485] 
PROBLEMS AND SOLUTIONS. 
557 
property of 4 concyclic foci is that taking any three of them A, B, C, the distances 
of a point P of the curve from these three foci are connected by a linear relation 
X. AP + y. BP + v. CP = 0, where X + y ± v = 0, or if as is more convenient the distances 
are considered as +, then where X + /u, + y = 0. A circular cubic may be determined 
so as to satisfy 7 conditions; having a focus at a given point is 2 conditions; hence 
a circular cubic may be determined so as to pass through three given points, and to 
have as foci two given points. 
2. A Cartesian, or bicircular cuspidal quartic (that is a quartic having a cusp 
at each of the circular points at infinity) has nine foci, but of these there are three 
which lie in a line with the centre of the Cartesian (or intersection of the cuspidal 
tangents), and which are preeminently the foci of the Cartesian. We may, therefore, 
say that the Cartesian has three foci, which foci lie in a line, the axis of the 
Cartesian. A Cartesian may be determined to satisfy 6 conditions; having a focus at 
a given point is 2 conditions; but having for foci three given points on a line is 
5 conditions; and hence a Cartesian may be found having for foci three given points 
on a line, and passing through a given point; there are in fact two such Cartesians, 
intersecting at right angles at the given point. 
3. The theorem at first sight appears impossible; for take any three points 
F, G, H in a line and any other point A ; then, as just remarked, there are, having 
F, G, H for foci and passing through A, two Cartesians. And we may draw through 
F, G, H, and with A for focus, a circular cubic depending upon two arbitrary 
parameters; the position of a second focus of the circular cubic is (on account of 
the two arbitrary parameters) primd facie indeterminate; and this is confirmed by the 
remark that the circular cubic can actually be so determined as to have for focus an 
arbitrary point B; and yet the theorem in effect asserts that the foci concyclic with A, 
of the circular cubic, lie on one or other of the two Cartesians. 
4. To explain this, it is to be remarked that the arbitrary point B is a focus 
which is either concyclic with A or else not concyclic with A. In the latter case, 
although B is arbitrary, yet the foci concyclic with A may and in fact do lie on 
one of the Cartesians; the difficulty is in the former case if it arises; viz., if we 
can describe a cubic through the points F, G, H in a line, and with A and B as 
concyclic foci; that is, if we can find a third focus G, such that the distances from 
A, B, G of a point P on the curve are connected by a relation X. AP + y . BP + v . GP = 0, 
where X + y + v = 0. It may be shown that this is in a sense possible, but that the 
resulting cubic is not a proper circular cubic, but is the cubic made up of the line 
FGH taken twice, and of the line infinity. To 
passes through the points F, G, H we have 
show 
this, 
since 
the 
required cubic 
X . AF + y. BF + v . GF = 0 and thence 
AF, 
A G, 
AH, 
1 
= 0, 
X . AG + y . BG +v.CG =0 
BF, 
BG, 
BH, 
1 
\.AH+y.BH + v.GH = 0 
GF, 
CG, 
GH, 
1 
X -P y -P v — 0