414] 
ON POLYZOMAL CURVES. 
523 
129. In the case of a bicircular quartic, the two tangents at / and the two 
tangents at J meet in four points, which (although not recognising them as foci) I 
call the nodo-foci ; these lie in pairs on two lines, diagonals of the quadrilateral formed 
by the four tangents (the third diagonal is of course the line IJ), which diagonals 
I call the “ nodal axes ; ” and the point of intersection of the two nodal axes is the 
“ centre ” of the curve. The nodo-foci are four points, two of them real, the other two 
imaginary, viz., they are two pairs of antipoints, the lines through the two pairs 
respectively being, of course, the nodal axes ; these are consequently real lines bisecting 
each other at right angles in the centre (with the relation 1 : i between the distances). 
The centre may also be defined as the intersection of the harmonic of IJ in regard 
to the tangents at I, and the harmonic of this same line in regard to the tangents 
at J. Speaking of the tangents as asymptotes, the nodo-foci are the angles of the 
rhombus formed by the two pairs of parallel asymptotes ; the nodal axes are the 
diagonals of this rhombus, and the centre is the point of intersection of the two 
diagonals ; as such it is also the intersection of the two lines drawn parallel to and 
midway between the lines forming each pair of parallel asymptotes. 
Article No. 130. Circular Cubic and Bicircular Quartic; the Axial or Symmetrical 
Case. 
130. In a circular cubic or bicircular quartic, the pencil of the tangents from 
I and that of the tangents through J, considered as corresponding to each other in 
some one of the four arrangements, may be such that the line IJ considered as 
belonging to the two pencils respectively shall correspond to itself, and when this is 
so, the four foci, A, B, C, D, which are the intersections of the corresponding tangents 
in question, will lie in a line (viz., the conic which exists in the general case will 
break up into a line-pair consisting of the line IJ and another line). The line in 
question may be called the focal axis; it will presently be shown that in the case of 
the circular cubic it passes through the centre, and that in the case of the bicircular 
quartic it not only passes through the centre, but 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 mentioned, 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. 
Article Nos. 131 to 140. Circular Cubic and Bicircular Quartic: Singular Forms. 
131. The circular cubic may have a node or a cusp. If this were at one of the 
points I, 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 
66—2