## The analytical geometry of the conic sections

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1908 Excerpt: ...of intersection of each pair is the same, and in this case the proposition becomes illusory. The proposition is thus seen to be always true, except in certain particular cases when both the conics are pairs of straight lines. 209. Contact of Conics. We will denote the four common points of two conics S and S' by Q, R, T, U. Now it may happen that two or more of these points coincide. Suppose that Q and R coincide while T and U are separate points. The conics are then said to touch at the point Q, or to have 'single contact.' Suppose now that Q and R coincide, as also T and U, but Q and T do not coincide. The conics then touch at two points Q and T and are said therefore to have 'double contact.' Suppose next that Q, R and T coincide but U is a separate point. The conics are then said to have 'three point contact' atQ. 'Three point contact' is sometimes called 'contact of the second order' but it must be most carefully discriminated from 'double contact.' Lastly suppose that Q, R, T and U all coincide, the conics are then said to have 'four point contact,' or as it is sometimes called ' contact of the third order.' When two conics have contact of any order at a point they will have a common tangent line at that point. Conics which have single contact may be looked upon as the limiting case of two conics which cut in four points, two of which are very near together. Such conics are sometimes said to have two ' consecutive points' common. So conics having three point contact may be regarded as the limiting case of two conics which cut in four points, three of which are very near together. Such conics are sometimes said to have three 'consecutive points' common. And in the same way conics with four point contact may be said to have four 'consecutive points' com...

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