PROFESSOR CAYLEY ON POLYZOMAL CURVES. 75 



that the centre lies on the axis), then the curve has a node at each of the points 

 of contact, viz., the curve breaks up into a line and circle intersecting at the two 

 points of contact, and the circle has its centre at the focus of the parabola. 



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

 that the circle is the circle of maximum curvature, touching the parabola at its 

 vertex), then the curve has a tacnode, viz., it breaks up into a line and circle 

 touching each other and the parabola at the vertex, that is, the line is the tangent 

 to the parabola at its vertex, and the circle is the circle having the focus of the 

 parabola for its centre, and passing through the vertex, or what is the same 

 thing, having its radius = ^ of the semi-latus rectum of the parabola. 



177. If the conic be a circle, then the curve is a bi-circular quartic such that 

 its four nodo-foci coincide together at the centre of the circle ; viz., the curve is a 

 cartesian having the centre of the conic for its cuspo- focus, that is, for the inter- 

 section of the cuspidal tangents of the cartesian. The intersections of the conic 

 with the other circle, or say with the orthotomic circle, are a pair of non-axial 

 foci of the cartesian ; viz., the anti-points of these are two of the axial foci. The 

 third axial focus is the centre of the orthotomic circle. 



Case of Double Contact, Casey s Equation in the Problem of Tactions — Art. No. 178. 



1 78. In the case where the conic has double contact with the orthotomic circle, 

 then (as we have seen) the envelope of the variable circle is a pair of circles, each 

 touching the variable circle ; or, if we start with three given circles and a conic 

 through their centres, then the envelope is a pair of circles, each of them touch- 

 ing each of the three given circles ; that is, we have a solution of the problem of 

 tactions. Multiplying by 2, the equation found ante, No. 173, for the variable 

 circle, and then for the moment representing it by (a, b, c, f, g, h) (u, v,wf — 0; 

 then attributing any signs at pleasure to the radicals %/a, s/\>, s/c, the equation 



< »f a conic through the centres of the given circles, and having double contact 

 with the orthotomic circle, will be 



(a, b, c, f, g, h) (u, v, wf — (u Ja + v J\> + iv Jc) 2 = , 

 viz . representing this equation as before by 



Ivvj + mwu + nuv = , 



we have 



I : m : n = f — Jbc : g — Jca, : h — Jab , 

 that is, substituting for a, b, c, f, g, h their values, and taking, for instance, a, b, c 



= dj% b"J% c"s/2, we find 



I : m : n = (b" - c"f - (b - cf - (b' - c?f 

 : (c" - a") 2 - (c - a) 2 - (c' - a'f 



■ (a" - b"f - (a - bf - {a' - b'f, 



