Geometrical Theorems oil the Rectification of the Conic Sections. 

 By JAMES Mac CULLAGH, Esq. Communicated hy the 

 Rev. Dii. Sadleir, S. F. T. C. D., Vice President. 



Read June 21st, 1830. 



Lemma 1 . Let T and t be two 

 points indefinitely near each other 

 on any given curve AT, and let 

 tangents at T and t meet in the 

 points P and p any other given 

 line MN, straight or curved, and 



draw Vq perpendicular to tp ; then the difference between the arc 

 AT and the tangent TP will exceed or fall short of the difference be- 

 tween the arc At and the tangent tp by a quantity which is ulti- 

 mately to pq in a ratio of equality. 



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

 mately equal to the sum of pq, VT, and V^, (V being the intersection 

 of pt and PT produced) ; and T^, or the increment of the arc AT, i.s 

 ultimately equal to the sum of VT and Yt ; therefore the difference 

 of the increments is ultimately equal to pq. Whence the proposition 

 is manifest. 



