383] 
PROBLEMS AND SOLUTIONS. 
561 
Taking the radius as unity, (a, /3) as the coordinates of M, and (x, y) as the 
coordinates of P, we have here 
(x 2 + y- — 1) + (a 2 + f3 2 — 1) = (x — a) 2 + (y — ftf, or ax+ (3y — 1=0; 
that is, the locus of P is a right line, the polar of M in regard to the circle. 
It may be remarked, that, when M is an inside point, then throughout the locus 
P is an outside point; and, replacing the negative quantity Do. If by its value, 
— — d i. M, we have do . P — di. M = (MP) 2 . If, however, M is an outside point, then 
in part of the locus P is an outside point, and we have do .P + do. M— (MP) 2 , while 
in the remainder of the locus P is an inside point, and, replacing the negative 
quantity do.P by its value, =— di.P, we have — di.P+do.M—(MP) 2 . For the 
case H—, the locus of P is a right line, but for each of the other two cases 
—h and the locus is a circle; the discussion of the several cases presents no 
particular difficulty. 
[Yol. I. pp. 43—45.] 
1387. (By W. K. Clifford.)—1. Four common tangents are drawn to a circle 
and an ellipse which passes through the centre (0) of the circle; if A, B be. opposite 
intersections of the tangents, prove that 0A and OB are equally inclined to the 
tangent at 0 to the ellipse. 
2. If a straight line A join the poles of B with respect to two conics, prove 
that the lines joining AB to a pair of opposite intersections of common tangents, 
form, with A, B, an harmonic pencil. 
3. If a point A be the intersection of the polars of B with respect to two 
conics, and AB be cut by a pair of common chords in G, D, prove that ACBB is 
an harmonic range. 
2. Solution by Professor Cayley. 
This elegant theorem is included as a particular case in the known theorem, 
“ Given three conics inscribed in the same quadrilateral, the tangents from any point 
to these conics form a pencil in involution.” 
Mr Clifford’s theorem is in fact as follows: viz., Four common tangents are drawn 
to a circle and an ellipse which passes through the centre 0 of the circle; if A, B 
be opposite intersections of the tangents, then OA, OB are equally inclined to the 
tangent at 0 to the ellipse. 
This comes to saying that the tangent at 0 to the ellipse, say OT, is the double 
or sibi-conjugate line of the involution of the pencil formed by the lines OA, OB, 
and the lines 01, OJ drawn from 0 to the circular points at infinity; and if we 
replace the circle by an arbitrary conic S, and the line at infinity by an arbitrary 
line IJ, the theorem will be as follows : 
C. V. 
71