PROFESSOR CAYLEY ON POLYZOMAL CURVES. 57 



133. The remaining seven foci have disappeared, viz., we may consider that 

 one of them has gone off to infinity on the focal axis, and that three pairs of foci 

 have come to coincide with the points 7, J respectively. The circle (as in the 

 general case of a symmetrical quartic) has become a line, the focal axis ; the circles 

 R, S, T (contrary to what might at first sight appear) continue to be determinate 

 circles, viz., these have their centres at A,B, C respectively, and pass through 

 the points (i? 1? C x ), (C 2 , A 2 ), and (A s , B 3 ) respectively, see ante, No. 83. But on 

 each of these circles we have not more than two proper foci, and it is only on the 

 axis as representing the circle that we have three proper foci, the axial foci 

 A, B,C : in regard hereto it is to be remarked that the equation of the curve 

 can be expressed not only by means of these three foci in the form 

 *JTK + s/mB + JtOG — ; but by means of any two of them in the form 

 y/IK + JmB + .K — 0, where K is a constant, or, what is the same thing 

 (z being introduced for homogeneity in the expressions of A and B respectively), 

 in the form JlK + JmE + Kz 2 = . 



134. Using for the moment the expression "twisted" as opposed to sym- 

 metrical— (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, 



f Ordinary, 



I Bicuspidal = 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, C, D we have two, 

 suppose B and C, coinciding with N ; and the sixteen foci are as follows, viz. 



B, C, A, D are N,N,A,D; 



B v C v A v B 1 „ N, N, Anti-pts. of (A, D) ; 



C 2 , A v B v Z> 2 „ Anti-pts. of (N, A) , Anti-pts. of (N, D) ; 



A 2> B 3 ,C Z ,B 3 „ Do. do. 



viz., we have the points {A, D) each once, the node N four times, the anti-points of 

 {A, D) once, and the anti-points of (N, A) and of (JV, D), each pair twice. But 

 properly there are only four foci, viz., the points A, D and their anti-points. The 

 circle subsists as in the general case, and so does the circle R (BC, 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 at right 

 angles : the circles S and T each reduce themselves each to the point iV considered 

 as an evanescent circle, or what is the same thing to the line-pair NI, NJ. 



VOL. XXV. PART I. P 



