Royal Irish Academy. 65 



II. Let the axis of the surface, perpendicular to the plane of the 

 conic, be considered analogous to the conjugate axis ; then, since 

 the square of the distance from focus to centre, in a conic, is equal 

 to the difference between the squares of the transverse and conju- 

 gate semi-axis, we may consider, as analogous to the transverse 

 semi-axis, the line drawn to the extremity of the perpendicular axis 

 from the point analogous to the focus. 



III. Since the square of the semiconjugate diameter is equal to 

 the sum of squares of semiaxes minus the square of central radius 

 vector, let the same be supposed true of the line analogous ; i. e. 

 if A be the line analogous to the transverse, and B to the conjugate 

 semi-axis, let 



B'= V A 2 + B 2 — A' 2 . 

 Assuming these definitions, we shall have the following theorems 

 analogous to those in piano. 



1 . The sum or difference (according as the focal conic is perpen- 

 dicular to a real or imaginary axis) of the distances from the points 

 analogous to the foci, to the corresponding point on the surface, is 

 equal to 2 A. 



2. The rectangle under them = B' 2 . 



3. The sine of the angle, made by either with the tangent plane, 

 . B 



13 W 



4. The rectangle under the perpendiculars from these points on 

 tangent plane = B 2 . 



5. The sine of the angle between the central radius vector and 



A J\ 

 tangent plane = -^rs ( A ' bein S the central radius vector). 



6. The portion of the normal intercepted between the surface 

 and the plane of the focal conic is -jr . B'. 



7. If a plane be drawn perpendicular to the line joining points 



A 2 

 analogous to the foci, and at a distance from the centre equal to -p- 



(C being the distance of one of the focal points from the centre), 

 the distance of a point in the surface from the corresponding focus 

 will be to its distance from this plane : : C : A. 



8. Hence, given a focal conic and the perpendicular axis, we can 

 find points and tangent planes ad libitum, by the following construc- 

 tion: — Take in the focal conic two diametrically opposite points; 

 with one as centre, and twice the distance from it to the extremity 

 of the perpendicular axis as radius, describe a sphere. Through the 

 other point draw a plane, normal to the focal conic ; it will cut the 

 sphere in a certain circle. Connect any point in this circle with the 

 two points on the focal conic, and at the middle point of the line 

 connecting it with the second point draw to it a perpendicular plane. 

 This is a tangent plane to the surface, and the point where it cuts 

 the first connecting line is a point on the surface. 



Another mode of generating the surface is easily derivable from (7 .). 

 Phil. Mag. S. 3. Vol. 2 1 . No. 1 35. July 1 842. F 



