On the Surfaces of the Second Order. 3 1 7 



to be central, let a plane drawn through the centre, parallel to 

 the plane which touches the surface at S, intersect any one of 

 these right lines. Then the distance of the point of intersec- 

 tion from the point S will always be equal to the primary semi- 

 axis of the surface.* 



If through any point S of a given central surface a right 

 line be drawn touching two other given surfaces confocal with 

 it, and if this right line be intersected by a plane drawn 

 through the centre parallel to the plane which touches the 

 first surface at S, the distance of the point of intersection from 

 the point S will be constant, wherever the point S is taken on 

 the first surface. If this constant distance be called /, and the 

 other denominations be the same as in the formula (7), the 

 wilue of I will be given by that formula, f 



* " Examination Papers," An. 1838, p. xlvii., question 9. 



f In the notes to the last-mentioned work of M. Chasles, on the History of 

 'Methods in Geometry^ will be found many theorems relative to surfaces of the 

 second order. Among them are some of the theorems which are given in the 

 present Paper ; but it is needless to specify these, as M. Chasles's work is so well 

 known. 





