our disposal will not, however, allow us to do more than indicate a few of the results.
§ 92. We establish a correspondence between the lines and planes in pencils in space, or reciprocally between the points and lines in two or more planes, but consider principally pencils.
In two pencils we may either make planes correspond to planes and lines to lines, or else planes to lines and. lines to planes. If hereby the condition be satisfied that to a flat, or axial, pencil corresponds in the first case a projective flat, or axial, pencil, and in the second a projective axial, or flat, pencil, the pencils are said to be pro ective in the first case and reciprocal in the second.
For instance, two pencils which join two points Si and Si to the different points and lines in a given plane ir are projective (and in perspective position), if those lines and planes be taken as
to the directrix than to the focus. In a parabola the vertex lies halfway between directrix and focus.
It follows in an ellipse the ratio between the distance of a point from the focus to that from the directrix is less than unity, in the parabola it equals unity, and in the hyperbola it is greater than unity.
It is here the same which focus we take, because the two foci lie symmetrical to the axis of the conic. If now P is any point on the conic having the distances r, and r2 from the foci and the distances di and d2 from the corresponding directrices, then ri/di=r2/d2=e,
where e is constant. Hence alsodi d2 =e.
In the ellipse, which lies between the directrices, dl+d2 is constant, therefore also ri+ri. In the hyperbola on the other hand di–d2 is constant, equal to the distance between the directrices, therefore in this case ri–r2 is constant.
If we call the distances of a point on a conic from the focus its focal distances we have the theorem:
In an ellipse the sum of the focal distances is constant; and in an hyperbola the difference of the fecal distances is constant.
This constant sum or difference equals in both cases the length of the principal axis.
End of Article: QUADRIC 

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