459 



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 



£^ + !i^ = l, (16) 



where ^ is the angle made by a directive plane with the 

 plane of the focal to which the modulus m belongs. One 

 modulus is greater than unity ; the other is less than unity, 

 but greater than the cosine of the angle which the plane of 

 the corresponding focal makes with a directive plane. In 

 the hyperboloid of one sheet, the less modulus is that which 

 belongs to the focal hyperbola. In the cone, the less 

 modulus belongs to the focal lines. Of the two moduli of a 

 cone, that which belongs to the focal lines may be termed 

 ^e linear modulus ; and the other, to which only a single 

 focus corresponds, may be called the singular modulus. 



§ G. Second Class of Surfaces. — In this class of sur- 

 faces, one of the quantities a, b vanishes, or both of them 

 vanish. 



I. When m zz cos <j), and <p is not zero, a vanishes, but b 



• If the first of the equations (10), when p, and q, are both negative, be 

 supposed 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 remaining 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 imaginary focal is perpendicular to the primary axis. A modular 

 focal may be imaginary, 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 two of 

 them are single points coinciding with the vertex. 



