of Surfaces of the Second Order. 437 



When the surface is one of revolution round the transverse 

 axe the conjugate directrix planes coalesce, p andp' there- 

 fore coincide, and are equal, and P = : hence = ; 



e re 



which is a fundamental property of surfaces of revolution and 

 of the conic sections. 



When the surface is an oblate spheroid the conjugate di- 

 rectrix planes become parallel to the plane of xy> and the 

 focal centre coincides with the centre of the surface, the 

 perpendiculars pp 1 are in the same right line but on opposite 

 sides of T, hence in an oblate spheroid, if a right line is drawn 

 perpendicular to the directrix planes meeting the surface in 

 T, and the directrix planes in m and m' ; the rectangle m T 

 x T m' is to the square of the semidameter O T, as the square 

 of the distance between the directrix planes is to the square 

 of the diameter of the central circular section, or 



m r X T m' m m 



r* 4 a* 



When the surface is an elliptic paraboloid, let / and /' be 

 the semiparameters of the parabolas in the planes of xy and 

 .rz, then p p i i\ 



~7*~ = T' 



When the surface is a cone, let a and /3 denote its semi- 

 angles ; a. > /3 ; then 



p p' sin 9 j3 

 r 2 " tan* ' 



or the square of the distance of any point on the surface of a 

 cone from the vertex is to the rectangle under the perpendi- 

 culars from the same point on two planes passing through 

 the vertex parallel to the circular sections of the cone in a 

 constant ratio. 



PROP. II. The cone whose vertex is a focal centre of (2), 

 and base any plane section of this surface, has its circular 

 sections parallel to the planes passing through the vertex of 

 the cone, and the right lines in which the base of the cone 

 intersects the conjugate directrix planes. 



When the base of the cone passes through the right line, 

 in which the conjugate directrix planes intersect, the planes 

 parallel to the circular sections coincide, and the cone is 

 therefore a surface of revolution. 



Hence the cone whose vertex is a focal centre of (2), and 

 base any plane section of this surface, passing through the 

 intersection of the conjugate directrix planes, is a surface of 



