Vol. 10 No 5 - Pi Mu Epsilon

399 PI MU EPSILON JOURNAL FROHLIGER, TUGGLE, VAN SISTINE, PARTICULAR POLARITY 400(See Figure 6.) At the point P on C with coordinates (t, fft)) the line 1 withequation y = fJ(t)x - (tfJ(t) - St)) is tangent to the graph. Under ourcorrelation, 1 is related to a point X, whose coordinates are (f '(t), t f '(t)- f(t)).Let C' be the set of all of these points Xi. That is, C' is the curve parameterizedby the equationsx = f '(I)y = tfl(t) - St).We will call C' the dual of C under this polarity. Notice that, at the point(x, y) = (f '(t), t f '(t) - f(t)), a tangent line 1 ' has slopedy _ dy/dt - tfl'(t) + fJ(t) - fJ(t) - ---- - t and equation y = tx - St). (Thisdx dddt f "(t)Figure 6also holds at those points where f "(t) = 0 since f " is continuous with isolatedzoos.) Under the correlation this X i is the original point P(t, Rt)) on C. Usingthe chain rule again, we see that if fl'(t) * 0 then, on C',We could have started by assuming that C was a curve parameterized by afinBtion gxh: I - RxR for some open interval I. We can compute and - d^dx dx 2in terms of the parameter. As long as we have die condition that, on C,dyd^dxdx2- is continuous and - has isolated zeros and discontinuities, we willachieve the same relationship between C and C' as the one described above.Therefore, it follows that C is also the dual of C' under the polarity.Figure 7