268 On t/i£ Problem of the In-and-circumscribed Triangle, 

 these tangents meet V, we must determine A, so that 



may represent right lines, and we get 



where F has the meaning given in the last Number, and ' Conies/ 

 p. 268. 



The condition that one of these lines should touch U is 



(2AU'+X0)'^=4A(AU'2 + X(©U' + AV) + ©'X^), 



which reduces to 



4A2V' + X(4A(H)'-02)=o. 



Eliminating \, the result becomes divisible by V, and is 

 16A3A'V + 4A(02_4A©')I' + (®^-4A©')2U=O, 



as was found before. 



It is obvious that the same method would readily give the 

 equation of the locus, if the third side instead of touching U 

 touch a new conic W. If the three sides touch U, one vertex 

 moves on V and another on a right line L, then the equation of 

 the locus of the third vertex is 



(A0-MU)2=AU>|r2, 

 where (^ is the result of elimination between the equations L, V, 



and X —. — 1- V —, — I- z —r- ', M is the condition that the pole of the 

 dx ^ dy dz 



line L with respect to U should lie on V ; and •x/r is the equation 



of the polar with respect to V of the pole of L with respect to U. 



I have formed the equation of the locus when two vertices 



move on conies, and the three sides touch the conic z'^-\-2xy; 



but I do not perceive the rule for forming the equation in the 



general case, except that I can verify (what Mr. Cayley has proved 



geometrically) that the equation is of the form ^^4- Uijfr=0, where 



, . , , V • .• u . dV d\J' dV , 



is obtamed by elimmatmg between x—j — I- y —^ — |- z -5— and 



the equations of the conies traversed by the two base angles. 



If the three vertices move on V, the three sides may touch 

 respectively the conies U + «V, U + 6V, U + cV, provided that 

 a, b, c are connected by the relation 



{e-{ab + ac + bc)A'}^={2®' + 2{a + b + c)A'}{2A + 2abcA'}. 



The problem to find the locus of the vertex of a quadrilateral, 

 whose other three vertices move on V, and whose sides touch U, 

 reduces itself to a problem noticed already. 



I use the following abbreviations : — 



4AA' = a; @2_4^@»_^. 2Aa + 0/3=y. 



