On the Siirfaces of the Second Order. 309 



surface, and that P', P" 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 



y* z* 



+ - = X 



P (1 



if x be the length of a bifocal chord making the angles ]3 and y 

 with the axes of y and 2 respectively, we have 



1 = cos 2 /3 + cos 2 7 

 X P 2 



14. At the point S on a given central surface expressed by 

 the equation (2), let a tangent plane be applied, and let k, k' 

 be the squares of the semiaxes of a central section made in the 

 surface by a plane parallel to the tangent plane ; each of the 

 quantities k, k f being positive or negative, according as the 

 corresponding semiaxis of the section is real or imaginary, that 

 is, according as it meets the given surface or not. Then the 

 equations* of two other surfaces confocal with the given one, 

 and passing through the point S, are 



f * 2 _-. ** f * , m 



P~~k Q-k -k~ ' P-k f+ Q-V R-k'~ 



The given surface is intersected by these two surfaces respec- 

 tively in the two lines of curvature which pass through the 

 point S ; the tangent drawn to the first line of curvature at S is 

 parallel to the second semiaxis of the section, and the tangent 

 drawn to the second line of curvature at S is parallel to the first 

 semiaxis of the section. 



When two confocal surfaces intersect, the normal applied to 



* " Exam, Papers," An. 1837, p.c., questions 4, 5, 6 ; An,1838,p. c., questions 71, 72. 



