On the Surfaces of the Second Order. 303 



gressive diminution of their mean or secondary axes, to become 

 flattened, and to approach more and more nearly to a plane 

 passing through the primary axis. And it will appear here- 

 after, that if a bifocal right line, that is, a right line passing 

 through both focal curves, be the intersection of two planes 

 touching these curves, those two planes will be at right angles 

 to each other. Therefore the locus of the point where a tangent 

 plane of a given central surface is intersected perpendicularly 

 by a bifocal right line is a sphere. The primary axis of the sur- 

 face is evidently the diameter of this sphere. 



Hence we conclude that the locus of the point where a tan- 

 gent plane of a paraboloid is intersected perpendicularly by a 

 bifocal right line is a plane touching the paraboloid at its vertex. 

 For a paraboloid is the limit of a central surface whose primary 

 axis is prolonged indefinitely in one direction, and a plane is 

 the corresponding limit of the sphere described on that axis as 

 diameter. As this consideration is frequently of use in deduc- 

 ing properties of paraboloids from those of central surfaces, it 

 may be well to state it more particularly. It is to be observed, 

 then, that the indefinite extension of the primary axis at one 

 extremity may take place according to any law which leaves the 

 other extremity always at a finite distance from a given point, 

 andj gives a finite limiting parameter to each of the principal 

 sections of the surface which pass through that axis. The 

 simplest supposition is, that one extremity of the axis and the 

 adjacent foci of those two principal sections remain fixed, while 

 the other extremity and the other foci move off, with the centre, 

 to distances which are conceived to increase without limit. Then, 

 at any finite distances from the fixed points, the focal curves 

 approach indefinitely to parabolas, as do also all sections of the 

 surface which pass through the primary axis, while the surface 

 itself approaches indefinitely to a paraboloid ; so that the limit 



focal curves, and the analogy between their points and the foci of conies. And I 

 regarded that analogy as fully established when I found (in March or April, 1832) 

 that the normal at any point of a surface of the second order is an axis of the cone 

 which has that point for its vertex and a focal for its base. 



