# Inflections Sentence Examples

- These slight
**inflections**of the cleavage may be sharp-sided, and may pass into small faults or steps along which dislocation has taken place. - Though many syllables have to do duty for the expression of more than one idea, the majority have only one or at most two meanings, but there are some which are used with quite a number of different
**inflections**, each of which gives the word a new meaning. - A collection of the various signs of the alphabet has shown thirty-two letters, four more than Arabic. De Slane, in his notes on the Berber historian Ibn Khaldun, shows the following points of similarity to the Semitic class: - its tri-literal roots, the
**inflections**of the verb, the formation of derived verbs, the genders of the second and Arab districts to build mills for the Arabs. - The books from the eighth to the tenth inclusive are devoted to the
**inflections**of words and their other modifications. - The curve (1 x, y, z) m = o, or general curve of the order m, has double tangents and
**inflections**; (2) presents itself as a singularity, for the equations dx(* x, y, z) m =o, d y (*r x, y, z)m=o, d z(* x, y, z) m =o, implying y, z) m = o, are not in general satisfied by any values (a, b, c) whatever of (x, y, z), but if such values exist, then the point (a, b, c) is a node or double point; and (I) presents itself as a further singularity or sub-case of (2), a cusp being a double point for which the two tangents becomes coincident. - In regard to the ordinary singularities, we have m, the order, n „ class, „ number of double points, Cusps, T double tangents,
**inflections**; and this being so, Pliicker's ” six equations ” are n = m (m - I) -2S -3K, = 3m (m - 2) - 6S- 8K, T=Zm(m -2) (m29) - (m2 - m-6) (28-i-3K)- I -25(5-1) +65K-1114 I), m =n(n - I)-2T-3c, K= 3n (n-2) - 6r -8c, = 2n(n-2)(n29) - (n2 - n-6) (2T-{-30-1-2T(T - I) -1-6Tc -}2c (c - I). - Seeking then, for this curve, the values, n, e, of the class, number of
**inflections**, and number of double tangents, - first, as regards the class, this is equal to the number of tangents which can be drawn to the curve from an arbitrary point, or what is the same thing, it is equal to the number of the points of contact of these tangents. - Secondly, as to the
**inflections**, the process is a similar one; it can be shown that the**inflections**are the intersections of the curve by a derivative curve called (after Ludwig Otto Hesse who first considered it) the Hessian, defined geometrically as the locus of a point such that its conic polar (§ 8 below) in regard to the curve breaks up into a pair of lines, and which has an equation H = o, where H is the determinant formed with the second differential coefficients of u in regard to the variables (x, y, z); H= o is thus a curve of the order 3 (m - 2), and the number of**inflections**is =3m(m-2). - The node or cusp is not an inflection, and we have thus for a node a diminution 6, and for a cusp a diminution 8, in the number of the intersections; hence for a curve with 6 nodes and cusps, the diminution is = 66+8K, and the number of
**inflections**is c= 3m(m - 2) - 66 - 8K. - We may further consider the
**inflections**and double tangents, as well in general as in regard to cubic and quartic curves. - The expression for the number of
**inflections**3m(rn - 2) for a curve of the order m was obtained analytically by Plucker, but the theory was first given in a complete form by Hesse in the two papers " Uber die Elimination, u.s.w.," and " Uber die Wendepuncte der Curven dritter Ordnung " (Crelle, t. - The whole theory of the
**inflections**of a cubic curve is discussed in a very interesting manner by means of the canonical form of the equation x +y +z +6lxyz= o; and in particular a proof is given of Plucker's theorem that the nine points of inflection of a cubic curve lie by threes in twelve lines. - It may be noticed that the nine
**inflections**of a cubic curve represented by an equation with real coefficients are three real, six imaginary; the three real**inflections**lie in a line, as was known to Newton and Maclaurin. - For an acnodal cubic the six imaginery
**inflections**disappear, and there remain three real**inflections**lying in a line. - For a crunodal cubic the six
**inflections**which disappear are two of them real, the other four imaginary, and there remain two imaginary**inflections**and one real inflection. - For a cuspidal cubic the six imaginary
**inflections**and two of the real**inflections**disappear, and there remains one real inflection. - A quartic curve has 24
**inflections**; it was conjectured by George Salmon, and has been verified by H. **Inflections**, in, n, 0, being connected by the Pluckerian equations, - the number of nodes or cusps may be greater for particular values of the parameter, but this is a speciality which may be here disregarded.- Branch may have
**inflections**and double tangents, or there may be double tangents which touch two distinct branches; there are also double tangents with imaginary points of contact, which are thus lines having no visible connexion with the curve. - It may be added that there are on the odd circuit three
**inflections**, but on the even circuit no inflection; it hence also appears that from any point of the odd circuit there can be drawn to the odd circuit two tangents, and to the even circuit (if any) two tangents, but that from a point of the even circuit there cannot be drawn (either to the odd or the even circuit) any real tangent; consequently, in a simplex curve the number of tangents from any point is two; but in a complex curve the number is four, or none, - f our if the point is on the odd circuit, none if it is on the even circuit. - We can by means of it investigate the class of a curve, number of
**inflections**, &c. - in fact, Pliicker's equations; but it is necessary to take account of special solutions: thus, in one of the most simple instances, in finding the class of a curve, the cusps present themselves as special solutions.