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The books from the eighth to the tenth inclusive are devoted to the inflections of words and their other modifications.

30These slight inflections of the cleavage may be sharp-sided, and may pass into small faults or steps along which dislocation has taken place.

20A 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.

20Though 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.

10The 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.

11All parts of speech, except adverbs, are declined by terminational inflections.

00Dealing next with accent, punctuation marks, sounds and syllables, it goes on to the different parts of speech (eight in number) and their inflections.

00The first, which has been called Oldest Danish, dating from about 1 ioo and 1250, shows a slightly changed character, mainly depending on the system of inflections.

00The 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.

00In 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).

00for the reciprocal curve these letters denote respectively the order, class, number of nodes, cusps, double tangent and inflections.

00Seeking 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.

00Secondly, 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).

00The 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.

00520) is that every singularity whatever may be considered as compounded of ordinary singularities, say we have a singularity =6' nodes, cusps, double tangents and c' inflections.

00We may further consider the inflections and double tangents, as well in general as in regard to cubic and quartic curves.

00The 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.

00It 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.

00For an acnodal cubic the six imaginery inflections disappear, and there remain three real inflections lying in a line.

00For 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.

00For a cuspidal cubic the six imaginary inflections and two of the real inflections disappear, and there remains one real inflection.

00A quartic curve has 24 inflections; it was conjectured by George Salmon, and has been verified by H.

00inflections, 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.

00branch 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.

00It 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.

00A non-singular quartic has only even circuits; it has at most four circuits external to each other, or two circuits one internal to the other, and in this last case the internal circuit has no double tangents or inflections.

00A very remarkable theorem is established as to the double tangents of such a quartic: distinguishing as a double tangent of the first kind a real double tangent which either twice touches the same circuit, or else touches the curve in two imaginary points, the number of the double tangents of the first kind of a non-singular quartic is =4; it follows that the quartic has at most 8 real inflections.

00We 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.

00The reggae rhythm combines well with the vocals, which also have true reggae inflections.

00Plants and Ghosts promises Davies' familiar sensuous, lucid dance yet given new inflections and textural change.

00Boult's timing of the Spanish rhythmic inflections is, perhaps surprisingly given his reputation for English music, near-perfect.

00For instance, the vocal inflections of a singer in Santa Fe modulated the lighting in New York.

00But Beautiful imparts a warm, after-hours feeling with Brooks ' subtle inflections behind the leader's comforting open horn.

00Soulful with jazz inflections, mature definitely not poppy or watered down.

00It includes a survey of grammar, with tables for verb conjugations and noun inflections.

00inflexionicular, it shows several distinctive Northern features, eg. the verbal inflections, which are generally a good indication of dialect localisation.

00All parts of speech, except adverbs, are declined by terminational inflections.

00Dealing next with accent, punctuation marks, sounds and syllables, it goes on to the different parts of speech (eight in number) and their inflections.

00Though 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.

00These slight inflections of the cleavage may be sharp-sided, and may pass into small faults or steps along which dislocation has taken place.

00The first, which has been called Oldest Danish, dating from about 1 ioo and 1250, shows a slightly changed character, mainly depending on the system of inflections.

00In its rudiments it is akin to the HamitoSemitic group. It possesses two grammatical genders, not masculine and feminine, but the human and the non-human; the adjective agrees in assonance with its noun, and euphony plays a great part in verbal and nominal inflections.

00A 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.

00The books from the eighth to the tenth inclusive are devoted to the inflections of words and their other modifications.

00The 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.

00In 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).

00for the reciprocal curve these letters denote respectively the order, class, number of nodes, cusps, double tangent and inflections.

00Seeking 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.

00Secondly, 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).

00The 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.

00520) is that every singularity whatever may be considered as compounded of ordinary singularities, say we have a singularity =6' nodes, cusps, double tangents and c' inflections.

00We may further consider the inflections and double tangents, as well in general as in regard to cubic and quartic curves.

00The 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.

00The 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.

00It 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.

00For an acnodal cubic the six imaginery inflections disappear, and there remain three real inflections lying in a line.

00For 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.

00For a cuspidal cubic the six imaginary inflections and two of the real inflections disappear, and there remains one real inflection.

00A quartic curve has 24 inflections; it was conjectured by George Salmon, and has been verified by H.

00inflections, 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.

00branch 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.

00It 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.

00A non-singular quartic has only even circuits; it has at most four circuits external to each other, or two circuits one internal to the other, and in this last case the internal circuit has no double tangents or inflections.

00A very remarkable theorem is established as to the double tangents of such a quartic: distinguishing as a double tangent of the first kind a real double tangent which either twice touches the same circuit, or else touches the curve in two imaginary points, the number of the double tangents of the first kind of a non-singular quartic is =4; it follows that the quartic has at most 8 real inflections.

00We 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.

00The problem is more complex with verb inflections and in languages other than English.

00Believe it or not, it is really the inflections of our voices that dogs understand.

00Damage to Broca's area results in problems with language fluency: shortened sentences, impaired flow of speech, poor control of rhythm and intonation, and a telegraphic style with missing inflections.

00In the early stages of simultaneous bilingual language development, a child may mix words, parts of words, and inflections from both languages in a single sentence.

00

The word usage examples above have been gathered from various sources to reflect current and historial usage. They do not represent the opinions of YourDictionary.com.