=36. Solution of Formulas.=—The solution of even the simplest of these formulas, such as Flamant’s, is laborious because of the exponents involved. Darcy’s and Kutter’s formulas are even more cumbersome because of the character of the coefficient. The labor involved in the solution of these formulas has resulted in the development of a number of diagrams and other short cuts. Since each formula involves three or more variables it cannot be represented by a single straight line on rectangular coordinate paper. The simplest form of diagram for the solution of three or more variables is the nomograph, an example of which is shown in Fig. 13 for the solution of Flamant’s formula. A straight-edge placed on any two points of the scales of two different vertical lines will cross the other line at a point on the scale corresponding to its correct value in the formula. Such a diagram is in common use for the solution of problems for the flow of water in cast-iron pipe. [Illustration: FIG. 13.—Diagram for the Solution of Flamant’s Formula for the Flow of Water in Cast-iron Pipe. ] Fig. 14 has been prepared to simplify the solution of Hazen and Williams’ formula. The scales of slope for different classes of material are shown on vertical lines to the left of the slope line. For use these scales must be projected horizontally on the slope line. The scales for other factors are shown on independent reference lines. For example let it be required to find the loss of head in a 12 inch pipe carrying 1 cubic foot per second when the coefficient of roughness is 100. A straight-edge placed at 1.0 cubic feet per second on the quantity scale, and 12 inches on the diameter scale crosses the slope line at .00092 opposite the slope scale for _c_ = 100. It crosses the velocity line at 1.31 feet per second. Kutter’s formula is the most commonly used for sewer design and has been generally accepted as a standard in spite of its cumbersomeness. Fig. 15 is a graphical solution of Kutter’s formula for small pipes, and Fig. 16 for larger pipes. The diagrams are drawn on the nomographic principle and give solutions for a wide range of materials, but they are specially prepared for the solution of problems in which _n_ = .015. In their preparation the effect of the slope on the coefficient has been neglected. Fig. 17 is drawn on ordinary rectangular coordinate paper and can be used only for the solution of problems in which _n_ = .015. Both diagrams are given for practice in the use of the different types. [Illustration: FIG. 14.—Diagram for the Solution of Hazen and Williams’ Formula. ] [Illustration: FIG. 15.—Diagram for the Solution of Kutter’s Formula. For values of _n_ between 0.010 and 0.020. Specially arranged for _n_ = 0.015. Values of Q from 0.1 to 10 second-feet. ] [Illustration: FIG. 16.—Diagram for the Solution of Kutter’s Formula.