474 



It may be well to remark that the excepted surfaces are 

 limits of surfaces which can be generated modularly, as the 

 circle is the limit of the ellipse in the analogous generation 

 of the conic sections. Thus the sphere is the limit of an 

 oblate spheroid, one of whose axes remains constant, while 

 its focal circle is indefinitely diminished ; and the right cir- 

 cular cylinder is the limit of an elliptic cylinder, whose focal 

 lines are conceived to approach indefinitely to coincidence 

 with each other and with the axis of the cylinder, while one 

 of the axes of the principal elliptic section remains constant. 

 In these cases the dirigent lines, along with the directrices, 

 move off to infinity. The other three excepted surfaces 

 correspond to the supposition ^ = 90°, which was excluded 

 in the discussion of the general equation (1). For if we 

 make m sec ^ = n, the quantity which constitutes the right- 

 hand member of that equation may be written 



w^ (x - rco) 2 + ri" {y - y^^ cos^ ; 



and if we suppose n to remain finite and constant, while ^ 

 approaches to 90°, and m indefinitely diminishes, this quan- 

 tity will approach indefinitely to r^ {x — x^, which will be 

 its limiting value when ^ =: 90°. But a; — ^72 is the distance 

 of the point S from a fixed plane intersecting the axis of x 

 perpendicularly at the distance x^ from the origin of coordi- 

 nates ; and therefore, in the limit, the equation expresses 

 that the distances of any point S of the surface from the 

 focus F and from this fixed plane, are to each other as n to 

 unity, that is, in a constant ratio ; which is a common pro- 

 perty of the three surfaces in question. This property also 

 belongs to the right cone, but the right cone does not rank 

 among the excepted surfaces. 



§ 12. We have seen that, when the modulus is unity, 

 any plane parallel to either of the directive planes intersects 

 the surface in a right line ; whence it follows, that through 

 any point on the surface of a hyperbolic paraboloid two right 



