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XXIII. Note on a Theorem relating to a Triangle, Line, and Conic. 

 By A. Cayley, Esq.* 



I FIND, among my papers headed " Generalization of a Theo- 

 rem of Steiner's," an investigation leading to the following 

 theorem, viz. : — 



Consider a triangle, a line, and a conic ; with each vertex of 

 the triangle join the point of intersection of the line with the 

 polar of the same vertex in regard to the conic ; in order that 

 the three joining lines may meet in a point, the line must be a 

 tangent to a curve of the third class; if, however, the conic 

 break up into a pair of lines, or in a certain other case, the curve 

 of the third class will break up into a point, and a conic inscribed 

 in the triangle. 



Let the equations of the sides of the triangle be 

 x=zO, 2/ = 0, z=0, 

 the equation of the conic 



and that of the line 



\x + fiy + vz =0 ; 



then the polar of the vertex (y=0, z=0) has for its equation 



acc + hy+gz=0; 

 it therefore meets the line \x + fiy + vz=0 in the point 



x : y : z-=.hv—gn> \g\—av : a/j, — h\, 

 and the equation of the line joining this point with the vertex 

 (y = 0, 2 = 0) is (afjL—h\)y=(g\—av)z. And the equations of 

 the three joining lines therefore are 



(afj,-h\)y={g\—av)z, 

 (bv—ffi) z=(hfM—b\)cc, 

 (c\-gv) z = {fv-cn)y, 

 lines which will meet in a point if 



{ af jL-h\){bv -fr){c\-gv) - {g\-av){hfi-b\) (fi-cfi) =0 

 or, multiplying out and putting as usual 



K = abc-aP-bg*-ch* + 2fgh, 

 % = bc-p &c, 



if 



2(abc-fgh)\fiv ^ 



that is, the line must touch a curve of the third class. 

 * Communicated by the Author. 



