470 



series of hyperboloids have the same asymptotic cone, and 

 their primary axes be indefinitely diminished, they will ap- 

 proach indefinitely to the cone ; and, in the limit, the focal 

 elhpse and hyperbola of the hyperboloid of one sheet will 

 pass into the vertex and the focal lines of the cone, thus 

 making the vertex doubly modulai", while the focal ellipse 

 of the hyperboloid of two sheets will also be contracted 

 into the vertex, and will make that point umbilicar. 



When the two directive planes coincide, and become one 

 directive plane, the umbilicar property is reduced to this, 

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

 F and from the directive plane are in a constant ratio to 

 each other ; and therefore the surface becomes one of re- 

 volution round an axis passing through F at right angles to 

 that plane, the point F being a focus of the meridional section, 

 or the vertex if the surface be a cone. When the directive 

 planes are supposed to be parallel, but separated by a finite 

 interval, we get the same class of surfaces of revolution, with 

 the addition of the surface produced by the revolution of an 

 ellipse round its minor axis ; the point F being still on the 

 axis of revolution, but not having any fixed relation to the 

 surface. 



§ 10. If in the equation (30) we supposed the right-hand 

 member to have an additional terra containing the product 

 of the quantities £c — 0:2 and y — 1/2, with a constant coeffi- 

 cient, all the foregoing conclusions regarding the geometrical 

 meaning of that equation would remain unchanged, because 

 the additional term could always be taken away by assigning 

 proper directions to the axes of x and y. If, after the re- 

 moval of this term, the coefficients of the squares of the 

 aforesaid quantities were both positive, the locus of F would 

 be a modular focal of the surface expressed by the equation; 

 but if one coefficient were positive and the other negative, 

 the locus of F would be an umbilicar focal. The equation 

 in its more general form is evidently that which we should 



