Mr. Stubbs on a new Method in Geometry. 341 



the other extremity of the diameter passing through the pole 

 is a circle passing through the cusp or pole and touching the 

 curve at the opposite point, and consequently the locus of its 

 centre is a circle. 



In a conic section, if a point be taken inside the curve, and 

 any chord be drawn, if we join the points in which it meets 

 the curve with the focus, and also the given point with the 

 focus, the product of the tangents of the half angles formed 

 by those lines at the focus is constant; hence by inversion, if 

 through a fixed point outside an inverse focal conic section we 

 describe a polar circle, and join the points where it meets the 

 curve with the pole, and also the given point with the pole, 

 the product of the tangents of the half angles is constant. 



In a conic section, if a chord subtend a constant angle at 

 the focus, the envelope of the chord is a conic section with 

 same focus and directrix ; hence by inversion, if the arc 

 of a polar circle contained between the points where it cuts 

 the curve subtends a constant angle at the pole of a focal 

 inverse conic, the envelope of this circle is an inverse focal 

 conic with same pole and circular directrix. 



If three tangents be drawn to a parabola, so as to form a 

 triangle, the three angles and focus are in a circle ; by inver- 

 sion, if three circles be drawn through the pole of a cardioide 

 touching the cardioide, the points of intersection are in a right 

 line. 



Every property, in fact, of a curve, with respect to any pole, 

 has its analogous property in the inverse curve with respect 

 to the same pole ; to an asymptote in one, correspondsa circle 

 passing through the pole and having its tangent at that point 

 parallel to the asymptote, which the curve tends to approach 

 as the radius diminishes ; to a point of inflexion in one curve 

 corresponds the property of the osculating circle at the conju- 

 gate or inverse point, passing through the pole; to a tangent 

 in one corresponds a polar circle tangent to the other at the 

 inverse point ; to a cusp corresponds a cusp, and the osculating 

 circle of the inverse curve is the inverse of that of the direct 

 curve ; so from the known properties of curves we can find 

 the singular points of their inverse curves. 



I shall not dwell any longer on those properties, as they 

 are all obvious when the principle is explained. I shall merely 

 show what Pascal's celebrated Theorem of the Hexagon in- 

 scribed in a conic section* becomes by inversion. 



If in any inverse conic section six points be taken and six 

 polar circles be described through each two consecutive points 

 and the pole, the intersections of each opposite pair lie in a 

 [* See Phil. Mag. S. 3. vol. xxii. p. 168.— Edit.] 



