[485 
485] 
PROBLEMS AND SOLUTIONS. 
583 
r Professor 
transversal 
mg double 
a* a more 
f the two 
and more- 
ies of the 
Cartesians 
vely, these 
also thus; 
,ss through 
ß, 7) and 
Paßy and 
ametrically 
; of itself 
h of their 
problem:— 
■tesians, to 
“conic of 
nics which 
¡ren points 
to find a 
Solution by the Proposer. 
1. Consider four given points, and in connection therewith a given line IJ; the 
locus of the poles of IJ, in regard to the several conics which pass through the four 
points, is a conic, the “ conic of poles.” Consider a particular conic ®, through the 
four points; the pole of IJ in regard to the conic 0 is a point C on the conic of 
poles, and the tangents from G to the conic © meet the conic of poles in two points 
H, K; the chord of intersection HK passes through the point II which is the pole 
of IJ in regard to the conic of poles. Moreover, the polars of a point C', in regard 
to the several conics through the four points, meet in a point IP, the “ common 
pole ” of G', and in particular if G' be the point G on the conic of poles, then the 
common pole is a point il on the line IJ; this being so, the line HK passes (as 
already mentioned) through II, and the lines HK and IIil are harmonics in regard 
to the conic of poles. 
2. Assuming the foregoing properties, then, given the four points, the line IJ, 
the conic of poles, and the point G on this conic; we may construct II the pole of 
IJ in regard to the conic of poles; and also il the common pole of G; the line HK 
is then given as a line passing through II, and harmonic to IIil in regard to the 
conic of poles; this line meets the conic of poles in the points H, K; and then 
GH, GK are the tangents from G to a conic © which passes through the four points. 
3. In particular if IJ be the line infinity, then the conic of poles is the conic 
of centres; II is the centre of this conic; il is as before the common pole of G; 
HK is given as the diameter of the conic of centres, conjugate to nil; H, K are 
the extremities of this diameter; and then GH, GK are the asymptotes of the conic 
through the four points, which has the point G for its centre; and the asymptotes 
are therefore constructed as required. If the points H, K are imaginary, the asymptotes 
will be also imaginary; the conic © is in this case an ellipse. 
4. It is hardly necessary to remark, in regard to the construction of the point il, 
that we have among the conics through the four points, three pairs of lines meeting 
in points P, Q, R respectively (it is clear that the conic of poles passes through these 
three points); the harmonics of CP, CQ, GR in regard to the three pairs of lines 
respectively meet in a point, which is the required point il. In the particular case 
where the point G is on the conic of centres, the three harmonics are parallel; it 
is therefore sufficient to construct one of them; and the line HK is then the diameter 
of the conic of poles, conjugate to the harmonic so constructed. 
5. It remains to prove the properties assumed in (1). We may take z = 0 for 
the equation of the line IJ, x — 0, y = 0 for the equations of the tangents to the 
conic © at its intersections with the line IJ, so that we have (x = 0, y = 0) for the 
coordinates of the point G; the equation of the conic © will be of the form z 2 — xy = 0, 
and the four points may then be taken to be the intersections of the conic z 2 — xy=- 0, 
and the arbitrary conic 
(a, b, c, f, g, hjx, y, z) 2 = 0. 
The equation of the conic of centres is found to be 
x (ax + liy + gz) — y (hx + by + fz) — 0, or ax 2 — by 2 + gzx — hxy = 0 ;