308 On the Surfaces of the Second Order. 



to either of its non-directive axes, the projections will be a series 

 of confocal conies ; and when the surface is umbilicar, the foci 

 of all these conies will be the corresponding projections of the 

 umbilics.* When the surface is not umbilicar, its directive axis 

 will be parallel to the primary axis of the projections. 



The same line of curvature has two projections, according as 

 it is projected by right lines parallel to the one or to the other 

 non-directive axis. In the ellipsoid these projections 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 hyperbolas. 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 ellipse or a hyperbola. 



The corresponding projections of two lines of curvature which 

 pass through a given point of the surface are confocal conies in- 

 tersecting each other in the projection of that point, and of 

 course intersecting at right angles. 



13. A bifocal chord is a bifocal right line terminated both 

 ways by the surface.f In a central surface, the length of a 

 bifocal chord is proportional to the square of the diameter which 

 is parallel to it ; the square of the diameter being equal to the 

 rectangle under the chord and the primary axis. 



More generally, if a chord of a given central surface touch 

 two other given surfaces confocal with it, the length of the chord 

 will be proportional to the square of the parallel diameter of the 

 first surface, the square of the diameter being equal to the rect- 

 angle under the chord and a certain right line 2/, determined by 

 the formula 



~ (P-F) (P-P")' 

 wherein it is supposed that the equation (2) represents the first 



* "Exam. Papers," An. 1838, p. xlvi., question 4; p. xcix., question 70. 

 t The theorems in 13 are now stated for the first time. 



