Elements of analysis as applied to the mechanics of engineering and machinery . verse occurs. If, in the first case, the exponent n be assumed as becoming lessand less and finally vanishing, or approximating zero, the ordinateswill approximate more and more to the constant value y = x^ = 1,and the corresponding curves above AX^ to the broken line ANF-^X^;but if, in the second case, the exponent 7i become greater and greater,the ordinates will gradually aj^proxim^ate to the limiting value y = x^= SCO = 00, whilst the abscissas will gradually approximate to thelimit X =y^^1] and hence, the corre
Elements of analysis as applied to the mechanics of engineering and machinery . verse occurs. If, in the first case, the exponent n be assumed as becoming lessand less and finally vanishing, or approximating zero, the ordinateswill approximate more and more to the constant value y = x^ = 1,and the corresponding curves above AX^ to the broken line ANF-^X^;but if, in the second case, the exponent 7i become greater and greater,the ordinates will gradually aj^proxim^ate to the limiting value y = x^= SCO = 00, whilst the abscissas will gradually approximate to thelimit X =y^^1] and hence, the corresponding curves wiU approachnearer and nearer to the broken line A MP^ Z^. 12 ELEMENTS OF ANALYSIS. [Art. 9. If we assume n-= — 1, tlius putting y = x ^ = —, we shall have for X 0; and we have therefore 0, ^ = 00, and for x = oo^ y to consider a curve 1 P^ 1 (treated of in Art. 3, and illustrated inFig. 5,) which always approaches the axis of ordinates on the oneside, and the axis of abscissas on the other, but without ever reach-ing either the one or the other. Fig. If the exponent (— n) of the function y =z x ^ =—-^ be a proper 1 1 fraction, we shall have for ^ <^ 1, y <C — and for ^ ^ 1, ly ^ —; but if this exponent be greater than unity, we shall have for x > — , and for x ^ 1, y <C - • Therefore, the curves correspond-ing to the function y = x^ run, at the beginning, below or above,and afterwards, from the point P, above or below the curve y = x~^ = -- -, according as n is less or greater than unity. Whilst generallythe curves corresponding to the positive values of n run first below, [Art. 9. ELEMENTS OF ANALYSIS. 13 and ftom P^^ above, the straight line X^ X^. those proceeding fromnegative exponents (— n)\ pass first above, and from P^, below, theline X^ IX^. In the former curves, we have for ^ = 0, also y = 0^and for ^ = oo, also y = cc ] whilst in the latter, we have for ^ = 0,^ = oo, and for ^ = go, ?/ = 0. If the former depart more
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