1902-10-01 10.2307/j100069 10 223 10.2307/j100069 222 1902 10.2307/2968883 amermathmont 185 222-223 GEOMETRY. 185. Proposed by W. J. GREENSTREET, M. A., Editor of The Mathematical Gazette, Stroud, England. Given the tangential equations to two conics S, S', find the tangential co-ordinates of the join of the poles of two given parallel lines with respect to S. Deduce the tangent- ial equation of the center of 8, and find that of the intersection of S and S'. Solution by G. B. M. ZERR, A.M., Ph.D., Professor of Chemistry and Physics, The Temple College, Philadel- phia, Pa. Let bc -f2=A, ca-g2- B, ab-h2- G, gh-af=:f, hf-bg=-G, fg-ch-=H. Then ==AA2 +B,I2 + Gv2 +2F,--v+2GvA+2HAi%. Similarly, S=A'A2 +B','2 + G'v2 + 2F,i.v+2 G'sA +2TAYv. Let Aa+.,IP+?sr and Aa+ifL+vr++m be the two given parallel lines; p, q, t and p', q', t' their poles with respect to S. Then for the first line ap+hq+gt_A, hp+bq+ft=Ij_, gp+fq+ct=v. Solving these equations for p, q, t, p-(AA+H, + G1v)/ AI q= (I +BBi+FY )/A, t=-(GA+F,u+GCv)/A , where A -abe+2fgh-af2 -bg2 _ch2. For the second line, ap'+hq' + gt' +J m A, hp'+ bq'+ft'+mrn,u gp'+fq'+et'+m=r. xml-12 p-12 223 p'=[AA+HtL4+Gv-m(A+H+G)]1/A. q'=[ RA+Bt+FY-m(H+B+_F] A . t'=-[GA+Fj.+C-m(G+F+C)]/ A. (p, q, t), (p', q', t') are the tangential co-ordinates of the join of the poles. Let A', B', 0' be the angles of the triangle of reference. The center is the pole of the line at infinity asinA'+AsinB'-trsinG'=O. The tangential co-ordin- ates of the center are obtained by substituting sinA', sinB', sinC' for A, u., v in p, q, t and are S1 =(AsinA'+IsinB'+ Gsin') /A , S2 =(HsinA'+BsinB'+FsinG')/ A, S3 ( GsinA' +FsinB' + Csin0')/ A . The tangential equation of the center is AS, +,'S2 +vS3 =O. Write a + ka' for a, b -+ kb' for b, c+kck' for c, f + kf for f, g+kg' for g, h+ kh' for h in AA2 +BIL 2 + Cv 2 +2FI,v + 2G?vA + 2HAIWi _O. Then the tangential equation of the four points of intersection of S and S' is S+kO+k28' O where k is undetermined, and Ai=(bc'+b'c-2ff)A2 +(ca'+c'a-2gg'),tl2 +(ab'+a'b-2hh')v2 +2 (gh' + g'h-af -a'f ),v+2 (hf +h'f- bg'- b'g) rA + 2(fg' +fg- ch'-o'h) AA. The condition for equal roots for k is 02 =4SS', which is the equation of the four points of intersection. xml-13 p-13 eng 10 Oct., 1902 The American Mathematical Monthly 9 Greenstreet W. J. Zerr G. B. M. 1 10.2307/i349203 research-article 00029890