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In this case the centre is a crunode and the curve resembles fig.

00A form which presents itself is when two ovals, one inside the other, unite, so as to give rise to a crunode - in default of a better name this may be called, after the curve of that name, a limacon.

00It 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.

00As 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.

00Secondly, 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.

00Thirdly, 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.

00In this case the centre is a crunode and the curve resembles fig.

00A form which presents itself is when two ovals, one inside the other, unite, so as to give rise to a crunode - in default of a better name this may be called, after the curve of that name, a limacon.

00It 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.

00As 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.

00Secondly, 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.

00Thirdly, 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.

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