56 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



coincides with one or other of the nodal axes, viz., with that passing through the 

 real or the imaginary nodo-foci ; that is, the curve may have on the focal axis two 

 real or else two imaginary nodo-foci. The focal axis contains, as has been men- 

 tioned, four foci— the remaining twelve foci are situate symmetrically, six on each 

 side of the focal axis, the arrangement of the sixteen foci being as mentioned 

 ante, No. 81 et seq. ; the focal axis is in fact an axis of symmetry of the curve, 

 and if preferred it may be named the axis of symmetry, transverse axis, or simply 

 the axis. And the curve (circular cubic, or bicircular quartic) is in this case a 

 " symmetrical" or " axial" curve. 



Circular Cubic and Bicircular Quartic: Singular Forms — Art. Nos. 131 to 140. 



131. The circular cubic may have a node or a cusp. If this were at one of the 

 points /, J the curve would be imaginary, and I do not attend to the case ; and 

 for the same reason, for the bicircular quartic I do not attend to the case where 

 one of the points I, J is a cusp. There remain then for the circular cubic and 

 for the bicircular quartic the cases where there is a node or a cusp at a real point 

 of the curve ; and for the bicircular quartic the case where each of the points /. J 

 is a cusp — in general the curve has no other node or cusp, but it may besides 

 have a node or cusp at a real point thereof. 



132. I consider first the case of the bicircular quartic where each of the points 

 /, J is a cusp. The curve is in this case of necessity symmetrical* — it is in fact a 

 Cartesian; viz., the Cartesian may be taken by definition to be a quartic curve 

 having a cusp at each of the circular points at infinity. But in this case, as dis- 

 tinguished from the general case of the bicircular quartic, there is an essential 

 degeneration of all the focal properties, and it is necessary to explain what these 

 become. The centre is evidently the intersection of the cuspidal tangents ; the 

 nodo-foci (so far as they can be said to exist) coalesce with the centre, and they 

 do not in so coalescing determine any definite directions for the nodal axes ; 

 that is, there are no nodal axes, and the only theorem in regard to the focal 

 axis or axis of symmetry is, that it passes through the centre. Of the four 

 tangents through the point I, one has come to coincide with the line IJ ; and 

 similarly, of the four tangents through the point J one has come to coincide with 

 the line JI: there remain only three tangents through / and three tangents 

 through J, and these by their intersections determine nine foci — viz., three foci 

 A, B, C on the axis, and besides (B v C x ) the anti-points of (B, C) : (C 2 , A 2 ) the 

 anti-points of (C, A) and (A s , B 3 ) the anti-points of (A, B). 



* It will appear, post Nos. 161-164, that if starting with three given points as the foci of a 

 bicircular quartic, we impose the condition that the nodes at I, J shall be each of them a cusp, then 

 either the quartic will be the circle through the three points taken twice, in which case the assumed 

 focal property of the given three points disappears altogether, or else the three points must be in lined. 

 or the curve be symmetrical, that is, a Cartesian. 



