507 



§ 18. Through any point S of a given surface four bifo- 

 cal right lines may in general be drawn. Supposing the 

 surface 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 intersection from the point S will always be equal to the 

 primary semiaxis 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 value of I will be given by that formula. f 



Professor Mac Cullagh communicated the following note 

 relative to the comparison of arcs of curves, particularly of 

 plane and spherical conies. 



The first Lemma given in my paper on the rectification 

 of the conic sections (Transactions of the Royal Irish Aca- 

 demy, vol. xvi., p. 79) is obviously true for curves described 

 on any given surface, provided the tangents drawn to these 

 curves be shortest lines on the surface. The demonstration 

 remains exactly the same ; and the Lemma, in this general 

 form, may be stated as follows. 



Understanding a tangent to be a shortest line, and sup- 

 posing two given curves E and F to be described on a given 



* Exam. Papers, An. 1838, p. xlvii., quest. 9. 



t 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. 



