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430 ON THE CONICS WHICH TOUCH FOUR GIVEN LINES. [280 
the left-hand region. No conic touching the four lines lies wholly or in part in the 
shaded regions, and every conic touching the four lines lies wholly in the inner region 
or wholly in the upper region, or partly in the upper and partly in the lower region, 
or partly in the right-hand and partly in the left-hand region. It will be convenient 
to consider, 1°. the conics which lie in the inner region; 2°. the conics which lie in 
the upper and lower regions, or in the upper region only; 3°. the conics which lie 
in the right-hand and left-hand regions. 
1°. The conics in the inner region are obviously ellipses; an extreme term is the 
finite right line BD, considered as an indefinitely thin ellipse; this gradually broadens 
out and there is (as a mean term) an ellipse which touches the four lines in the 
points in which they are intersected two and two by the lines joining the points E, F 
with the point of intersection of the diagonals AG and BD, the ellipse then narrows 
in the transverse direction and at length reduces itself to the finite line AG considered 
as an indefinitely thin ellipse. 
2°. Considering first the conics which lie in the upper and lower regions, these 
are of course hyperbolas, an extreme term is the infinite portions A oo and G oo of 
the line AG, considered as an indefinitely thin hyperbola: we have then hyperbolas 
such that for each of them, one branch lies in the lower region and touches the two 
lines through A, while the other branch lies in the upper region and touches the 
two lines through G; the points of contact of the lower branch with the lines through 
A gradually recede from A, but the point of contact on the line AD, which adjoins 
the left-hand region, recedes with the greater rapidity, and it at last becomes infinite 
while the point of contact with the line AB which adjoins the right-hand region 
remains finite: we have thus a hyperbola having the line AD for its asymptote; the 
point at infinity of this line belongs of course indifferently to the upper afid lower 
regions; and we may therefore consider one branch as lying in the lower region and 
touching AB and (at infinity) AD; the other branch as lying in the upper region 
and touching the two lines through G, and besides (at infinity) the line AD. We have 
next a series of hyperbolas such that for each of them, one branch lies in the lower 
region and touches only the line AB, the other branch lies in the upper region and 
touches the two lines through C and besides the line AD. We arrive again at a 
limiting case when the point of contact on the line AB passes off to infinity, or AB 
becomes an asymptote; we have here in the lower region a branch touching (at infinity) 
AB and in the upper region a branch touching the two lines through G, the line AD, 
and besides (at infinity) the line AB. To this succeeds a series of hyperbolas such 
that for each of them, one branch lies in the lower region but does not touch either 
of the lines through A, the other branch lies in the upper region and touches as well 
the two lines through G as also the two lines through A. Finally the branch in the 
lower region passes, off to infinity, or the conic becomes a parabola lying wholly in 
the upper region. 
We have thus arrived at the conics which lie wholly in the upper region, the 
extreme term being a parabola: this passes into an ellipse touching of course the 
four lines, and gradually reducing itself to the finite line EF considered as an inde 
finitely thin ellipse.