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cusps

cusps Sentence Examples

  • It differs from typical rats of the genus Mus by its broader incisors, and the less distinct cusps on the molars.

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  • It differs from typical rats of the genus Mus by its broader incisors, and the less distinct cusps on the molars.

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  • When the summits of these are worn by mastication their surfaces present circles of dentine surrounded by a border of enamel, and as attrition proceeds different patterns are produced by the union of the bases of the cusps, a trefoil form being characteristic of some species.

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  • presence of from four to five sharp cusps or tubercles on the crown of the molars.

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  • Molars in general characters resembling those of Sarcophilus, but of more simple form, the cusps being less distinct and not so sharply pointed.

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  • First two premolars with compressed and sharp-pointed crowns, and slightly developed anterior and posterior accessory basal cusps.

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  • Molars with numerous sharp-pointed cusps.

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  • In the lower jaw the molars more compressed, with longer cusps; the last not notably smaller than the others.

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  • As a sub-order, the Paucituberculata are characterized by the presence of four pairs of upper and three of lower incisor teeth; the enlargement and forward inclination of the first pair of lower incisors, and the presence of four or five sharp cusps on the cheek-teeth, coupled with the absence of "syndactylism" in the hind limbs.

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  • The molar-like teeth slightly diminishing in size from the first to the fourth, with square crowns, each bearing four pyramidal cusps.

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  • Of the Old World forms, the family Triconodontidae is typified by the genus Triconodon, from the English Purbeck, in which the cheek-teeth carry three cutting cusps arranged longitudinally.

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  • 15), of the Dorsetshire Purbeck; the latter having the three cusps of the cheek-teeth rotated so as to assume a tritubercular type.

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  • the direction of the prevailing wind, and the cusps to leeward.

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  • Zeno; it is trefoilshaped in section, with a tie-beam joining the cusps.

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  • The upper molars, which may be either selenodont or buno-selenodont, carry five cusps each, instead of the four characteristic of all the preceding groups; and they are all very low-crowned, so as to expose the whole of the valleys between the cusps.

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  • The molars are partially selenodont in the typical genus Anthracotherium, with five cusps, or columns, on the crowns of those of the upper jaw, which are nearly square.

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  • In Ancodon (Hyopotamus) the cusps on the molars are taller, so that the dentition is more decidedly selenodont; the distribution of this genus includes not only Europe, Asia and North Africa, but also Egypt where it occurs in Upper Eocene beds in company with the European genus Rhagatherium, which is nearer Anthracotherium.

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  • The essential characteristic of the Cricetines is to be found in the upper cheek-teeth, which (as shown in the figure of those of Cricetus in the article RODENTIA) have their cusps arranged in two longitudinal rows separated by a groove.

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  • This independence of adaptation applies to every detail of structure; the six cusps of a grinding tooth may all evolve alike, or each may evolve independently and differently.

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  • The morainic belts are arranged in groups of concentric loops, convex southward, because the ice sheets advanced in lobes along the lowlands of the Great Lakes; neighboring morainic loops join each other in re-entrants (north-pointing cusps), where two adjacent glacial lobes came together and formed their moraines in largest volume.

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  • The first and second molars have quadrate crowns, with four principal obtuse conical cusps, around which numerous accessory cusps are clustered.

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  • The crown of the third molar is nearly as long as those of the first and second together, having, in addition to the four principal lobes, a large posterior heel, composed of clustered conical cusps, and supported by additional roots.

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  • In the European Miocene we have Hyotherium and Palaeochoerus, and in the Upper Oligocene Propalaeochoerus, which have square molars without any tendency to a selenodont structure in their cusps.

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  • In this the crowns of the molars are more or less shortened, with their cusps either arranged in longitudinal lines, or forming four upper and three lower more or less distinct oblique ridges.

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  • - The mole-rats (Spalacidae) bring us to the mouselike section, or Myoidea, in which there are no premolars and the molars may be occasionally reduced to z; these teeth being either rooted or rootless, with either cusps or enamel-folds, and the first generally larger than the second.

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  • 13, B); in the upper teeth the outer cusps and in the lower the inner ones are the higher, and when worn the crown surfaces show oblique dentineareas; in shape the third molar is like the second, but it is smaller.

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  • the hollow tympanic bullae, they have the clavicles imperfect, the first front toe opposable to the rest, the temporal region of the skull roofed with bone, and the crowns of the molars with cusps arranged in rows but eventually covered by a layer of enamel.

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  • 13, A) are rooted and have a plate-like structure, with the cusps or tubercles forming three longitudinal rows in those of the upper jaw, but only two distinct ones in the lower.

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  • The large-eared African Otomys and the allied Oreomys (Oreinomys), often made the type of a distinct sub-family, may be included in this section; as well as the small African tree-mice, Dendromys, allied to which is Deomys, peculiar in the circumstance that only the first molar has three rows of cusps, the other two having only a couple of such rows, as in cricetines.

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  • In Europe these form the genus Ischyrornys and the family Ischyromyidae, and have premolars i, and all the cheek-teeth low-crowned, with simple cusps or ridges.

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  • ANTHRACOTHERIUM ("coal-animal," so called from the fact of the remains first described having been obtained from the Tertiary lignite-beds of Europe), a genus of extinct artiodactyle ungulate mammals, characterized by having 44 teeth, with five semi-crescentic cusps on the crowns of the upper molars.

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  • The dentition comprises the typical 44 teeth, of which the molars are short-crowned, with four crescentic cusps on those of the upper jaw (selenodont type).

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  • The last premolar and the molars have quadrate crowns, provided with two strong transverse ridges, or with four obtuse cusps.

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  • Since the circumference of a circle is proportional to its radius, it follows that if the ratio of the radii be commensurable, the curve will consist of a finite number of cusps, and ultimately return into itself.

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  • In the particular case when the radii are in the ratio of I to 3 the epicycloid (curve a) will consist of three cusps external to the circle and placed at equal distances along its circumference.

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  • The hypocycloid derived from the same circles is shown as curve d, and is seen to consist of three cusps arranged internally to the fixed circle; the corresponding hypotrochoid consists of a three-foil and is shown in curve e.

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  • The method by which the cycloid is generated shows that it consists of an infinite number of cusps placed along the fixed line and separated by a constant distance equal to the circumference of the rolling circle.

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  • It may be remarked that in Poncelet's memoir on reciprocal polars, above referred to, we have the theorem that the number of tangents from a point to a curve of the order m, or say the class of the curve, is in general and at most = m(m - 1), and that he mentions that this number is subject to reduction when the curve has double points or cusps.

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  • 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).

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

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  • The expression 2 is that of the number of the disposable constants in a curve of the order m with nodes and cusps (in fact that there shall be a node is I condition, a cusp 2 conditions) and the equation (9) thus expresses that the curve and its reciprocal contain each of them the same number of disposable constants.

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  • With regard to the demonstration of Pliicker's equations it is to be remarked that we are not able to write down the equation in point-co-ordinates of a curve of the order m, having the given numbers 6 and of nodes and cusps.

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  • We can only use the general equation (*fix, y, z) m = o, say for shortness u= o, of a curve of the mth order, which equation, so long as the coefficients remain arbitrary, represents a curve without nodes or cusps.

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  • But, as is evident, the node or cusp is not a point of contact of a proper tangent from the arbitrary point; we have, therefore, for a node a diminution and for a cusp a diminution 3, in the number of the intersections; and thus, for a curve with 6 nodes and K cusps, there is a diminution 26+3K, and the value of n is n= m (m - I)-26-3K.

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  • 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.

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  • Thirdly, for the double tangents; the points of contact of these are obtained as the intersections of the curve by a curve II = o, which has not as yet been geometrically defined, but which is found analytically to be of the order (m-2) (m 2 -9); the number of intersections is thus = m(rn - 2) (m 2 - 9); but if the given curve has a node then there is a diminution =4(m2 - m-6), and if it has a cusp then there is a diminution =6(m2 - m-6), where, however, it is to be noticed that the factor (m2 - m-6) is in the case of a curve having only a node or only a cusp the number of the tangents which can be drawn from the node or cusp to the curve, and is used as denoting the number of these tangents, and ceases to be the correct expression if the number of nodes and cusps is greater than unity.

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  • Hence, in the case of a curve which has 6 nodes and K cusps, the apparent diminution 2(m 2 - m-6)(26+3K) is too great, and it has in fact to be diminished by 2 1(25(5 - I) +66K+ 2 K(K - I)1, or the half thereof is 4 for each pair of nodes, 6 for each combination of a node and cusp, and 9 for each pair of cusps.

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  • 520) 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.

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  • By means of Pliicker's equations we may form a table - The table is arranged according to the value of in; and we have m=o, n= r, the point; m =1, n =o, the line; m=2, n=2, the conic; of m = 3, the cubic, there are three cases, the class being 6, 4 or 3, according as the curve is without singularities, or as it has 1 node or r cusp; and so of m =4, the quartic, there are ten cases, where observe that in two of them the class is = 6, - the reduction of class arising from two cusps or else from three nodes.

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  • The ten cases may be also grouped together into four, according as the number of nodes and cusps (5+ic) is = o, r, 2 or 3.

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  • Suppose that in general the variable curve is of the order m with S nodes and cusps, and therefore of the class n with double tangents and E.

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  • 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.

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  • Considering the variable curve corresponding to a given value of the parameter, or say simply the variable curve, the consecutive curve has then also 6 and nodes and cusps, consecutive to those of the variable curve; and it is easy to see that among the intersections of the two curves we have the nodes each counting twice, and the cusps each counting three times; the number of the remaining intersections is = m 2 - 263 K.

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  • The branch, whether re-entrant or infinite, may have a cusp or cusps, or it may cut itself or another branch, thus having or giving rise to crunodes or double points with distinct real tangents; an acnode, or double point with imaginary tangents, is a branch by itself, - it may be considered as an indefinitely small re-entrant branch.

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  • As mentioned with regard to a branch generally, an infinite branch of any kind may have cusps, or, by cutting itself or another branch, may have or give rise to a crunode, &c.

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  • It is an old and easily proved theorem that, for a curve of the order m, the number 6+K of nodes and cusps is at most = Ernr) (m2); for a given curve the deficiency of the actual number of nodes and cusps below this maximum number, viz.

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  • 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.

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  • Imagine a curve of order m, deficiency D, and let the corresponding points P, P' be such that the line joining them passes through a given point 0; this is an (m - m-1) correspondence, and the value of k is=1, hence the number of united points is =2m-2+2D; the united points are the points of contact of the tangents from 0 and (as special solutions) the cusps, and we have thus the relation or, writing D=2(m - i)(m-2) - S - K, this is n=m(m - i)-23-3K, which is right.

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  • Thus assuming that the penultimate curve is without nodes or cusps, the number of the tangents to it is=m2 - m, (alms+a2m2+

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  • Taking P 1 =o to have 3 1 nodes and K 1 cusps, and therefore its class n 1 to be=m 1 2 - m 1 25 1 -3K,, the expression for the number of tangents to the penultimate curve is = (a1 2 - a,) m1 2 + (a2 2 - a2)m2 2 +

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  • limit the proper tangents to P i =o each a l times, the lines to its nodes each 2a 1 times, and the lines to its cusps each 3a 1 times; the remaining terms (a1 2 - al)m12+ (a22 - a2)m22 + ...

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  • In the insectivorous type, as exemplified in moles and shrew-mice, the middle pair of incisors in each jaw are long and pointed so as to have a forceps-like action for seizing insects, the hard coats of which are broken up by the numerous sharp cusps surmounting the cheek-teeth.

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  • A more generally accepted view - especially among palaeontologists - is the tritubercular theory, according to which the most generalized type of tooth consists of three cusps arranged in a triangle, with the apex pointing inwards in the teeth of the upper jaw.

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  • Additions of extra cusps form teeth of a more complicated type.

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  • Each cusp of the primitive triangle has received a separate name, both in the teeth of the upper and of the lower jaw, while names have also been assigned to super-added cusps.

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  • ascendant changes, the sign on the cusps of the other eleven houses may change, also.

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  • Transits and progressions, too, are reported when crossing cusps.

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  • tooth jaws bore small chisel like incisors, small canines, and low-crowned cheek teeth with rounded conical cusps.

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  • In other species the summits of the ridges are divided into conical cusps, and may have accessory cusps clustering around them (as in M.

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  • When the summits of these are worn by mastication their surfaces present circles of dentine surrounded by a border of enamel, and as attrition proceeds different patterns are produced by the union of the bases of the cusps, a trefoil form being characteristic of some species.

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  • presence of from four to five sharp cusps or tubercles on the crown of the molars.

    0
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  • Molars in general characters resembling those of Sarcophilus, but of more simple form, the cusps being less distinct and not so sharply pointed.

    0
    0
  • First two premolars with compressed and sharp-pointed crowns, and slightly developed anterior and posterior accessory basal cusps.

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  • Molars with numerous sharp-pointed cusps.

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  • In the lower jaw the molars more compressed, with longer cusps; the last not notably smaller than the others.

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  • As a sub-order, the Paucituberculata are characterized by the presence of four pairs of upper and three of lower incisor teeth; the enlargement and forward inclination of the first pair of lower incisors, and the presence of four or five sharp cusps on the cheek-teeth, coupled with the absence of "syndactylism" in the hind limbs.

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    0
  • The molar-like teeth slightly diminishing in size from the first to the fourth, with square crowns, each bearing four pyramidal cusps.

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    0
  • Of the Old World forms, the family Triconodontidae is typified by the genus Triconodon, from the English Purbeck, in which the cheek-teeth carry three cutting cusps arranged longitudinally.

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  • 15), of the Dorsetshire Purbeck; the latter having the three cusps of the cheek-teeth rotated so as to assume a tritubercular type.

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  • the direction of the prevailing wind, and the cusps to leeward.

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  • Zeno; it is trefoilshaped in section, with a tie-beam joining the cusps.

    0
    0
  • The upper molars, which may be either selenodont or buno-selenodont, carry five cusps each, instead of the four characteristic of all the preceding groups; and they are all very low-crowned, so as to expose the whole of the valleys between the cusps.

    0
    0
  • The molars are partially selenodont in the typical genus Anthracotherium, with five cusps, or columns, on the crowns of those of the upper jaw, which are nearly square.

    0
    0
  • In Ancodon (Hyopotamus) the cusps on the molars are taller, so that the dentition is more decidedly selenodont; the distribution of this genus includes not only Europe, Asia and North Africa, but also Egypt where it occurs in Upper Eocene beds in company with the European genus Rhagatherium, which is nearer Anthracotherium.

    0
    0
  • The essential characteristic of the Cricetines is to be found in the upper cheek-teeth, which (as shown in the figure of those of Cricetus in the article RODENTIA) have their cusps arranged in two longitudinal rows separated by a groove.

    0
    0
  • This independence of adaptation applies to every detail of structure; the six cusps of a grinding tooth may all evolve alike, or each may evolve independently and differently.

    0
    0
  • The morainic belts are arranged in groups of concentric loops, convex southward, because the ice sheets advanced in lobes along the lowlands of the Great Lakes; neighboring morainic loops join each other in re-entrants (north-pointing cusps), where two adjacent glacial lobes came together and formed their moraines in largest volume.

    0
    0
  • The first and second molars have quadrate crowns, with four principal obtuse conical cusps, around which numerous accessory cusps are clustered.

    0
    0
  • The crown of the third molar is nearly as long as those of the first and second together, having, in addition to the four principal lobes, a large posterior heel, composed of clustered conical cusps, and supported by additional roots.

    0
    0
  • In the European Miocene we have Hyotherium and Palaeochoerus, and in the Upper Oligocene Propalaeochoerus, which have square molars without any tendency to a selenodont structure in their cusps.

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  • In this the crowns of the molars are more or less shortened, with their cusps either arranged in longitudinal lines, or forming four upper and three lower more or less distinct oblique ridges.

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  • - The mole-rats (Spalacidae) bring us to the mouselike section, or Myoidea, in which there are no premolars and the molars may be occasionally reduced to z; these teeth being either rooted or rootless, with either cusps or enamel-folds, and the first generally larger than the second.

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  • The first group is that of the hamsters, or cricetines (Cricetinae), in which the molars are rooted and tuberculated, with the cusps of the upper ones arranged in two longitudinal rows (fig.

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  • 13, B); in the upper teeth the outer cusps and in the lower the inner ones are the higher, and when worn the crown surfaces show oblique dentineareas; in shape the third molar is like the second, but it is smaller.

    0
    0
  • the hollow tympanic bullae, they have the clavicles imperfect, the first front toe opposable to the rest, the temporal region of the skull roofed with bone, and the crowns of the molars with cusps arranged in rows but eventually covered by a layer of enamel.

    0
    0
  • 13, A) are rooted and have a plate-like structure, with the cusps or tubercles forming three longitudinal rows in those of the upper jaw, but only two distinct ones in the lower.

    0
    0
  • The large-eared African Otomys and the allied Oreomys (Oreinomys), often made the type of a distinct sub-family, may be included in this section; as well as the small African tree-mice, Dendromys, allied to which is Deomys, peculiar in the circumstance that only the first molar has three rows of cusps, the other two having only a couple of such rows, as in cricetines.

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  • In Europe these form the genus Ischyrornys and the family Ischyromyidae, and have premolars i, and all the cheek-teeth low-crowned, with simple cusps or ridges.

    0
    0
  • ANTHRACOTHERIUM ("coal-animal," so called from the fact of the remains first described having been obtained from the Tertiary lignite-beds of Europe), a genus of extinct artiodactyle ungulate mammals, characterized by having 44 teeth, with five semi-crescentic cusps on the crowns of the upper molars.

    0
    0
  • The dentition comprises the typical 44 teeth, of which the molars are short-crowned, with four crescentic cusps on those of the upper jaw (selenodont type).

    0
    0
  • The last premolar and the molars have quadrate crowns, provided with two strong transverse ridges, or with four obtuse cusps.

    0
    0
  • Since the circumference of a circle is proportional to its radius, it follows that if the ratio of the radii be commensurable, the curve will consist of a finite number of cusps, and ultimately return into itself.

    0
    0
  • In the particular case when the radii are in the ratio of I to 3 the epicycloid (curve a) will consist of three cusps external to the circle and placed at equal distances along its circumference.

    0
    0
  • The hypocycloid derived from the same circles is shown as curve d, and is seen to consist of three cusps arranged internally to the fixed circle; the corresponding hypotrochoid consists of a three-foil and is shown in curve e.

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  • The method by which the cycloid is generated shows that it consists of an infinite number of cusps placed along the fixed line and separated by a constant distance equal to the circumference of the rolling circle.

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  • The name cycloid is usually restricted to the portion between two consecutive cusps (fig.

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  • It may be remarked that in Poncelet's memoir on reciprocal polars, above referred to, we have the theorem that the number of tangents from a point to a curve of the order m, or say the class of the curve, is in general and at most = m(m - 1), and that he mentions that this number is subject to reduction when the curve has double points or cusps.

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  • 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).

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

    0
    0
  • The expression 2 is that of the number of the disposable constants in a curve of the order m with nodes and cusps (in fact that there shall be a node is I condition, a cusp 2 conditions) and the equation (9) thus expresses that the curve and its reciprocal contain each of them the same number of disposable constants.

    0
    0
  • With regard to the demonstration of Pliicker's equations it is to be remarked that we are not able to write down the equation in point-co-ordinates of a curve of the order m, having the given numbers 6 and of nodes and cusps.

    0
    0
  • We can only use the general equation (*fix, y, z) m = o, say for shortness u= o, of a curve of the mth order, which equation, so long as the coefficients remain arbitrary, represents a curve without nodes or cusps.

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    0
  • But, as is evident, the node or cusp is not a point of contact of a proper tangent from the arbitrary point; we have, therefore, for a node a diminution and for a cusp a diminution 3, in the number of the intersections; and thus, for a curve with 6 nodes and K cusps, there is a diminution 26+3K, and the value of n is n= m (m - I)-26-3K.

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  • 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.

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    0
  • Thirdly, for the double tangents; the points of contact of these are obtained as the intersections of the curve by a curve II = o, which has not as yet been geometrically defined, but which is found analytically to be of the order (m-2) (m 2 -9); the number of intersections is thus = m(rn - 2) (m 2 - 9); but if the given curve has a node then there is a diminution =4(m2 - m-6), and if it has a cusp then there is a diminution =6(m2 - m-6), where, however, it is to be noticed that the factor (m2 - m-6) is in the case of a curve having only a node or only a cusp the number of the tangents which can be drawn from the node or cusp to the curve, and is used as denoting the number of these tangents, and ceases to be the correct expression if the number of nodes and cusps is greater than unity.

    0
    0
  • Hence, in the case of a curve which has 6 nodes and K cusps, the apparent diminution 2(m 2 - m-6)(26+3K) is too great, and it has in fact to be diminished by 2 1(25(5 - I) +66K+ 2 K(K - I)1, or the half thereof is 4 for each pair of nodes, 6 for each combination of a node and cusp, and 9 for each pair of cusps.

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  • 520) 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.

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  • By means of Pliicker's equations we may form a table - The table is arranged according to the value of in; and we have m=o, n= r, the point; m =1, n =o, the line; m=2, n=2, the conic; of m = 3, the cubic, there are three cases, the class being 6, 4 or 3, according as the curve is without singularities, or as it has 1 node or r cusp; and so of m =4, the quartic, there are ten cases, where observe that in two of them the class is = 6, - the reduction of class arising from two cusps or else from three nodes.

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  • The ten cases may be also grouped together into four, according as the number of nodes and cusps (5+ic) is = o, r, 2 or 3.

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  • Suppose that in general the variable curve is of the order m with S nodes and cusps, and therefore of the class n with double tangents and E.

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  • 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.

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  • Considering the variable curve corresponding to a given value of the parameter, or say simply the variable curve, the consecutive curve has then also 6 and nodes and cusps, consecutive to those of the variable curve; and it is easy to see that among the intersections of the two curves we have the nodes each counting twice, and the cusps each counting three times; the number of the remaining intersections is = m 2 - 263 K.

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  • The branch, whether re-entrant or infinite, may have a cusp or cusps, or it may cut itself or another branch, thus having or giving rise to crunodes or double points with distinct real tangents; an acnode, or double point with imaginary tangents, is a branch by itself, - it may be considered as an indefinitely small re-entrant branch.

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  • As mentioned with regard to a branch generally, an infinite branch of any kind may have cusps, or, by cutting itself or another branch, may have or give rise to a crunode, &c.

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  • It is an old and easily proved theorem that, for a curve of the order m, the number 6+K of nodes and cusps is at most = Ernr) (m2); for a given curve the deficiency of the actual number of nodes and cusps below this maximum number, viz.

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  • 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.

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  • Imagine a curve of order m, deficiency D, and let the corresponding points P, P' be such that the line joining them passes through a given point 0; this is an (m - m-1) correspondence, and the value of k is=1, hence the number of united points is =2m-2+2D; the united points are the points of contact of the tangents from 0 and (as special solutions) the cusps, and we have thus the relation or, writing D=2(m - i)(m-2) - S - K, this is n=m(m - i)-23-3K, which is right.

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  • Thus assuming that the penultimate curve is without nodes or cusps, the number of the tangents to it is=m2 - m, (alms+a2m2+

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  • Taking P 1 =o to have 3 1 nodes and K 1 cusps, and therefore its class n 1 to be=m 1 2 - m 1 25 1 -3K,, the expression for the number of tangents to the penultimate curve is = (a1 2 - a,) m1 2 + (a2 2 - a2)m2 2 +

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  • limit the proper tangents to P i =o each a l times, the lines to its nodes each 2a 1 times, and the lines to its cusps each 3a 1 times; the remaining terms (a1 2 - al)m12+ (a22 - a2)m22 + ...

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  • In the insectivorous type, as exemplified in moles and shrew-mice, the middle pair of incisors in each jaw are long and pointed so as to have a forceps-like action for seizing insects, the hard coats of which are broken up by the numerous sharp cusps surmounting the cheek-teeth.

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  • A more generally accepted view - especially among palaeontologists - is the tritubercular theory, according to which the most generalized type of tooth consists of three cusps arranged in a triangle, with the apex pointing inwards in the teeth of the upper jaw.

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  • Additions of extra cusps form teeth of a more complicated type.

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  • Each cusp of the primitive triangle has received a separate name, both in the teeth of the upper and of the lower jaw, while names have also been assigned to super-added cusps.

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  • The jaws bore small chisel like incisors, small canines, and low-crowned cheek teeth with rounded conical cusps.

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  • By six weeks of gestation the tips or cusps of the primary teeth appear.

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  • This idea can work especially well if you are on one of the astrological cusps of Virgo.

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