The collected mathematical papers of Arthur Cayley.

-522

ON POLYZOMAL CURVES.

[414

125. Hence taking the points on the curve to be the circular points at infinity,
we have the sixteen foci lying in fours upon four different circles-that is, we have
four tetrads of concyclic foci. Let any one of these tetrads be A, B, C, D, then if
Antipoints of (B, C)(A, D) are (B1, C(), (Al, D1),
(G, A)(B, D),, (, A2), (B, 2),,, (A, B) (, D),,(A, 3), (C3, 3),
the four tetrads of concyclic foci are
A, B,,D;
Al, B1, C1, D;
A2, B2, C2, D,;
A3, B3, C3, D3.
It is to be observed that if A, B, C, D are any four points on a circle, then if, as
above, we pair these in any manner, and take the antipoints of each pair, the four
antipoints lie on a circle, and thus the original system A, B, C, D, of four points on
a circle, leads to the remaining three systems of four points on a circle. The theory
is in fact that already discussed ante, No. 72 et seq.
126. The preceding theory applies without alteration to the bicircular quartic,
viz., the quartic curve which has a node at each of the circular points at infinity.
The class is here =8, but among the tangents from a node each of the two tangents
at the node is to be reckoned twice, and the number of the remaining tangents is
=4: the number of foci is =16. And, by the general theorem that in a binodal
quartic the pencils of tangents from the two nodes respectively are homologous, the
sixteen foci are related to each other precisely in the manner of the foci of the
circular cubic. The latter is in fact a particular case of the former, viz., the bicircular
quartic may break up into the line infinity, and a circular cubic.
Article Nos. 127 to 129. Centre of the Circular Cubic, and Nodo-Foci, &c. of the
Bicircular Quartic.
127. The tangents at I, J have not been recognised as tangents from I, J, giving
by their intersection a focus, but it is necessary in the theory to pay attention to the
tangents in question. It is clear that these tangents are in fact asymptotes-viz., in
the case of the circular cubic they are the two imaginary asymptotes of the curve,
and in the case of a bicircular quartic, the two pairs of imaginary parallel asymptotes;
but it is convenient to speak of them as the tangents at I, J.
128. In the case of a circular cubic, the tangents at I and J meet in a point
which I call the centre of the curve, viz., this is the intersection of the two imaginary
asymptotes.