218 Mr. J. J. Sylvester on a new Class of Theorems. 



What is the meaning of all three roots of this determinant 

 becoming equal, i. e. of only one conoid being capable of being 

 drawn through the intersection of U and V to touch the plane 



Ix + my + nz -\-pt ? 



Evidently {ex vi analogize) that this plane shall pass through 

 three consecutive points of the curve of intersection, i, e* that 

 it shall be the osculating plane to the curve. 



If we return to the intersection of two co-planar conies, and 

 if we suppose a line to be drawn through two of the points of 

 intersection, the conies capable of being drawn through the 

 four points of intersection to touch the line, besides becoming 

 coincident, evidently degenerate each into a pair of right lines. 

 It would seem, therefore, by analogy, that if a plane be drawn 

 including any two tangent lines to the curve of intersection of 

 two surfaces of the second degree, this should be touched by two 

 coincident cones drawn through the curve of intersection, and 

 consequently every such double tangent plane to the intersec- 

 tion of two conoids (and it is evident that one or more of these 

 can be taken at every point of the curve) must pass through 

 one of the vertices of the four cones in which the intersection 

 may always be considered to lie ; and it would appear from 

 this, that in general four double tangent planes admit of being 

 drawn to the curve, which is the intersection of two conoids, 

 at each point thereof. At particular points a tangent plane 

 may be drawn passing through more than one of the vertices, 

 and then of course the number of double tangent planes that 

 can be drawn will be lessened. These results, indicated by 

 analogy, become immediately apparent on considering the 

 curve in question as traced upon any one of the four containing 

 cones. For the plane drawn through a tangent at any point, 

 and the vertex of the cone being a tangent plane to the cone, 

 must evidently touch the curve again where it meets it. We 

 thus have an additional confirmation of the analogy between 

 a poi t of intersection of two curves and the tangent at any 

 point of the intersection of two surfaces. 



I might extend the analytical theorems which have been esta- 

 blished for functions of three and four to functions of a greater 

 number of variables ; but enough has been done to point out 

 the path to a new and interesting class of theorems at once in 

 elimination and in geometry, which is all that I have at present 

 leisure or the disposition to undertake. 



26 Lincoln's-Inn-Fields, 

 August 13, 1850. 



