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surface, let tangents drawn to the first curve at two points 

 T, t, indefinitely near each other, meet the second curve in 

 the points P, p. Then taking a fixed point A on the curve 

 E, if we put 5 to denote (according to the position of this 

 point with respect to T) the sum (or difference) of the arc 

 AT and the tangent TP, and s + ds to denote the sum 

 (or difference) of the arc A^ and the tangent tp, we shall 

 have ds equal to the projection of the infinitesimal arc Pp 

 upon the tangent ; that is, if a be the angle which the tan- 

 gent TP makes with the curve F at the point P, we shall 

 have ds equal to Vp multiplied by the cosine of a. 



Now through the points P, p conceive other tangents 

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

 T', i' ; 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 al- 

 ways 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' or s — s' is a, con- 

 stant quantity. 



A simple example of this theorem is afforded by the 

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

 confocal 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 internally or externally according as the 

 conies intersect or not. Hence we have the two following 

 properties* of confocal conies : — 



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

 the Rev. Charles Graves, Fellovir of Trinity College, in the " notes and addi- 

 tions'" to his translation of M. Chasles's Memoirs on Cones and Spherical Conies, 



