526 
ON POLYZOMAL CURVES. 
[414 
viz., Ave have the point D once, the point K nine times, and the antipoints of K, D 
three times. But properly the point D is the only focus. The circle 0 is, it would 
appear, any circle through K, D, but possibly the particular circle which touches the 
cuspidal tangent may be a better representative of the circle 0 of the general case— 
the circles R, S, T reduce themselves each to the point K considered as an evanescent 
point. 
138. The like is the case if the curve be symmetrical, but in the case of the 
bicircular quartic excluding the Cartesian; the circle 0 is here the axis, which is in 
fact the cuspidal tangent. 
139. For the Cartesian, if there is a node N; then of the three foci A, B, C, 
two, suppose B and G, coincide with N; the nine foci are A once, JV four times, and 
the antipoints of JV, A twice: but properly the point A is the only focus. And if 
there be a cusp K; then all the three foci A, B, C coincide with K; and the nine 
foci are K nine times; but in fact there is no proper focus. 
140. A circular cubic cannot have two nodes unless it break up into a line and 
circle; and similarly a bicircular quartic cannot have two nodes (exclusive of course 
of the points I, J) unless it break up into two circles; the last-mentioned case will 
be considered in the sequel in reference to the problem of tactions. 
Article No. 141. As to the Analytical Theory for the Circular Cubic and the Bicircular 
Quartic respectively. 
141. It may be remarked in regard to the analytical theory about to be given, 
that although the investigation is very similar for the circular cubic and for the 
bicircular quartic, yet the former cannot be deduced from the latter case. In fact if 
for the bicircular quartic, using a form somewhat more general than that which is 
ultimately adopted, we suppose that for the two nodes respectively (£ = 0, 2 = 0) and 
(y = 0, 2 = 0), then if f-\-mz = 0, lf + m'z — 0, nr)+pz = 0, nrj-\-p'z=0 are the tangents 
at the two nodes respectively, the equation Avill be 
{If + mz) {If + m'z) (nr) +pz) (nr) +p'z) + ezfrj + 2 3 (a% + bij) + cz i = 0, 
and if (in order to make this equation divisible by 2, and the curve so to break up 
into the line 2=0 and a cubic) Ave write 1 = 0 or n = 0, then the curve Avill indeed 
break up as required, but Ave shall have, not the general cubic through the two points 
(|=0, 2 = 0), (rj = 0, 2 = 0), but in each case a nodal cubic, viz., if ¿ = 0 there Avill be 
a node at the point (77 = 0, 2 = 0), and if n = 0 a node at the point (£ = 0, 2 = 0). 
Article Nos. 142 to 144. Analytical Theory for the Circular Cubic. 
142. I consider then the tAvo cases separately; and first the circular cubic. The 
equation may be taken to be 
fy (pZ + qy) + ez%y 4- 2 2 (ag + brj + cz) = 0,