This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1870 Excerpt: ...which, if c be positive, becomes a positive constant, and is of an even order; hence, y = b is a minimum ordinate. Art. (70). Since for x = a, we have = 0, the tangent line at the extremity of this minimum ordinate is parallel to the axis of X; and since (m and m--2 being even) for all values of x except x = a, d'y y ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1870 Excerpt: ...which, if c be positive, becomes a positive constant, and is of an even order; hence, y = b is a minimum ordinate. Art. (70). Since for x = a, we have = 0, the tangent line at the extremity of this minimum ordinate is parallel to the axis of X; and since (m and m--2 being even) for all values of x except x = a, d'y y and are positive, the curve at all of its points is convex towards the axis of X, Art. (8G). If c be negative; the with differential coefficient will be negative; and x = a and y = b will be the co-ordinates of a point at which the ordinate is a maximum. In this case, the second differential coefficient for all values of x, except x = a, is negative, and the curve, for all positive values of y, concave, and for all negative values of y, convex, towards the axis of X. 118. Second: Let m be entire and odd. In this case, each value of x gives one real value for y; and each value of y, a real value for x; hence, the curve is unlimited in either direction. When x = a, the first differential coefficient, as before, is equal to 0; as also the second, third, fec. The wth, however, if c be positive, is a positive constant, and of an odd order; there is then in this case, neither a maximum nor a minimum, Art, (TO). By examining the second differential coefficient, we see (since m--2 is odd), that for every value of x a, it is negative; that for x = a, it is 0; and when x a, it is positive: hence, for all values of x a, which give y positive, the curve is concave towards the axis of X; and for all values of x a, it is convex, as in the figure. Therefore, at the point whose co-ordinates are x = a and y = b, as x increases, the curve changes from being concave, and becomes convex, towards the axis of X. If c be negative; the reverse will be the case, and as...
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