132. I consider first the case of the bicircular quartic where each of the points 
I, J is a cusp. The curve is in this case of necessity symmetrical ( x )—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 I and three tangents through J, and these by 
their intersections determine nine foci—viz., three foci A, B, G on the axis, and besides 
(B 1} Cj) the antipoints of (B, G): (G 2 , A 2 ) the antipoints of (C, A) and (A s , B 3 ) the 
antipoints of {A, B). 
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 I, J respectively. The circle 0 (as in the general 
case of a symmetrical quartic) has become a line, the focal axis; the circles B, S, T 
(contrary to what might at first sight appear) continue to be determinate circles, viz., 
these have their centres at A, B, G respectively, and pass through the points (B u CJ, 
(C 2 , A 2 ), and (.dg, 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 0 that we have three proper foci, the axial foci A, B, G: 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 VZ21 + Vm23 + Vw(£ = 0; but by means of any 
two of them in the form V151 + Vra23 + K = 0, where K is a constant, or, what is the 
same thing (z being introduced for homogeneity in the expressions of 21 and 23 
respectively), in the form V721 + Vm23 + Kz- = 0. 
134. Using for the moment the expression “twisted” as opposed to symmetrical— 
1 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, and thus the curve be 
symmetrical, that is, a Cartesian.