132 
ANALYTICAL THEOREM RELATING TO THE EOUR CONICS, &C. [331 
the equations of the four conics are 
U+(K-abc)~ = 0, 
U+(K-dbc)£ = 0, 
U+(K— abc)~ = 0, 
U + (K-abc)(j+l + $ = 0. 
It is in fact easy to verify directly that each of these conics passes through the three 
given points ; but the equations may also be exhibited in the form proper for putting 
this in evidence. Putting for shortness 
X 
ah’ 
h + f’ 
Z = 
/V 
the equations of the sides of the triangle formed by the given points are X = 0, Y =0, Z = 0, 
and the foregoing equations of the four conics may be expressed in the form 
(-bg 2 -ch 2 + 2fgh)YZ+ bg 2 . ZX + ch?.XY= 0, 
a/ 2 . YZ + (— ch 2 - a/ 2 + 2fgh) ZX + ch 2 .XY= 0, 
af 2 . YZ + bg 2 . ZX + (— af 2 — bg 2 + 2fgh) XY = 0, 
(_ bg 2 - ch 2 + 2fgh) YZ + (- ch 2 - a/ 2 + 2fgh)ZX + (-af 2 - bg 2 + 2\fgh) 17=0, 
which is the required form. 
Cambridge, November 28, 1863.