58 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



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, C 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 1 = RA . RD . And the circles S and T reduce themselves 

 each to the point N 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, C, D, three, 

 suppose A, B, C, coincide with K ; and the sixteen foci are as follows, viz., 



B, C, A, D are K,K,K,D, 



B v C v A v D x „ K, K, Anti-points of {K, B) , 

 C 2 ,A 2 ,B 2 ,I) i „ Do. do. 



A 3 ,B 3 ,C 3 ,I> 3 „ Do. do. 



viz., we have the point D once, the point K nine times, and the anti-points of 

 K, D three times. But properly the point D is the only focus. The circle 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 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 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 C, coincide with N; the nine foci are A once, iVfour times, and the 

 anti-points of N, 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 /, 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. 



As to the Analytical Theory for the Circular Cubic and the Bicircular Quartic respectively — 



Art. No. 141. 



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 



