On the Surfaces of the Second Order. 273 



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 correspond- 

 ing 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 the linear modulus ; and the other, to 

 which only a single focus corresponds, may be called the singu- 

 lar modulus. 



6. Second Class of Surfaces. In this class of surfaces, one 

 of the quantities A, B vanishes, or both of them vanish. 



I. When m = cos 0, and is not zero, A vanishes, but B 

 does not ; and the surface is either a paraboloid or a cylinder. 



1. If the surface is a paraboloid, we may suppose the origin 

 of co-ordinates to be at its vertex, in which case both H and K 

 vanish, and we have the relations 



(r = # 3 -#i, yi = y 2 cos 2 0, 



(17) 

 x? + y-i cos 2 tf> - x? - y? = ; 



the equation of the surface being 



y z sin 2 + s 2 + 2G% = 0, (18) 



which shows that the paraboloid is elliptic, having its axis in the 

 direction of x, and the plane of xy for that of its greater prin- 

 cipal section. From the relations (1 7) we obtain the following : 



(19) 

 G* = ; 



from which we see that the focal and dirigent curves are para- 

 bolas, having their axes the same as that of the surface ; and 

 their vertices equidistant from the vertex of the surface, but at 

 opposite sides of it. The cavity of each curve is turned in the 



T 



