Demonslralion of a Problem in Conic Section. 28 J 



aisserted by Dr. Hutton ; and the principle which he has 

 adopted as general, obtains in one case only. As I have 

 not seen this error corrected, nor any other method laid 

 down by authors on conic sections for obtaining conjugate 

 hyperbolas by nneans of intersecting four cones situated as 

 above, 1 send you the following, thinking that it may possi- 

 bly merit a place among the mathematical articles of your 

 Journal. 



Let LAC, HAF, HAL, and 

 CAF,be four cones, having a 

 common vertex A, their axes 

 in the plane of the paper, and 

 touching each other in the 

 right-lined elements, CAH, 

 and LAP. 



If the two cones CAL and 

 FAH be cut by a plane 

 QCBDE, parallel to the line 

 PO, it will intersect the 

 cones in opposite hyperbo- 

 las ; and if we take the plane 

 perpendicular to that of the 

 paper, these hyperbolas will 

 be orthographically project- [ 

 cdinthelineEQ. If through I 

 either of the points C, or D, 

 in which the cutting plane — 

 meets the right-lined ele- 

 ments AC, AD, another 

 plane be passed parallel to 

 the hne KAB, this plane will intersect the two cones 

 LAH and CAF, in opposite hyperbolas, and these hyper- 

 bolas will be conjugates to the former. 



Demonstration. 



Pass any plane as GF perpendicular to the axis of the 

 cone GAF, it will intersect its surface in a circle, and the 

 plane QE in a right line, which will be a common ordinate 

 at the point E, to the transverse axis of the hyperbola, and 

 the diameter GF of the circle. Since the triangles CAB. 

 CGE and DEF are similar, CB is to AB as CE to EG, and 

 CB is to AB as DE to EF; and by multiplying together 



Vol. VI. — No. 2. 36 



