. Lectures on the differential geometry of curves and surfaces. Fig- i^- Fig. V. The results and the diagi-ams were given first by Darboux. The preceding investigation is the same as Darbouxs, already cited {% 71),in substance though it is formally different in analysis. Darboux refers thesurface to the tangent plane at the umbilicus, so that its equation has theform z = U {a^^ + y-) + i (cu:^ + Shx-t/ + Sbxf + af) + .... The equation for the values of t is bt + (2b - a) t- + (a - 26) t-b=0; and ^^^ ^ _ (1+^1^(1 + ^:^3)(^-^.)(^i-^3) with similar values for vio and m^. 74. It was assumed that w
. Lectures on the differential geometry of curves and surfaces. Fig- i^- Fig. V. The results and the diagi-ams were given first by Darboux. The preceding investigation is the same as Darbouxs, already cited {% 71),in substance though it is formally different in analysis. Darboux refers thesurface to the tangent plane at the umbilicus, so that its equation has theform z = U {a^^ + y-) + i (cu:^ + Shx-t/ + Sbxf + af) + .... The equation for the values of t is bt + (2b - a) t- + (a - 26) t-b=0; and ^^^ ^ _ (1+^1^(1 + ^:^3)(^-^.)(^i-^3) with similar values for vio and m^. 74. It was assumed that wii, vi., m^ were all real. The alternative is thatonly one of them, say m^, is real; then nio and ms are conjugate. As miniums = A,{t,-t.^-{t,-t,y{t,-t,y 100 LINES OF CURVATURE NEAR AN UMBILICUS [CH. IV it follows that rthm^m^ is negative. Now monh is the product of two con-jugate quantities and therefore is positive; hence iih is negative. Then for values of t nearly equal to t^, we have p, q, p-t^q all large,while there is a parabolic asymptote p - qti = Kq
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