282 Demons Iralion of a Problem in Conic Sections, 



the correspondinej terms of these proportions, we have 

 CB^^ to AB- as CED to GEF. 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 in a 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 MIN 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 AO^ 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 of 

 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 hy- 

 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 cutout by one plane parallel 

 to the axes of the four cones. 



I am, Sir, with great 



respect and consideration, 

 you obedient servant 



C. DAVIES, 

 Asst. Prof. Nat. and Ex. Phi'y. 



To Prof. B. Silliman, New-Haven. 



