497 



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 surfaces 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 paraboloids, the 

 two elliptic paraboloids are distinguished by the circum- 

 stance 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 ofits directive planes by right lines paral- 

 lel to either ofits non-directive axes, the projections will be 

 a series of confocal conies ; and when the surface is umbili- 

 car, the foci of all these conies will be the corresponding 

 projections of the umbilics.f When the surface is not um- 

 bilicar, its directive axis will be parallel to the primary axis 

 of the projections. 



The same line of curvature has two projections, accord- 

 ing as it is projected by right lines parallel to the one or to 

 the other non-directive axis. In the ellipsoid these projec- 

 tions are always curves of different kinds, the one being an 

 ellipse when the other is a hyperbola; but in a hyperboloid 

 the projections are either both ellipses or both hyper- 

 bolas. In the hyperbolic paraboloid the projections are 

 parabolas. In the elliptic paraboloid, one of the projections 

 is always a parabola, and the other is either an elHpse or a 

 hyperbola. 



* See Dupin's DeTcloppements de Geometrie. 



t Exam. Papers, An. 1838, p. xlvi., quest. 4; p. xcix., quest. 70. 



