Note relative to the comparison of Arcs of Curves. 319 



t'p to be drawn, touching the curve E in the points T', t' ; and 

 let s' and ds' have for these tangents the same signification which 

 s and ds have for the former tangents. Supposing the nature of 

 the curve F to be such that it always bisects, either internally or 

 externally, the angle made at the point P by the tangents TP 

 and T'P, it is evident that ds = + ds', and therefore either s + s f 

 or s - s' is a constant quantity. 



A simple example of this theorem is afforded by the plane 

 and spherical conies. If the curves E and F be two cqnfocal 

 conies, either plane or spherical, and tangents TP, T'P be 

 drawn to F from any point P of E (the tangents being of 

 course right lines when the curves are plane, and arcs of great 

 circles when they are spherical; in both cases shortest lines), 

 it is well known that the angle TPT' made by the tangents is 

 always bisected by the conic E. The angle is bisected inter- 

 nally or externally according as the conies intersect or not. 

 Hence we have the two following properties* of confocal 

 conies : 



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

 them 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 focal 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 difference between the tangents 



* The first of these properties was originally given for spherical conies by the 

 Rev. Charles Graves, Fellow of Trinity College, in the "notes and additions" to 

 his translation of M. Chasles's Memoirs on Cones and Spherical Conies, p. 77 

 (Dublin, 1841). Mr. Graves obtained it as the reciprocal of the proposition, 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 afterwards given in the University 

 Calendar. See " Examination Papers," An. 1841, p. xli., questions 3-6; An. 1842, 

 p. Ixxxiii., questions 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. 



