562 
PROBLEMS AND SOLUTIONS. 
[383 
Consider a conic S; a line meeting this conic in the points I, J; and the point 
0, the intersection of the tangents at I, J, or (what is the same thing) the pole of 
the line IJ in regard to the conic. If through the point 0 there be drawn any 
other conic ©, and if A, B be opposite intersections of the common tangents of the 
conics S, ©; then the tangent OT at the point 0 to the conic © is the double or 
sibi-conjugate line of the involution of the pencil formed by the lines OA, OB, and 
the lines 01, OJ; or, as we may also express it, the lines OT, OT, the lines OA, OB, 
and the lines 01, OJ form a pencil in involution. 
Now, considering the two points or point-pair (A, B) as a conic inscribed in the 
quadrilateral formed by the common tangents of the conics S and ©, the conics 
S and © and the point-pair (A, B) are a system of three conics inscribed in the 
same quadrilateral; and hence, by the general theorem above referred to, if O' be any 
point whatever, the tangents from 0 to the conic 8, the tangents from O' to the 
conic ©, and the tangents from 0' to the point-pair (that is, the two lines 
O'A, O'B) form a pencil in involution. But, if O' coincide with 0, then the tangents 
to the conic S are the lines 01, OJ; and the tangents to the conic © are the 
coincident lines OT, OT; and we have thence the theorem in question; viz., that the 
lines OT, OT, the lines 01, OJ, and the lines OA, OB form a pencil in involution. 
[Yol. i. pp. 77—79.] 
1409. (By W. K. Clifford.)—For every point 4 on a conic section there exists 
a straight line BO, not meeting the curve, such that, if through any other point K 
on the conic there be drawn any two straight lines meeting BO in B, 0, and the 
curve in D, E, the angles BAG, DAE are either equal or supplementary. 
Solution by Professor Cayley. 
I find that this very elegant theorem depends on the lemma to be presently 
stated, and that it is intimately connected with Newton’s theorem for the organic 
description of a conic, or, what is the same thing, with the theorem of the anharmonic 
relation of the points of a conic. 
Lemma. If AT be the tangent, and any other line through a point i of a 
conic, and if two lines equally inclined to AT and AS respectively meet the conic