On the Surfaces of the Second Order. 271 



with a focal line and its dirigent, eu harmonically any right 

 line which crosses them. 



5. From this discussion it appears, that the ellipsoid and 

 the hyperboloid of two sheets can be generated modularly, each 

 in one way only, the modular focal being the ellipse for the 

 former, and the hyperbola for the latter ; but that the hyperbo- 

 loid of one sheet can be generated in two ways, each of its focals 

 being modular, and each focal having its proper modulus. The 

 cone also admits two modes of generation,* in one of which, 

 however, the focus is limited to the vertex of the cone, and the 

 directrix to its internal axis. But when the hyperboloid of one 

 sheet, or the cone, is a surface of revolution, it has only one 

 mode of modular generation. 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 ob- 

 served (and the remark applies equally to surfaces of the second 

 class), is distinguished by the circumstance that it does not in- 

 tersect 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 inter- 

 sects 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 hyper- 



* The double generation of the cone, when its vertex is the focus, may be proved 

 synthetically by the method indicated in the Examination Papers of the year 1838, 

 p. xlvi. (published in the University Calendar for 1839). Supposing the cone to 

 stand on a circular base (one of its directive sections), and to be circumscribed by a 

 sphere, the right lines joining its vertex with the two points where a diameter per- 

 pendicular to the plane of the base intersects the sphere, will be its internal and 

 mean axes. Then if P be either of these points, Fthe vertex, C the point where 

 the axis PFcuts the plane of the base, and B any point in the circumference of 

 the base, the triangles PVB and PB C will be similar, since the angles at Fand 

 are equal, and the angle at P is common to both triangles ; therefore V will be to 

 JBO as PV to PB, that is, in a constant ratio. It is not difficult to complete the 

 demonstration, when the focus is supposed to be any point on one of the focal 

 lines. 



