499 



§ 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' being positive or negative accord- 

 ing 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 



v-k^ ^-k^ K-k~ ' v-k'^^-k'^ Vi-k'- ^' 



The given surface is intersected by these two surfaces re- 

 spectively 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 one of them at any point S of the line of curvature formed 

 by their intersection lies in the tangent plane of the other, 

 and is parallel to an axis of any section made in the 

 latter by a plane parallel to the tangent plane. Sup- 

 posing the surfaces to be central, if two normals be applied 

 at the point S, and a diameter of each surface be drawn pa- 

 rallel to the normal of the other, the two diameters so 

 drav/n will be equal and of a constant length, wherever the 

 point S is taken on the line of curvature ; the square of that 

 length being equal to the difference of the squares of the 

 primary axes of the surfaces, and the diameter of the sur- 

 face which has the greater primary axis being real, while 

 that of the other surface is imaginary. As the point 

 S moves along the line of curvature, each constant diameter 



* Exam. Papera, An. 1837, p.c, quests. 4, 5, 6 ; An. 1838, p. c, quests. 71, 72. 



