509 



1 . When two confocal conies do not intersect, if one of 

 tliem be touched in the points T, T' by tangents drawn from 

 any point P of the other, the sum of the tangents TP, T'P 

 will exceed the convex arc TT' lying between the points of 

 contact, by a constant quantity. 



2. When two confocal conies intersect in the point A, if 

 one of them be touched in the points T, T' by tangents 

 drawn from any point P of the other, the difllerence between 

 the tangents TP, T'P will be equal to the difference be- 

 the arcs AT, AT'. 



These properties give the readiest and most elegant so- 

 lution of problems concerning the comparison of different 

 arcs of a plane or spherical conic. Any arc being given on 

 a conic, we may find another arc beginning from a given point, 

 which shall differ from the given arc by a right line if the 

 conic be plane, or by a circular arc if the conic be sphe- 

 rical. 



I 



DONATIONS. 



Memoires de la Societe G eologique de France. Tom. 5. 

 Parts 1, 2. Presented by the Society. 



The Tenth Anmial Report of the Royal Cornwall Poly- 

 technic Society. (1842.) Parts 1 and 2. Presented by the 

 Society. 



Bulletin de V Academic Royale de Briixelles, from 5th of 

 November, 1842, to 8th of July, 1843. 



p. 77 (Dublin, 1841). Mr. Graves obtained it as the reciprocal of the pro- 

 position, that when two spherical conies have the same directive circles, any 

 tangent arc of the inner conic divides the outer one into two segments, each of 

 which has a constant area. Both properties, with the general theorem relative 

 to curves described on any surface and touched by shortest lines, were after- 

 wards given in the University Calendar. See Exam. Papers, An. 1841, p. xli., 

 quests. 3-6; An. 1842, p. Ixxxiii., quests. 30-34. These two properties of 

 conies were communicated, in October 1843, to the Academy of Sciences of 

 Paris, by M. Chasles, who supposed them to be new. See the Comptes rendus, 

 torn. xvii. p. 838. 



