We live at a defining moment for humanity, as the compounding effects of technology and civilization reach an inflection point.
The Klong is rhythmic, the play being on the inflection of the voice in speaking the words, which inflection is arranged according to fixed schemes; the rhyme, if it can so be called, being sought not in the similarity of syllables but of intonation.
It is usual to distinguish between the general coast-line measured from point to point of the headlands disregarding the smaller bays, and the detailed coast-line which takes account of every inflection shown by the map employed, and follows up river entrances to the point where tidal action ceases.
Xxviii., 1844); in the latter of these the points of inflection are obtained as the intersections of the curve u = o with the Hessian, or curve A = o, where A is the determinant formed with the second derived functions of u.
It will readily be understood how the like considerations apply to other cases, - for instance, if the line is a tangent at an inflection, passes through a crunode, or touches one of the branches of a crunode, &c.; thus, if the line S2 passes through a crunode we have pairs of hyperbolic legs belonging to two parallel asymptotes.
It may be mentioned that the single sheet is a sort of wavy form, having upon it three lines of inflection, and which is met by any plane through the vertex in one or in three lines; the twin-pair sheet has no lines of inflection, and resembles in its form a cone on an oval base.
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 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.
Indic.) is closely parallel to the inflection of the same person in Sanskrit and of quite unique linguistic interest.
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.
Thirdly, the three intersections by the line infinity may be coincident and real; or say we have a threefold point: this may be an inflection, a crunode or a cusp, that is, the line infinity may be a tangent at an inflection, and we have the divergent parabolas; a tangent at a crunode to one branch, and we have the trident curve; or lastly, a tangent at a cusp, and we have the cubical parabola.