An elementary treatise on curve tracing . F. ASYMPTOTES NOT PARALLEL TO THE AXES 97 PLATEThe cross asymptotes do not meet the curve except at an v. infinite distance; that parallel to Oy cuts the curve where 10?/=-27a. Also (0, — ^a) and (fa, 0) are points in the curve, and nearthe origin 3x^ = 4<ay. These considerations are sufficient to give the form of thecurve. Fig. 8. 131. These methods of obtaining the cross asymptotesare, I think, the best to use in almost any case, when thepractical application is well understood; but it is obviousthat a more direct method of approximation must some
An elementary treatise on curve tracing . F. ASYMPTOTES NOT PARALLEL TO THE AXES 97 PLATEThe cross asymptotes do not meet the curve except at an v. infinite distance; that parallel to Oy cuts the curve where 10?/=-27a. Also (0, — ^a) and (fa, 0) are points in the curve, and nearthe origin 3x^ = 4<ay. These considerations are sufficient to give the form of thecurve. Fig. 8. 131. These methods of obtaining the cross asymptotesare, I think, the best to use in almost any case, when thepractical application is well understood; but it is obviousthat a more direct method of approximation must some-times have the advantage. Thus, when y can be expressed explicitly in terms of x,it can be expanded in descending powers of x, as was donein Art. 127 (Ex. 1). Take, for instance, the following curve : x(x + l)y = (x2 + x+l)(x-2), or x{x-{-l){y — x + 2) = x — 2; .-. y-x + 2 = x-\x-2)/(x-\-l) = x-^, when x and y are both infinite; the asymptotes parallel toOy are x = 0 and — 1; near (0, co),x= —2y-^; near (— 1, oo ),x + l = Sy-^\
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