74 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



Properties depending on the relation <■ ' and Circle — 



Art. Nob. 175 to 177. 



175. I refer to the conic of the theorem simply as the conic, and to the fixed 

 circle simply as the circle, or when any ambiguity might otherwise arise, then 

 as the orthotomic circle. This being so, I consider the effect in regard to the 

 trizomal curve, of the various special relations which may exist between the 

 circle and the conic. 



If the conic touch the circle, the curve has a node at the point of contact. 



If the conic has with the circle a contact of the second order, the curve has a 

 cusp at the point of contact. 



If the centre of the circle lie on an axis of the conic, then the four intersec- 

 tions lie in pairs symmetrically in regard to this axis, or the curve has this axis 

 as an axis of symmetry. 



If the conic has double contact with the circle (this implies that the centre of 

 the circle is situate on an axis of the conic) the curve has a node at each 

 of the points of contact, viz., it breaks up into two circles intersecting in these 

 two points. The centres of the two circles respectively are the two foci of the 

 conic, which foci lie on the axis in question. Observe that in the general case 

 there are at each of the circular points at infinity two tangents, without any cor- 

 respondence of the tangents of the one pair singly to those of the other pair, and 

 there are thus four intersections, the four foci of the conic ; in the present case, 

 where the curve is a pair of circles, the two tangents to the same circle corre- 

 spond to each other, and intersect in the two foci on the axis in question. The 

 other two foci, or anti-points of these, are each of them the intersection of a 

 tangent of the one circle by a tangent of the other circle 



If the conic has with the circle a contact of the third order (this implies that 

 the circle is a circle of maximum or minimum curvature, at the extremity of an 

 axis of the conic), then the curve has at this point a tacnode, viz., it breaks up into 

 two circles touching each other and the conic at the point in question, and having 

 their centres at the two foci situate on that axis of the conic respectively. 



176. If the conic is a parabola, then the curve is a circular cubic having the 

 four intersections of the parabola and circle for a set of concyclic foci, and having 

 the focus of the parabola for centre. The like particular cases arise, viz., 



If the circle touch the parabola, the curve has a node at the point of contact. 



If the circle has, with the parabola, a contact of the second order, the curve has 

 a cusp at the point of contact. 



If the centre of the circle is situate on the axis of the parabola, then the four 

 intersections are situate in pairs symmetrically in regard to this axis, and the 

 curve has this axis for an axis of symmetry. 



If the circle has double contact with the parabola (which, of course, implies 



