604 
PROBLEMS AND SOLUTIONS. 
[383 
and we hence have the values 
X : Y : Z = a? if + y 6 z 3 + ¡Pot? — SA*y 3 z 3 : a?y 6 + y 3 z 6 + z 3 x 6 — ‘3oc?y 3 z 3 
: xyz (x 6 + y 6 + z 6 — y 3 z 3 — z 3 a? — x?if) 
for the coordinates of the point in question. 
[Yol. iv. pp. 38, 39.] 
1751. (Proposed by Professor Cayley.)—Let ABGD be any quadrilateral. Construct, 
as shown in the figure, the points F, G, H, I: in BG find a point Q such that 
BG GQ = l^ 
BG' GQ~ V2’ 
ellipse may be drawn passing through the eight points F, G, H, /, a, ¡3, 7, S, and 
having at these points respectively the tangents shown in the figure. 
and complete the construction as shown in the figure. Show that an 
{Professor Cayley remarks that if ABGD is the perspective representation of a square, 
then the ellipse is the perspective representation of the inscribed circle; the theorem 
gives eight points and the tangent at each of them; and the ellipse may therefore 
be drawn by hand with an accuracy quite sufficient for practical purposes. The 
demonstration is immediate, by treating the figure as a perspective representation: the 
gist of the theorem is the very convenient construction in perspective which it furnishes.} 
[Yol. iv. pp. 60—67.] 
1775. (Proposed by W. K. Clifford.)—If a straight line meet the faces of the 
tetrahedron ABGD in the points a, b, c, d, respectively; the spheres whose diameters 
are Aa, Bb, Gc, Dd have a common radical axis.