If the values of x and y are tabulated in columns, and their logarithms X and Y are looked up and written in parallel columns opposite, then the points (X, Y) should lie on a straight line to justify the assumption of equation (16). And if they do lie fairly on a line, its slope and y-intercept determine the constants m and b of equation (16). This can often be done graphically from the drawing with sufficient accuracy, but if greater accuracy is required they can be determined from the data by least squares. EXAMPLE. The points (X, Y) lie nearly on a line BD, Fig. 127. Graphically, we scale off from the figure, B m By least squares, putting the data into equation (19), we find hence = y 1.574x1.914 m = 1.914, In case the quantities x and y are connected by a relation of the form it is advantageous to compute Y = logy and plot x and Y. If these new values when plotted appear to be on a straight line we write and determine k and log c by the method of least squares. 177. Logarithmic Paper. Paper, called logarithmic paper, may be bought that is ruled in lines whose distances, horizontally and vertically, from a point O are proportional to the logarithms of the numbers 1, 2, 3, etc. Such paper may be used instead of actually looking up the logarithms in a table. For if the given values be plotted on this new paper, the resulting figure is identically the same as that obtained by plotting the logarithms of the given values on ordinary squared paper. The use of logarithmic paper is however not essential; it is merely convenient when one has a large number of problems of this type to solve. EXERCISES 1. A strong rubber band stretched under a pull of p kg. shows an elongation of E cm. The following values were found in an experiment: p. 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.1 0.3 0.6 0.9 1.3 1.7 2.2 2.7 3.3 3.9 E Find a relation of the form E = kp". Ans. E = .3p1.6 2. The amount of water A, in cu. ft., that will flow per minute through 100 feet of pipe of diameter d, in inches, with an initial pressure of 50 lbs. per sq. in., is as follows: 3. In testing a gas engine corresponding values of the pressure p, measured in lbs. per sq. ft., and the volume v, in cubic feet, were obtained as follows: V. p. 4. Find a relation between p and v from the following data: 5. The intercollegiate track records for foot-races are as follows, where d means the distance run, and t the record time: 6. In each of the following sets of data find a relation of the form y=kx" connecting the quantities. 8. Find an empirical equation connecting the x and y values given 9. Given age in years and diameter in inches of a tree 11⁄2 feet from the ground as follows. Age. Diameter. 19 13 58 114 140 181 229 7 13.2 17.9 24.5 33 kxn. Plot the data and determine a relation of the form y = Plot the data and determine a relation of the form y = kxn. 11. Following are vapor pressures, in mm. of mercury, of methyl alcohol at various temperatures: Represent these by an empirical formula. 12. The specific gravity of dilute sulphuric acid at different concentrations is given in the following table: Represent these by an empirical formula. 13. The temperature of a heated body, cooling in the air, was taken each minute, the results being tabulated as shown: 0 1 2 3 4 5 6 7 8 9 10 84.9 79.9 75.0 70.7 67.2 64.3 61.9 59.9 57.6 55.6 53.4 The temperature of the air was 20°. Deduce an equation expressing 0 in terms of t. |