A Treatise On the Higher Plane Curves; Intended As a Sequel to a Treatise On Conic Sections
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In our eReader you can find the full English version of the book. Read A Treatise On the Higher Plane Curves; Intended As a Sequel to a Treatise On Conic Sections Online - link to read the book on full screen. Our eReader also allows you to upload and read Pdf, Txt, ePub and fb2 books. In the Mini eReder on the page below you can quickly view all pages of the book - Read Book A Treatise On the Higher Plane Curves; Intended As a Sequel to a Treatise On Conic Sections
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249, that if ah, cd be any two of these six pairs of bitangents, the equation of the quartic may be transformed to ahcd= V, the eight points of contact lying on a conic V. Thus we see that the form X^U+2W+ W includes six pairs of the bitangents of the quartic, these twelve bitangents all touching a curve of the third class, viz. the Cayleyan of the system, and the intersections of each pair lying on the Jacobian. So again, if the points of contact of any of these pairs of bitangents be joined d...irectly or transversely, the joining lines also touch the Cayleyan, and the intersection of each pair lies on the Jacobian. This may be stated in a slightly diflferent form by considering the cubic 8, of which U, V, W are polar conies. Then if the equation of a quartic is a function of the second degree in U, V, W, since the vanishing of such a function expresses the condition that the line x U+ y V+ z W= should touch a fixed conic, it is easy to see that the quartic may be defined as the locus of a point whose polar with respect to 8 touches a fixed conic, or, in other words, the locus of the poles with respect to 8 of the tangents of that fixed conic; or, it will come to the same thing if it be defined as the envelope of the polar conies of the points of that conic.
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To quickly assess the difficulty of the text, read a short excerpt:
249, that if ah, cd be any two of these six pairs of bitangents, the equation of the quartic may be transformed to ahcd= V, the eight points of contact lying on a conic V. Thus we see that the form X^U+2W+ W includes six pairs of the bitangents of the quartic, these twelve bitangents all touching a curve of the third class, viz. the Cayleyan of the system, and the intersections of each pair lying on the Jacobian. So again, if the points of contact of any of these pairs of bitangents be joined d...irectly or transversely, the joining lines also touch the Cayleyan, and the intersection of each pair lies on the Jacobian. This may be stated in a slightly diflferent form by considering the cubic 8, of which U, V, W are polar conies. Then if the equation of a quartic is a function of the second degree in U, V, W, since the vanishing of such a function expresses the condition that the line x U+ y V+ z W= should touch a fixed conic, it is easy to see that the quartic may be defined as the locus of a point whose polar with respect to 8 touches a fixed conic, or, in other words, the locus of the poles with respect to 8 of the tangents of that fixed conic; or, it will come to the same thing if it be defined as the envelope of the polar conies of the points of that conic.
What to read after A Treatise On the Higher Plane Curves; Intended As a Sequel to a Treatise On Conic Sections?
You can find similar books in the "Read Also" column, or choose other free books by George Salmon to read onlineMoreLess
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