54 



of Gudius, and by Dr. Smith. The second inscription has 

 lost some letters ; instead of THN NEAN, the words with 

 which its first line then commenced, the last two letters of 

 these words only are now discernible. He also exhibited to 

 the Academy fac-similes of the inscriptions, and made some 

 remarks on the differences observable in the characters in 

 which they are written. 



The Rev. Charles Graves, F.T.C.D., read a paper On 

 certain general Properties of the Cones of the Second De- 

 gree. 



Let a sphere be described whose centre is at the vertex 

 of a cone of the second degree, and through the vertex let 

 two planes be drawn parallel to the planes of the circular 

 sections of the cone ; the curve formed by the intersection 

 of the cone and sphere is called a spherical conic, and the 

 two planes meet the surface of the sphere in two great circles 

 which are called the cyclic arcs of the conic. These arcs, 

 as M. Chasles has observed, possess properties relative to 

 the conic exactly analogous to those of the asymptotes of 

 a hyperbola. Moreover, many of their properties depend 

 on the most elementary ones of the circle; but, as all the 

 properties of cones, and therefore of spherical conies, are 

 double, each theorem relative to the cyclic arcs furnishes a 

 corresponding one relative to the foci of the supplementary 

 conic, formed by the intersection of the sphere with a cone 

 whose generatrices are perpendicular to the tangent planes 

 of the cone on which the proposed conic is traced. And 

 further, the theorems relating to spherical conies become 

 applicable in general to the plane conic sections, by suppo- 

 sing the radius of the sphere to become infinite. 



These considerations, for which we are indebted to M. 

 Chasles, are calculated to direct the attention of geometers 

 to the cyclic arcs of the spherical conies. In following this 

 track, Mr. Graves has been led to many new and general 



