SIMSON. 503 



lines from two given points to the circumference of an 

 ellipse, the sum of which lines shall be equal to a 

 given line, the solution will give four lines, two on each 

 side of the transverse axis. But in one case there will be 

 innumerable lines which answer the conditions, namely, 

 when the two points are in the axis, and so situated 

 that the distance of each of them from the farthest ex- 

 tremity of the axis is equal to the given line, the points 

 being the foci of the ellipse. It is, then, a porism to 

 affirm that an ellipse being given, two points may be 

 found such that if from them be inflected lines to any 

 point whatever of the curve, their sum shall be equal 

 to a straight line which may be found ; and so of the 

 Cassinian curve, in which the rectangle under the in- 

 flected lines is given. In like manner if it be sought in 

 the cubic hyperbola (y oe~=oc ) to inflect from two 

 given points in a given straight line, two lines to a 

 point in the curve, so that the tangent to that point 

 shall, with the two points and the ordinate, cut the 

 given line in harmonical ratio ; this, which is only 

 capable of one solution in ordinary cases, becomes 

 capable of an infinite number when the two points 

 are in the axis, one of them the curve's apex, and the 

 other at the distance equal to the given line a from 

 the apex ; for in that case every tangent that can be 

 drawn, and every ordinate, cut the given line har- 

 monically with the curve itself and the given point.* 



* This curve has many curious and elegant properties : for ex- 

 ample All the lines Avhich can be drawn in every direction from 

 any point out of the curve are cut harmonically by the tangent, 

 the ordinate, and the lines joining the two given points. This 



