III. JSTOTE RELATIVE TO THE COMPARISON OF ARCS OF 

 CURVES; PARTICULARLY OF PLANE AND SPHERI- 

 CAL CONICS. 



[Proceedings of the Royallrish Academy, VOL. n. p. 446. Read Nov. 30, 1843.] 



THE first Lemma given in my Paper on the rectification of the 

 conic sections* is obviously true for curves described on any 

 given surface, provided the tangents drawn to these curves be 

 shortest lines on the surface. The demonstration remains exactly 

 the same ; and the Lemma, in this general form, may be stated 

 as follows : 



Understanding a tangent to be a shortest line, and sup- 

 posing two given curves E and F to be described on a given 

 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 s 

 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 P<? upon the tangent ; that is, if a be the 

 angle which the tangent TP makes witji the curve F at the 

 point P, we shall have ds equal to Pp multiplied by the cosine 

 of a. 



Now through the points P, p conceive other tangents TT, 



* Transactions of the Royal Irish Academy, VOL. xvi. p. 79 (supra, p. 20). 



