squared
or 4x squared = y squared + D squared-2Dx + x squared
[Hence] y squared-3x squared + D squared-2Dx = o [I.]

This is the equation of an hyperbola whose center is on the axis of
abscisses. In order to determine the position of the center, eliminate
the x term, and find the distance from the origin o to a new origin
o'.

Let E = distance from o to o'
[Hence] x = x' + E

Substituting this value of x in equation I.

y squared-3(x' + E) squared + D squared-2D(x' + E) = o
or y squared-3x squared-6Ex'-3E squared + D squared-2Dx'-2DE = o [II.]

In this equation the x' terms should disappear.

[Hence] -6Ex' - 2Dx' = o
[Hence] -E = - D/3

That is, the distance from the origin o to the new origin or the
center of the hyperbola o' is equal to one-third of the distance
from o to d; and the minus sign indicates that the measurement
should be laid off to the left of the origin o. Substituting this
value of E in equation II., and omitting accents--

We have

y squared - 3x squared + 2Dx - D squared/3 + D squared - 2Dx + 2D squared/3 = o
[Hence] y squared - 3x squared = - 4D squared/3

[Illustration: Fig I]

[Illustration: Fig II]

This is the equation of an hyperbola referred to its center o' as
the origin of co-ordinates. To write it in the ordinary form, that is
in terms of the transverse and conjugate axes, multiply each term by
C, i.e.,
__
Let \/C = semi-transverse axis.

[TEX: \sqrt{C} = \text{semi-transverse axis.}]

Thus Cy squared - 3Cx squared = - 4CD squared/3. [III.]

When in this form the product of the coefficients of the x squared and y squared
terms should be equal to the remaining term.

That is

3C squared = - 4CD squared/3.
[Hence] C = 4D squared/9.

And equation III. becomes:

4D squared 4D squared 16D^{4}
----- y squared - ----- x squared = - ---------
9 3 27

[TEX: \frac{4D^2}{9} y^2 - \frac{4D^2}{3} x^2 = -\frac{16D^4}{27}]
____
/ 4D squared 2D
The semi-transverse axis = \/ ----- = ----
9 3

[TEX: \text{The semi-transverse axis} = \sqrt{\frac{4D^2}{9}}
= \frac{2D}{3}]