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- In this case the centre is a
**crunode**and the curve resembles fig. - A 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. - It 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. - 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. - 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.

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