of Surfaces of the Second Degree. 



189 



may be easily understood. The equations of their bases are 

 respectively as follows, it being supposed that a>-h> c : — 



iA 



arc 



2-^ + 



■y' = 1, 





62, 



a^b^ 



The first is an ellipse, and lies in the plane of the least and 

 mean axes of the surface, its major axis being coincident with 

 the latter; the second is a hyperbola in the plane of the great- 

 est and least axes of the surface, its transverse axis being si- 

 tuated on the latter. When one of these cylinders is given, 

 the other may be constructed. 



As two coaxal surfaces of the second degree are called con- 

 focal when they have the same focal conies, so they may be 

 called biconcyclic when they have the same cyclic cylinders. 

 [This term will not include all central surfaces of the second 

 order the circular sections of which are coincident; those, for 

 instance, which are similar and similarly placed, are excluded 

 by the definition.] The cyclic cylinders will be found to have 

 their cyclic planes coincident with those of the surfaces ; and, 

 as the focal conies are the limits of a series of confocal surfaces, 

 to which they are ultimately reduced, the mean or least axis 

 diminishing down to zero; so the cyclic cylinders are the limits 

 of a system of biconcyclic surfaces, with which they ultimately 

 coincide, the greatest or mean axis being indefinitely increased. 



I subjoin a number of the properties of the cyclic cylinders 

 and of biconcyclic surfaces, together with those of the focal 

 conies and of confocal surfaces, from which they are respect- 

 ively deduced. 



Known Theorems. 



1. An ellipsoid being given, 

 through any point may be de- 

 scribed three surfaces of the 

 second degree, coaxal and con- 

 focal with it ; and of these one 

 is an ellipsoid, and the other 

 two hyperboloids of each kind. 



2. And these three confocal 

 surfaces are mutually orthogo- 

 nal. 



3. If any point be made the 

 common vertex of two cones, 



Derived Theorems. 



1. An ellipsoid being given, 

 three surfaces of the second 

 degree may be described, 

 touching a given plane, coaxal 

 and biconcyclic with it; and 

 of these one is an ellipsoid, and 

 the other two hyperboloids of 

 each kind. 



2. And the three lines drawn 

 from the common centre to the 

 three points of contact with the 

 given plane are, two by two, at 

 right angles. 



3. If a plane be drawn cut- 

 ting a surface of the second 



