The ratio of c to A is usually, as in the ellipse, denoted
by the symbol e, and is called the eccentricity of the hy-
perbola. Thus,

-^ = e / .•. c = Ae, or c^ = A?e'^.
But c2— A2=A2(e2-l),

where ci=A^+B'^,

or B=±A(e2-l)*.
(69.) In order to transfer the origin from the centre
to the vertex of the transverse axis, we have simply to
substitute (a!+A) for A in (8), and reduce. We then
have

■R2

2/2=Jj(2Aa!+a;2),

the equation of the hyperbola referred to the extremity
of the transverse axis as the origin of co-ordinates,
(10.) If in (8) we make a!=c=CF, then

y^=g(c^-A^)^gxB-

B2

•••^=A-
Putting this into a proportion, we have

A:B::B:y,
or 2A:2B::2B:2y.
But 2y is the double ordinate passing through a focus,
and is called the parameter of the transverse axis ; there-
fore,

(71.) The parameter of the transverse axis is a third
proportional to that axis and its conjugate.
(jQj C3&

{12.) In the equations r'^A+x and r=—A-\-j-, sub-
stitute e for — , and we have r'=A.-\-ex and r= —A-^ex

A.
for the distance from the focus to any point on the hy-
perbola.
(73.) The properties of the hyperbola may be divided
into two classes : in the first class may be comprehended



Om THE ELLIPSE AND HTPEEBOLA. 53

all such properties as are analogous to those of the el-
lipse, and in the second class all such as are derived
from its relation to the asymptotes. We shall therefore
discuss the analogous properties of the ellipse and hy-
perbola together, but the properties of the hyperbola
which are derived from its relation to the asymptotes
■we shall discuss separately.
Peoposition- III.
