282° Demonstration of a Problem in Conic Sections. 
the corresponding terms of these proportions, we have 
CB? to AB? as CED toGEF. But the rectangle GEF is 
equal to the square of the ordinate of the circle or hyper- 
bola at the point E, therefore AB is the semi-conjugate ax- 
is of the opposite hyperbolas, whose transverse axis is CD. 
Let now a plane HL be passed perpendicular to the axis 
AK of the cone HAL, it will intersect its surface ina circle, 
and the plane DI in a line ; this line will be a common or- 
dinate of the hyperbolas whose transverse axis is DN, and to 
the circle whose diameter is HL. Since the triangle DAO, 
DLI and HIN are similar, DO is to AO as DI to IL, and 
DO is to AO as IN to IH; by multiplying the correspond- 
ing terms of these proportions, we obtain DO? to AQ? as 
DIN to HIL. But the rectangle HIL is equal to the 
square of the ordinate of the circle orhyperbola at the point 
I, and therefore the hyperbolas having DN for a transverse 
axis have AO for a semi-conjugate. _ Since CB is equal to 
AO, and AB to DO, the transverse axis of the hyperbolas 
whose vertices are C and D, is equal to the conjugate axis 
of the hyperbolas whose vertices are D and N, as they are 
respectively equal to QCB and QAO, and the transverse 
axis of the latter hyperbolas, is equal to the conjugate © 
the former; the four hyperbolas are therefore conjugates. 
It follows from this demonstration that, if either two of 
the opposite cones, be intersected by a plane parallel to 
their common axis, the distance of this plane from the axis, 
is always equal to the semi-transverse axis of conjugate by- 
perbolas, and that these hyperbolas may be formed by in- 
tersecting the other two cones by a plane parallel to their 
common axis, and at a distance from it equal to the semi- 
transverse axis of the first hyperbolas. The cutting planes 
are at unequal distances from the axes of the cones, to 
which they are respectively parallel, unless those axes make 
with the right-lined elements of their corresponding cones, 
angles of forty-five degrees, in which case the hyperbolas 
are equilateral, and may be cut out by one plane paralle! 
to the axes of the four cones. 
I am, Sir, with great 
respect and consideration, 
you obedient servant 
C.D 
Asst. Prof. Nat. and Ex. Phi’y. 
To Prof. B. Silliman, New-Hayen. 
