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PROBLEMS AND SOLUTIONS. 
[383 
above-mentioned property holds; but the in-and-circumscribed hexagon has the additional 
property that the three diagonals meet in a point, and it is therefore a less general 
figure than the hexagon of the foregoing theorem. It would, I think, be worth while 
to study further the hexagon of the theorem. 
{Note. In the solution of Question 1548 it is shown that if two pairs of 
opposite sides of any hexagon intersect each on a diagonal produced, so likewise will 
the third pair. 
A slight variation of Professor Cayley’s proof may be obtained by finding the 
equations of P5, Ql, and thence of 36, which are respectively 
axe — (X — /x) z = 0, bx + (A, — g) z = 0, ax+by = 0, 
showing that 36 passes through 0. Ed. [W. J. M].}. 
[Yol. ii. pp. 70—72.] 
1562. (Proposed by F. D. Thomson, M.A.)—Find the locus of the points of contact 
of tangents drawn from a given point to a conic circumscribing a given quadrangle. 
The quadrangle being supposed convex, trace the changes of form of the locus for 
different positions of the given point. 
Solution by Professor Cayley ; and the Proposer. 
Let 0 be the given point ; 1, 2, 3, 4 the vertices of the given quadrangle ; 
A, B, G the centres of the quadrangle, viz., A the intersection of the lines 14, 23 ; 
B of 24, 31 ; G of 34, 12. The polars of 0 in regard to the several circumscribed 
conics intersect in a point O'. This being so, the locus is a cubic passing through 
the nine points 1, 2, 3, 4, A, B, C, 0, O', and which is moreover such that the 
tangents at the four points 1, 2, 3, 4 meet the cubic in the point 0, and the 
tangents at the four points A, B, G, 0 meet the cubic in the point O'. It is to be 
remarked that the nine points are so related to each other that a cubic through 
any eight of these points passes through the remaining ninth point ; say a cubic 
through 1, 2, 3, 4, A, B, C, 0 passes through O'; the nine points consequently do 
not determine the cubic ; but the cubic will be determined, e.g., by the conditions 
that it passes through 1, 2, 3, 4, A, B, G, 0, and has 01 for the tangent at 1. 
The series of cubics corresponding to different positions of the point 0 is identical 
with the series of cubics passing through the seven points 1, 2, 3, 4, A, B, G