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 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.
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 cusps, there is a diminution 26+3K, and the value of n is n= m (m - I)-26-3K.
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 cusps, the diminution is = 66+8K, and the number of inflections is c= 3m(m - 2) - 66 - 8K.
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.
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.
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 two cusps or else from three nodes.
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.
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.