On the Surfaces of the Second Order. 307 



other, they intersect everywhere at right angles ; and that 

 through any given point three surfaces may in general be de- 

 scribed, which shall have the same focal curves. If three con- 

 focal surfaces pass through the point S, the normal to each of 

 them at S is an axis of each of the cones which stand on the 

 f ocals and have S for their common vertex. The normals to the 

 three surfaces are therefore the three axes of each cone. 



If the points at which a series of confocal surfaces are touched 

 by parallel planes be the vertices of cones having one of the 

 f ocals for their common base, each of these cones will have one 

 of its axes perpendicular to the tangent planes. Therefore when 

 an axis of a cone which stands on a given base is always parallel 

 to a given right line, the locus of the vertex is an equilateral 

 hyperbola or a right line, according as the base is a central conic 

 or a parabola. 



12. A system of three confocal surfaces intersecting each 

 other consists of an ellipsoid, a hyperboloid of one sheet, and a 

 hyperboloid of two sheets, if the focals be central conies ; but it 

 consists of two elliptic paraboloids and a hyperbolic paraboloid, 

 if the focals be parabolas. In the central system, the ellipsoid 

 has the greatest primary axis, and the hyperboloid of two sheets 

 the least ; and the focal which is modular in one of these sur- 

 faces is umbilicar in the other. The asymptotic cones of the 

 hyperboloids are confocal, the focal lines of each cone being the 

 asymptotes of the focal hyperbola. In the system of parabo- 

 loids, the two elliptic paraboloids are distinguished by the cir- 

 cumstance that the modular focal of the one is the umbilicar 

 focal of the other. 



The curve in which two confocal surfaces intersect each other 

 is a line of curvature of each, as is well known ;* and a series of 

 lines of curvature on a given surface are found by making a 

 series of confocal surfaces intersect it. 



Now if a series of the lines of curvature of a given surface 

 be projected on one of its directive planes by right lines parallel 



* See Dupin's Developments de Geometries 



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