498 



The corresponding projections of two lines of curvature 

 which pass through a given point of the surface, are con fo- 

 cal conies intersecting 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.* In a central surface, the length 

 of a bifocal chord is proportional to the square of the dia- 

 meter which is parallel to it ; the square of the diameter 

 being equal to the rectangle under the chord and the pri- 

 mary 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 dia- 

 meter of the first surface, the square of the diameter being 

 equal to the rectangle under the chord and a certain right 

 line 21, determined by the formula 



p - ^^. (7) 



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

 first surface, and that p', v" are the quantities corresponding 

 to p in the equations of the other two surfaces. 



In any surface of the second order, the lengths of two 

 bifocal chords are proportional to the rectangles under the 

 segments of any two intersecting chords to which they are 

 parallel. 



In the paraboloid expressed by the equation 



— + — = aj, 



p q 



if X be the length of a bifocal chord making the angles /3 

 and y with the axes of y and z respectively, we have 



I cos^B , cos ^7 .„. 



- = 1 . (») 



X p q 



* The theorems in § 1 3 are now stated for the first time. 



