463 



law of the modular focals ; the law being, that the focus of 

 the parabola shall be the focus of the principal section in the 

 plane of which the parabola lies, and its vertex the focus of the 

 principal section in the perpendicular plane. The parabola 

 so described will have its concavity opposed to that of the 

 surface ; it will cut the surface in the umbilics, and will be 

 its umbilicar focal, the only such focal to be found among 

 the surfaces of the second class. We shall of course sup- 

 pose further, that this focal has a dirigent parabola connected 

 with it by the same law as in the other cases, the vertices of 

 the focal and dirigent being equidistant from that of the 

 surface and at opposite sides of it, while the parameter of 

 the dirigent is a third proportional to the parameters of the 

 focal and of the principal section in the plane of which 

 the curves lie. The two focals of a paraboloid are so re- 

 lated, that the focus of the one is the vertex of the other. 

 The cylinders have no other focals than those which occur 

 above. 



§ 8. In this, as in the first class of surfaces, the right 

 line FA, joining a focus F with the foot of its corresponding 

 directrix, is perpendicular to the focal line ; and the focal 

 and dirigent are reciprocal polars with respect to the section 

 xy of the surface. These properties are easily inferred from 

 the preceding results ; but, as they are general, it may be 

 well to prove them generally for both classes of surfaces. 

 Supposing, therefore, the origin of coordinates to be any 

 where in the plane of a:y, and writing the equation of the 

 surface in the form 



(^x-:c,)'' + {y-y,y + :,^ = -L{x-x^f + M(iy-y.,)\ (30) 

 which, when identified with (3), gives the relations 



A=l— L, Bzzl— M, 



G = hX.i — Xx, H=M7j2-y,, (31) 



K = LXa' -f Mya' — x,"^ - y,^ 



