of Surfaces of the Second Order. 435 



(2) in the plane of x y, C G = , C X = a, C Z = c, C Y 



= 5, CD = , Cs = e, CO = ae, CA = . 



(5.) Let = D A C, be the angle which a directrix plane 

 (or which a circular section of the surface as being parallel to 

 it) makes with the plane of xy, or with the plane of the prin- 

 cipal section, whose semiaxes are a and b ; then 



cos = . 







(6.) Let <u be the angle which the umbilical diameter u 

 makes with the corresponding directrix plane, then 



a 2 c 1 ac 



tan 2 w = tan 2 GDI -r-s M , ?a ^; or sinco = -j-.. 



(a 2 b*)(o z c 2 ) 6w 



Hence in two cases, the directrix plane is perpendicular 

 to the diameter conjugate to it; either when the surface is one 

 of revolution round the transverse axe, when b = c, or when 

 (X) is an oblate spheroid, in which case a = b. 



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



semiparameters of the parabolas in the planes of xy and x z ; 



p 



then 6 = ), and sin 2 w -. . 



I 



(7.) When (2) is a surface of revolution round the transverse 

 axe, or the axis of X, >j = ; and in this case the conjugate 

 directrix planes AD, AD' coincide and become perpendicu- 

 lar to the transverse axe CX, at the distance -'- from the 



centre; and the foci of the surface coincide with the focal 

 centre ; but when (2) is an oblate spheroid e = 0, and e = >j ; 

 in this case then (5 = 0, or the conjugate directrix planes be- 







come parallel to the plane of xy> distant from it by ; and 



the focal centre coincides with the centre of the surface. 



(8.) When the surface is an elliptic paraboloid, one pair of 

 conjugate directrix planes is infinitely distant, so that this 

 surface has but two foci, and two directrix planes. 



(9.) When the surface is a cone, the conjugate directrix 

 planes pass through the vertex, and are parallel to the circu- 

 lar sections of the cone ; the foci of the surface, and the focal 

 centres all coincide with the vertex. 



(10.) The distance of a focus of the surface (X) from the 



focal centre, or from the plane of xy^ is = . 



2 F2 



