458 



But when the hyperboloid of one sheet, or the cone, is a 

 surface of revolution, it has only one mode of modular ge- 

 neration. In cases of double generation, the directive planes 

 of course remain the same, as they have a fixed relation to 

 the surface. A modular focal, it may be observed (and the 

 remark applies equally to surfaces of the second class), is 

 distinguished by the circumstance that it does not intersect 

 the surface. The only exception to this rule are the focal 

 lines of the cone, which pass through its vertex. A focal 

 which is not modular may be called umbilicar, because it 

 intersects the surface in the umbilics ; an umbilic being a 

 point on the surface where the tangent plane is parallel to 

 a directive plane. Thus the focal hyperbola of the ellipsoid, 

 and the focal ellipse of the hyperboloid of two sheets, are 

 umbilicar focals, and pass through the umbilics of these 

 surfaces ; but the hyperboloid of one sheet has no umbilics, 

 and accordingly both its focals are modular, and neither of 

 them intersects the surface. The umbilicar focals and di- 

 rigents have properties which shall be mentioned here- 

 after. 



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 coinci- 

 dent 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 primary axis is not directive, only one of the 

 focals can be modular. The plane of an umbilicar focal is 



