On the Surfaces of the Second Order. 305 



passing through the common centre of the surfaces, because 

 otherwise the parts of any such normal, which are intercepted 

 between each pair of principal planes, would not be in a constant 

 ratio to each other. 



The point S being the point at which any of these parallel 

 normals is applied, the plane touching the surface at S is parallel 

 to a given plane ; the perpendicular OS dropped upon it from 

 the centre has a given direction, the plane OSS is fixed, and the 

 directions of the lines OL, OM, ON, in which the plane intersects 

 the principal planes, are also fixed. And as the angle OSS is 

 always a right angle, and the normal at S is always parallel to 

 GS, the distance SS bears a given ratio to each of the distances 

 OL, OM, ON, and therefore also to each of the intercepts MN. 

 Hence, since the rectangle under OS and any one of these in- 

 tercepts is constant, the rectangle under OS and SS is constant. 



Therefore if a series of confocal central surfaces be touched 

 by parallel planes, the points of contact will all lie in one plane, 

 and their locus, in that plane, will be an equilateral hyperbola, 

 having its centre at the centre of the surfaces, and having one 

 of its asymptotes perpendicular to the tangent planes. This 

 hyperbola evidently passes through two points on each of the 

 focal curves, namely the points where the tangent to each curve 

 is parallel to the tangent planes. 



If a series of confocal paraboloids be touched by parallel 

 planes, it will be found that the points of contact all lie in a 

 bifocal right line, and that the normals at these points lie in a 

 plane parallel to the axis of the surfaces ; so that the part of 

 any normal which is intercepted by the two principal planes is 

 constant. This theorem may be proved from the two following 

 properties of the paraboloid : 1. A normal being applied to the 

 surface at the point S, the segments of the normal, measured 

 from S to the points where it intersects the planes of the two 

 principal sections, are to each other inversely as the parameters 

 of these sections. 2. Supposing the axis x to be that of the sur- 

 face, the difference between the co-ordinates x of the point S and 

 of the point where the normal meets the plane of the principal 



