582
PROBLEMS AND SOLUTIONS.
[383
and then seeking for the equation of the hyperboloid which passes through the three
lines, this is found to be Aw 2 + Bw + G= 0, where A, B G have the before-mentioned
values.
If in the foregoing theorem the cubic is considered as a given cubic curve, and
the three points as three arbitrary points on the cubic, the question then arises to
find the triangle ; or we have the problem proposed as Question 1607.
[Vol. II. p. 91.]
1542. (Proposed by Professor Cayley.)—If a given line meet two given conics
in the points (A, B) and (A', B') respectively; and if (A", B") be the sibi-conjugate
points (or foci) of the pairs (A, A') and (B, B'), or of the pairs (A, B') and (A', B),
then (A", B") lie on a conic passing through the four points of intersection of the
two given conics.
[Vol. ii. pp. 97—100.]
1606. (Proposed by the Editor, [W. J. M.}).—Solve the following problems:
(a) Through three given points to draw a conic whose foci shall lie in two given
lines.
(/3) Through four given points to draw a conic such that one of its chords of
intersection with a given conic shall pass through a given point.
(y) Through two given points to draw a circle such that its chords of inter
section with a given circle shall pass through a given point.
Solution by Processor Cayley.
(a) Through three given points to draw a conic whose foci shall lie in two given
lines.
The focus of a conic is a point such that the lines joining it with the two
circular points at infinity (say the points /, J) are tangents to the conic. Hence the
question is, in a given line to find a point A, and in another given line to find a
point B, such that there exists a conic touching the four lines AT, AJ, BI, BJ
(where /, J are any given points) and besides passing through three given points.
More generally, instead of the lines from A, B to the given points I, J, we may
consider the tangents from A, B, to a given conic 0; the question then is, in a
given line to find a point A, and in another given line to find a point B, such that
