272 On the Surfaces of the Second Order. 



boloid of one sheet has no umbilics, and accordingly both its 

 focals are modular, and neither of them intersects the surface. 

 The umbilicar focals and dirigents have properties which shall 

 be mentioned hereafter. 



An umbilicar focal and the principal section whose plane 

 coincides with that of the focal are curves of different kinds, 

 the one being an ellipse when the other is a hyperbola ; but a 

 modular focal is always of the same kind with the coincident 

 section of the surface, being an ellipse, a hyperbola, or a pair of 

 right lines, according as the section is an ellipse, a hyperbola, 

 or a pair of right lines ; and when the section is reduced to a 

 point, so likewise is the modular focal. 



The plane of a modular focal always passes through the 

 directive axis. When the directive axis is the primary, as in 

 the hyperboloid of one sheet, both focals are modular. But in 

 the ellipsoid and the hyperboloid of two sheets, where the pri- 

 mary axis is not directive, only one of the focals can be modular. 

 The plane of an umbilicar focal is always perpendicular to the 

 directive axis ; and therefore, when that axis is the primary, 

 there is no umbilicar focal.* 



When the surface is doubly modular, the two moduli m, m' 

 are connected by the relation 



1; (16) 



m m 



* If the first of the equations (10), when PI and Q\ are both negative, be sup- 

 posed to express an imaginary focal, there will, in a central surface, be three focals, 

 two modular and one umbilicar; the two modular focals being in the principal 

 planes which pass through the directive axis, and the umbilicar focal in the remain- 

 ing principal plane. Then, when we know which of the axes is the directive axis, 

 we know which of the three focals is imaginary, because the plane of the imagi- 

 nary focal is perpendicular to the primary axis. A modular focal may be imagi- 

 nary, and yet have a real modulus ; this occurs in the hyperboloid of two sheets. 

 In the ellipsoid, the imaginary focal has an imaginary modulus. In all cases the 

 two moduli are connected by the relation (16). 



It will appear hereafter that the vertex of the cone is an umbilicar focus. The 

 cone has therefore three focals, none of which is imaginary ; but tM r o of them are 

 single points coinciding with the vertex. 



