414] 
ON POLYZOMAL CURVES. 
525 
(viz., the curve is twisted when there is not any axis of symmetry, but the foci lie 
only on circles)—then the classification is 
Circular Cubics, twisted, 
„ „ symmetrical, 
Bicircular Quartics, twisted, 
= Cartesian, 
and each of these kinds may be general, nodal, or cuspidal—viz., for the two last 
mentioned kinds there may be a node or a cusp at a real point of the curve. 
135. In the case of a node, say the point N; first if the curve (circular cubic 
or bicircular quartic) be twisted—then of the four foci A, B, G, D we have two, 
suppose B and G, coinciding with N; and the sixteen foci are as follows, viz. 
B , G , A , D are N, N, A, D; 
B u G 1} A 1} Dj „ N, N, Antipoints of (A, D); 
G 2 , A 2 , B 2 , D 2 „ Antipoints of (N, A), Antipoints of (AT, D); 
A 3 , B 3 , G 3t D 3 „ Do. do. 
do. 
viz., we have the points (A, D) each once, the node N four times, the antipoints of 
(A, D) once, and the antipoints of (N, A) and of (N, D), each pair twice. But 
properly there are only four foci, viz., the points A, D and their antipoints. The 
circle 0 subsists as in the general case, and so does the circle R(BG, AD), viz., this 
has for centre the intersection of the line AD by the tangent at N to the circle 0, 
and it passes through the point N, of course cutting the circle 0 at right angles: 
the circles S and T each reduce themselves each to the point A" considered as an 
evanescent circle, or what is the same thing to the line-pair N.I, NJ. 
136. The case is nearly the same if the curve be symmetrical, but in the case 
of the bicircular quartic excluding the Cartesian: viz., we have on the axis the foci 
B, G coinciding at N, and the other two foci A, D\ the sixteen foci are as above— 
and the circle R is determined by the proper construction as applied to the case in 
hand, viz., the centre R is the intersection of the axis by the radical axis of the 
point N (considered as an evanescent circle) and the circle on AD as diameter; that 
is RN 2 = RA . RD. And the circles S and T reduce themselves each to the point A T 
considered as an evanescent circle. 
137. Next if we have a cusp, say the point K: first if the curve (circular cubic 
or bicircular quartic) be twisted—then of the four foci A, B, G, D, three, suppose 
A, B, G, coincide with K; and the sixteen foci are as follows, viz., 
B , G , A , D are K, K, K, D, 
B 1 , G 3 , A lf A „ K, K, Antipoints of (K, D), 
G 2 , A 2 , B 2 , D, „ Do. do. 
A 3> B 3 , C 3 , D 3 „ Do. do.