On a Porismatic Property of two Conies. 439 



Using W to denote V — U, and (W) to denote what W be- 

 comes when ey is substituted for t, we see that W and (W) 

 are of the respective forms y 2 -\-tw and y9 ; showing that the 

 former is the characteristic of a cone which will be cut by any 

 plane t — ey drawn through the line (t, y) in a pair of right 

 lines; or, in other words, that one of the cones containing the 

 intersection of the two variable conoids (V and U) will have 

 its vertex in the invariable line which is the common tangent 

 to the two fixed conies : this proves the theorem stated by me 

 hypothetically in a foot note in one of my papers in the last 

 Number of the Magazine. The steps of the geometrical proof 

 there hinted at are as follows. 



The four consecutive points in which the two conies inter- 

 sect will be consecutive points in the curve of intersection of 

 the two variable conoids. This curve lies in each of four 

 cones of the second degree. Every double tangent plane to 

 it passes through the vertex of one amongst these. The plane 

 containing four, i. e. two (consecutive) pairs of consecutive 

 points,is a double tangent plane, and will therefore pass through 

 a vertex; but four consecutive points of a curve of the fourth 

 order described upon a cone, and lying in one tangent plane 

 thereto, can only be conceived generally as disposed in the form 

 of an /, of which the belly part will point to the vertex ; or, 



in other words, at any point where two consecutive osculating 

 planes coincide so that the spherical curvature vanishes, the 

 linear curvature will also vanish, i. e. there will be a point of in- 

 flexion at which, of course, the tangent line must pass through 

 the vertex of the cone. This is the assumption felt to be true, 

 but stated by me hypothetically in the paper referred to, be- 

 cause a ready demonstration did not at the moment occur to 

 me. The legitimacy of this inference is now vindicated by 

 the above analytical demonstration. 



The methods of general and correlative coordinates and of 

 determinants combined possess a perfectly irresistible force (to 

 which I can only compare that of the steam-hammer in the 

 physical world) for bringing under the grasp of intuitive 

 perception the most complicated and refractory forms of geo- 

 metrical truth. 



26 Lincoln's-Inn-Fields, 

 October 30, 1850. 



