I. GEOMETEICAL THEOEEMS ON THE EECTIFICATION 

 OP THE CONIC SECTIONS. 



[Transactions of the Royal Irish Academy, VOL. XTI. p. 79. Read, June 21, 1830.] 



Lemma 1. Let T and t be two points indefinitely near each 

 other on any given curve AT, and let tangents at Tand t meet 

 in the points P and p any 

 other given line MN, 

 straight or curved, and 

 draw Pq perpendicular to 

 tp\ then the difference 

 between the arc AT and 

 the tangent TP will ex- 

 ceed or fall short of the Fig> L 

 difference between the arc At and the tangent tp by a quantity 

 which is ultimately to pq in a ratio of equality. 



For the increment of TP, or the difference of TP and tp, is 

 ultimately equal to the sum of pq, VT, and Vt ( V being the 

 intersection of pt and PT produced) ; and Tt, or the increment 

 of the arc AT, is ultimately equal to the sum of FT and Vt', 

 therefore the difference of the increments is ultimately equal to 

 pq. Whence the proposition is manifest. 



PROBLEM. To find the Length of the Arc of a Parabola. 



Let F be the focus and A the vertex of a parabola AT: 

 draw AK perpendicular to AF, and let a tangent at the point 

 T meet it in P; take p indefinitely near to P, and let Fp 



