tremity 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 trans- 

 verse, and B to the conjugate semi-axis, let 



Assuming these definitions, we shall have the following theo- 

 rems analogous to those in piano. 



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

 perpendicular 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 2a. 



2. The rectangle under them =: b'^. 



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



plane, is —7 



* B. 



4. The rectangle under the perpendiculars from these 

 points on tangent plane :r b^. 



5. The sine of the angle between the central radius 



vector and tangent plane = -7—,, (a' being the central radius 



vector). 



6. The portion of the normal intercepted between the sur- 

 face and the plane of the focal conic is — . b'. 



A 



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

 points analogous to the foci, and at a distance from the centre 



A* 



equal to — (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. 



