Drawing two rectangular axes, and laying down a series of points whose abscissas represent, on any convenient scale, successive values of the sectorial measure, and whose ordinates represent, preferably on the same scale, the corresponding values of the function to be plotted, the locus traced out by this series of points will be a graphical representation of the variation of the function as the sectorial measure varies. The equations of the curves in the ordinary cartesian notation are:
Here is written for the sectorial measure , and for the numerical value of , etc. It is thus to be noted that the variables , are numbers, or ratios, and that the equation merely expresses that the relation between the numbers and is taken to be the same as the relation between a sectorial measure and its characteristic ratio. The numerical values of are given in the tables at the end of this chapter for values of between and . For greater values they may be computed from the developments of Art. 16.
The curves exhibit graphically the relations:
The slope of the curve is given by the equation , showing that it is always positive, and that the curve becomes more nearly vertical as becomes inﬁnite. Its direction of curvature is obtained from , proving that the curve is concave downward when is negative, and upward when is positive. The point of inﬂexion is at the origin, and the inﬂexional tangent bisects the angle between the axes.
The direction of curvature of the locus is given by , and thus the curve is concave downwards or upwards according as is negative or positive. The inﬂexions occur at the points , ; and the slopes of the inﬂexional tangents are .
The curve is asymptotic to both axes, but approaches the axis of more rapidly than it approaches the axis of , for when , is only , but it is not till that is so small as . The curves , cross at the points , .