50 
ON THE CONICS WHICH PASS THROUGH TWO GIVEN POINTS, &C. [392 
Hence as (a) passes from 1+c to \/ (1 — c 2 ), the coordinate of the centre passes 
from go to its minimum value £ {1 + V (1 — c 2 )} ; in the passage we have a = l giving 
the coordinate =1, the conic being in this case a pair of coincident lines (¿» — 1) 2 = 0. 
And as (a) passes from the foregoing value \J (1 — c 2 ) to 1 — c, the coordinate of the 
centre passes from the minimum value | {1 + ^(1 — c 2 )} to oo . 
The curve is a hyperbola if a lies without the limits 1+c, 1 — c, 
¿»-coordinate of centre 
(! — a) 2 — c 2 ’ 
which has the sign of — a, 
+ ca (1 — a) 
(1 - a) 2 - c 2 * 
¿»-senu-axis 
± cx 
y-semi-axis 
V{(l-a ) 2 -c 2 }’ 
semi-aperture of asymptotes 
which for a = 1 + c is =0 (parabola), but increases as 1 — a increases positively or 
negatively, becoming = 45° for a = + oo (the asymptotes being in this case the pair of 
lines y 2 — ¿» 2 = 0): 
a = + oo , coordinate of centre is = 0, 
® 1+C, ,, „ = GO , 
so that a diminishing from oo to 1 + c, the coordinate of the centre moves constantly 
in the same direction from 0 to — oo, 
a = 1 — c, coordinate of centre is = — oo , 
« = 0, „ „ 0, 
0, 
» 
the hyperbola being in this case the pair of lines ?/ 2 = (1 — c 2 ) ¿c 2 . 
a negative, the coordinate of centre becomes positive, viz. as a passes from a = 0 
to a = — V (1 — c 2 ), the coordinate of centre passes from 0 to a maximum positive value 
^ {1 — V(1 — c 2 )}, and then as a passes from — V(l — c 2 ) to — oo, the coordinate of 
centre diminishes from ^ {1 — ^/(1 — c 2 )} to 0. It is to be remarked that a being 
negative, the lines y 2 — x 2 = 0 are touched by the branch on the negative side of the 
origin, that is the branch not passing through the two points ¿» = 1, y = ± (l — c 2 ).