594 
PROBLEMS AND SOLUTIONS. 
[383 
The same conic will also intersect in the chord a'b'c', the three conics which pass 
through the intersection of Aa, Bb, Cc and touch any two sides of the triangle abc 
at the extremities of the third side. 
It will intersect in the chord a'bc the three conics which pass through the inter 
section oi Aa, Bb', Cc' and touch any two sides of the triangle ab'c' at the extremities 
of the third side. 
It will intersect in the chord ab'c the three conics which pass through the inter 
section of Aa, Bb, Cc' and touch any two sides of the triangle a'bc at the extremities 
of the third side. 
It will intersect in the chord abc' the three conics which pass through the inter 
section of Aa, Bb', Cc and touch any two sides of the triangle a'b'c at the extremities 
of the third side. 
Def. The critical conic of any quadrilateral is a circumscribed conic such that the 
tangent at any angular point forms a harmonic pencil with the sides and diagonal 
meeting at that point. 
It is obvious that if the quadrilateral be projected into a square, the critical 
conic will become the circumscribed circle. 
3. Solution by Professor Cayley. 
1. The equations of the sides of the quadrilateral may be taken to be respectively 
x = 0, y — 0, z — 0, w— 0, where the implicit constants are so determined that we have 
identically 
x+y+z+w= 0; 
this being so, the equations of the three diagonals are respectively 
x + y 
= 0, 
or 
z + w - 
= 0, 
or 
x + y—z—w = 
= 0 
(three 
equivalent 
forms) 
x + z 
= 0, 
or 
y + w = 
= 0, 
Ol 
x—y+z—w- 
= 0 
( » 
)) 
» ) 
X + w 
= 0, 
or 
y+z = 
= 0, 
or 
X—y — z + w- 
= 0 
( » 
» ) 
and the equations of the critical conics are respectively 
xy + zw = 0, xz 4- yw = 0, xw + yz = 0. 
Hence we see that the equation of the required conic is 
A = x 2 + y~ + z 2 + w 2 — 2yz — 2 zx — 2 xy — 2 xiu — 2 yw — 2zw = 0. 
In fact this equation may be written 
A = {x + y — z — w) 2 — 4 {xy + zw) = 0, 
A = (a? — y + z — w) 2 — 4 {xz + yw) = 0, 
A ={x — y — z + w) 2 — 4 {xw + yz) = 0,