OF LOCAL THEOREMS AND PORISMS. 351 



at those ordinates shall intersect each other in a line given in 

 position. 



In the curve ABC, taking any two ordinates, BE, CD, re- 

 quired its equation when the normals at these points intersect 



each other on the axis ; let x, y, and x, y, be the ordinates, 



i i 



then we must have 



AE + EP = AD + DP, 



ydy __ 9*y 



or x + Z-t-2- — x + 



JL 



d x i d x 



if the two points E and D are so chosen, that x s a. x, and if 



i 



a is a periodic function of the second order, we have, sup- 

 posing y — ^ x 



■{x.d^x . ^ « x . d^ ctx 



x — 5 — a. x -\ j 



ax dec x > 



. ^ x . d ^ x ,- — . 



or x + dTx — =*(*,« *) ; 



hence t/ 2 = — x 2 + 2 j d x % (x, ax)- 

 If we make ^ (x, a, x) — x, we have 



y 2 = — * 2 -{- 2 c x -f* c » 



i 



which is the equation to a circle, and it is well known, that 

 this curve possesses the property, since all its normals inter- 

 sect each other in the centre. 



This furnishes us with the following local theorems. Any 

 curve of this family being given, if from any point (P), in the axis 

 of the abscissce, we draw two lines cutting the same branch of the 

 curve perpendicularly {in B and C), and if we prolong the ordi- 



vol. ix. p. ii. y y note 



