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ON THE SYSTEM OF CONICS &C. 
137 
Fig. A. Four real points forming a convex quadrangle. The system contains two 
parabolas, and the pairs of lines and the parabolas divide the plane of the figure into 
five distinct regions, one of which contains only ellipses, and the other four contain 
each of them hyperbolas. 
Fig. A'. Four real points forming a triangle and interior point. The system does 
not contain any parabolas, the three pairs of lines divide the plane of the figure into 
three distinct regions, each of which contains only hyperbolas. 
A' 
Next, if the points are two of them real and two of them imaginary. The line 
joining the two imaginary points will be real and this line may meet the line joining 
the two real points, in a point outside the two real points, or included between them, 
i.e. the real centre of the quadrangle may lie outside the real points, or may be 
included between them; I consider the two cases separately. 
Fig. B. Two real and two imaginary points, the real centre of the quadrangle 
lying outside the real points. The system contains two parabolas, and these with the 
line joining the two real points and the line joining the two imaginary points divide 
the plane of the figure into three regions, one of which contains ellipses and the 
other two contain each of them hyperbolas. 
B 
c. III. 
18