436 Royal Irish Academy. 



during the Combination of Acids and Bases," was read, for an abs- 

 tract of which see p. 183 of the preceding Volume. 



January 25. — The Rev. Charles Graves, F.T.C.D., read a paper 

 " On certain general Properties of the Cones of the Second Degree." 



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 spheri- 

 cal 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 lias obsen^ed, 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 supple- 

 mentary 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 the- 

 orems relating to spherical conies become applicable in general to 

 the plane conic sections, by supposing the radius of the sphere to be- 

 come 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 properties of the cones of the second 

 degree, amongst which the following deserve to be noticed : — 



1 . If two fixed tangent arcs be drawn to a spherical conic, and 

 any third tangent arc be drawn meeting them in two points, the arcs 

 passing through these two points and through the pole of a cyclic 

 arc will intercept on that cyclic arc a portion of a constant length. 



2. If from two fixed points in a spherical conic, arcs be drawn to 

 any third point on the curve, and produced to meet one of the director 

 arcs, they will intercept between them on that director arc a portion 

 which will subtend a constant angle at the corresponding focus. 



3. A spherical conic and one of its cyclic arcs being given, if, 

 round the pole of this cyclic arc, as vertex, a spherical angle of varia- 

 ble magnitude be made to turn, whose sides intercept between them 

 on the cyclic arc a portion of a constant length, the arc joining the 

 points in which the sides of the moveable angle meet the given conic 

 will envelope a second spherical conic : the given cyclic arc will be 

 a cyclic arc of the new conic, and this arc will have the same pole 

 with relation to the two curves. 



4. A spherical conic and one of its foci being given, if round that 

 focus, as vertex, a constant spherical angle be made to turn, and from 

 the points in which its sides meet the director arc corresponding to 

 the given focus, two arcs be drawn touching the given conic, their 

 point of concourse will generate a second spherical conic : the given 

 focus will be a focus of the new conic, and the corresponding director 

 arc will be the same in the two curves. 



