308 PROCEEDINGS OF SECTION A. 



Case 2. — To construct conic touching give given straight lines. 



Taking three of the given lines as the triangle of reference, assume 

 the equation of the conic touching the five lines to be 2 and let the two 

 remaining lines be 



^ + 1, + ^=° 



a B y 



The two points {o-ilS^y^) (a.^^yi) must lie on L-, hence 



"•o / o /o 



The equation of the line joining (a,/3|yi) {a.jf3,y.,) is — 



a(/3iy2 — f^iyi) + ^(yi^a — y^ai) + y(ai/?2 — (hfti) = '> 



i.e., ^ + g- + I, = 0. 

 "o Ho yo 

 Hence to construct the required conic, find the two points whose axes 

 are two of the given lines with respect to the triangle formed by the 

 remaining three lines. The line joining these points will be the axis of 

 {aoPoyo) with respect to the same triangle. 



Case 3. To construct a parabola touching four given straight lines. 



Taking three of the lines as the triangle of reference, construct the 

 point whose axis with respect to this triangle is the fourth given line. 

 Join this point to the centroid of the triangle of reference : this joining 

 line is the axis of («o/^oyo)- 



Case 4. To construct conic through three given points and touching 

 a given line at a given point. 



Let A B C be the given points and D the point of contact of the 

 tangent DE. Construct the axis of A with respect to the triangle BCD; 

 construct a line making with DB, DC, DE an harmonic pencil. The 

 intersection of the lines so drawn will be the point (ao/^oyo)- 



Case 5. To find conic through a given point and touching two given 

 straight lines at given points. 



Let A be the given point and B and C the points of contact on the 

 tangents DB, DC. Draw BO, so that BD. BC, BOi, BA form an har- 

 monic pencil ; draw CO^ so that CD, CB, CO2, CA form an harmonic 

 pencil. BOi and CO2 will intersect in (ao/^oyo)- 



Case 6. To construct conic through two given points and having a 

 given self-conjugate triangle. 



[If the conic pass through the point (a,^,')',) it will also pass through 

 the points (-ai/3,7i) („, -^jy,) {a^/3^-y^).^ 



If D and E be the two given points, ABC being the self-conjugate 

 triangle, construct the points D, Dj D3 which form with D a standard 

 quadrangle. The axes of ! ) and E with respect to the triangle Dj D2 D3 

 will intersect in the point {a^f^^y^). 



