## Cusp Sentence Examples

- As regards the so-called hyperbolisms, observe that (besides the single asymptote) we have in the case of those of the hyperbola two parallel asymptotes; in the case of those of the ellipse the two parallel asymptotes become imaginary, that is, they disappear; and in the case of those of the parabola they become coincident, that is, there is here an ordinary asymptote, and a special asymptote answering to a
**cusp**at infinity. - Similarly a cubic through the two circular points is termed a circular cubic; a quartic through the two points is termed a circular quartic, and if it passes twice through each of them, that is, has each of them for a node, it is termed a bicircular quartic. Such a quartic is of course binodal (m = 4, 6= 2, K = o); it has not in general, but it may have, a third node or a
**cusp**. Or again, we may have a quartic curve having a**cusp**at each of the circular points: such a curve is a " Cartesian," it being a complete definition of the Cartesian to say that it is a bicuspidal quartic curve (m= 4, 6 = o, K= 2), having a**cusp**at each of the circular points. - The curve is symmetrical about the axis of x, and consists of two infinite branches asymptotic to the line BT and forming a
**cusp**at the origin. - Summits of the lower incisors, before they are worn, with a deep transverse groove, dividing it into an anterior and a posterior
**cusp**. Canines long, strong and conical. - On the other hand, in Merycopotamus, of the Lower Pliocene of India and Burma, the upper molars have lost the fifth intermediate
**cusp**of Ancodon; and thus, although highly selenodont, might be easily modified, by a kind of retrograde development, into the trefoil-columned molars of Hippopotamus. - There is symmetry about the initial line and a
**cusp**at the origin. - 9) in a lower horizon a
**cusp**is adumbrated in shadowy form, in a slightly higher horizon it is visible, in a still higher horizon it is full-grown; and we honour this final stage by assigning to the animal which bears it a new specific name. - In the case of a particle oscillating under gravity on a smooth cycloid from rest at the
**cusp**the hotlograph is a circle through the pole, described with constant velocity. - Molars with quadrate crowns and a blunt conical
**cusp**at each corner, the last notably smaller than the rest, sometimes rudimentary or absent. - The double point larities -
**cusp**or spinode; or node; Line-singu5 3. - The
**cusp**: the point as it travels along the line may come to rest, and then reverse the direction of its motion. - 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. - The expression 2 is that of the number of the disposable constants in a curve of the order m with nodes and
(in fact that there shall be a node is I condition, a**cusps****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. - But it can be shown, analytically or geometrically, that if the given curve has a node, the first polar passes through this node, which therefore counts as two intersections, and that if the curve has a
**cusp**, the first polar passes through the**cusp**, touching the curve there, and hence the**cusp**counts as three intersections. - 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, there is a diminution 26+3K, and the value of n is n= m (m - I)-26-3K.**cusps** - But if the given curve has a node, then not only the Hessian passes through the node, but it has there a node the two branches at which touch respectively the two branches of the curve; and the node thus counts as six intersections; so if the curve has a
**cusp**, then the Hessian not only passes through the**cusp**, but it has there a**cusp**through which it again passes, that is, there is a**cuspidal**branch touching the**cuspidal**branch of the curve, and besides a simple branch passing through the**cusp**, and hence the**cusp**counts as eight intersections. - 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, the diminution is = 66+8K, and the number of inflections is c= 3m(m - 2) - 66 - 8K.**cusps** - 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 andis greater than unity.**cusps** - Hence, in the case of a curve which has 6 nodes and K
, 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**cusps****cusp**, and 9 for each pair of.**cusps** - The most simple case is when three double points come into coincidence, thereby giving rise to a triple point; and a somewhat more complicated one is when we have a
**cusp**of the second kind, or node-**cusp**arising from the coincidence of a node, a**cusp**, an inflection, and a double tangent, as shown in the annexed figure, which represents the singularities as on the point of coalescing. - 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 twoor else from three nodes.**cusps** - The branch, whether re-entrant or infinite, may have a
**cusp**or, 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.**cusps** - Secondly, if two of the intersections coincide, say if the line infinity meets the curve in a onefold point and a twofold point, both of them real, then there is always one asymptote: the line infinity may at the twofold point touch the curve, and we have the parabolic hyperbolas; or the twofold point may be a singular point, - viz., a crunode giving the hyperbolisms of the hyperbola; an acnode, giving the hyperbolisms of the ellipse; or a
**cusp**, giving the hyperbolisms of the parabola. - 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. - With a given
**cusp**2.