Practical Curve Tracing with Chapters on Differentiation and Integration (Paperback)

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. 1910 Excerpt: ... some slight degree of difficulty. If the scale of X is chosen such that the scales of x and of log y are equal--that is, if the length of one sheet of the paper (which is square) in the X direction is taken equal to unity (i.e. to log 10)--the slope may be measured, as in the case of logarithmic paper, by measuring the vertical and horizontal distances to any, the same, scale of equal parts. If, however, any other scale is taken for x, the slope obtained by measuring the vertical and horizontal dimensions to the same scale of equal parts must be divided by the scale value of the length of one sheet of paper. Thus, in Fig. 39, the scale value of the X axis on one sheet is 0-5. Therefore measuring PM and QM to any scale of equal parts we have--PM 4-80 which gives the same equation for the curve as was derived by the other method in the last paragraph. The Logarithmic Curves.--At the beginning of the chapter we referred to the logarithmic family of curves, given by the general equation y = a. log bx, and showed that they were of precisely the same nature as the exponential curves, with the axes interchanged; that is, the curves will be asymptotic to the negative direction of the axis of Y, and will rise steadily as x increases, being contained wholly in the fourth and first quadrants (a being positive), intersecting the axis of X at the point x = j-, y = 0, whatever the value of a. This may be seen by putting y = 0 in the equation above, when we have--a. log bx = 0 i.e. log bx = 0 or bx = 1 1 x=b As an example of this family we will plot the curve y = 3 log10 2x The necessary points are calculated in the table below, and the curve is plotted in Fig. 40. To find the equation of a curve of this type it is obvious that y must be plotted with log x, or x and y mu...