492 



by the progressive climinution 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 hereafter, that if a bifocal right line, that is, a 

 right line passing through both focal curves, be the inter- 

 section 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 surface is evidently the 

 diameter of this sphere. 



Hence we conclude that the locus of the point where 

 a tangent plane of a paraboloid is intersected perpendicu- 

 larly by a bifocal right line is a plane touching the parabo- 

 loid 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 consi- 

 deration is frequently of use in deducing properties of para- 

 boloids 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, and 

 gives a finite limiting parameter to each of the principal sec- 

 tions 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 with- 

 out limit. Then, at any finite distances from the fixed 



conies. And I regarded that analogy as fully established when I foiiud (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. 



