494 



for a given surface, but for all surfaces which are confocal 

 with it. 



Hence the part of the normal intercepted between two 

 principal planes bears a given ratio to the part of it inter- 

 cepted between one of these and the third principal plane, 

 whether the normal be applied at any point of a given sur- 

 face, or at any point of a surface confocal with it. 



If therefore normals to a series of confocal surfaces be all 

 parallel to a given right line, they must all lie in the same 

 plane 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 paral- 

 lel 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 this 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 OS, the distance SS 

 bears a given ratio to each of the distances OL, OM, ON, 

 and therefore also to each of the intercepts MN, LN, LM. 

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

 intercepts 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 equila- 

 teral hyperbola, having its centre at the centre of the sur- 

 faces, 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 



