The collected mathematical papers of Arthur Cayley.

557]

13

557.
ON CERTAIN CONSTRUCTIONS FOR BICIRCULAR QUARTICS.
[From the Proceedings of the London llMathenatical Society, vol. v. (1873-1874),
pp. 29-31. Read March 12, 1874.]
I CALL to mind that if F, G are any two points and F', G' their antipoints;
then the circle on the diameter FG and that on the diameter F'G' are concentric
orthotomics, viz. they have the same centre, and the sum of the squared radii is
= 0. Moreover, if the circles B, B' are concentric orthotomics, and the circle A is
orthotomic to B, then it is a bisector of B', viz. it cuts B' at the extremities of a
diameter of B'; and B' is then said to be a bifid circle in regard to A.
Given two real circles, these have an axial orthotomic, the circle, centre on the
line of centres at its intersection with the radical axis, which cuts at right angles the
given circles; viz. this axial orthotomic is real if the circles have no real intersection;
but if the intersections are real, then the axial orthotomic is a pure imaginary, and
instead thereof we may consider its concentric orthotomic, viz. this is the axial bifid
of the two circles, or circle having its centre on the line of centres at the intersection thereof with the radical axis or common chord of the two circles, and having
this common chord for its diameter.
If one of the circles is a pure imaginary, then we have still an axial orthotomic;
viz. the pure imaginary circle is replaced by the concentric orthotomic; and the axial
orthotomic is a bisector of the substituted circle; and so if each of the circles is a
pure imaginary, then we have still an axial orthotomic, viz. each circle is replaced by
the concentric orthotomic, and the axial orthotomic is a bisector of the substituted
circles. And in either case the axial orthotomic of the original circles (one or each of
them pure imaginary) is real; viz. this is given either as the axial bisector of one
real circle and orthotomic of another real circle; or as the axial bisector of two circles,
from which the reality thereof easily appears. Or we may verify it thus: Suppose