_L_ = ((_d__{ 2} − _d__{ 1}) − (_H__{ 1} − _H__{ 2}))⁄(_S_ − S_{ 1}) = (_d_′ − _H_′)⁄_S_′,
in which _L_ = the distance between cross-sections;
_d__{ 1} = the depth of flow at the lower section;
_d__{ 2} = the depth of flow at the upper section;
_H__{ 1} = the velocity head at the lower section;
_H__{ 2} = the velocity head at the upper section;
_S_ = the hydraulic slope of the stream surface;
_S__{ 1} = the slope of the invert of the sewer.
In order to solve such problems with a satisfactory degree of accuracy the difference between _d__{ 1} and _d__{ 2} should be taken sufficiently small to divide the entire length of the sewer to be investigated into a large number of sections. The solution of the problem requires the determination of the wetted area, the hydraulic radius, and other hydraulic elements at many sections. The labor involved can be simplified by the use of diagrams, such as Fig. 19, or by specially prepared diagrams such as those accompanying the original article by C. D. Hill. The solution of the problem can be simplified by tabulating the computations as follows:
┌───────────────────────────────────────────────────────────────────────┐ │ _D_ = _Q_ = _A_ = _V_ = │ │ _Q_⁄_A_ = _S__{ 1} = _L_ = (_d__{ 1} − _H__{ 1})⁄_S__{ 1} │ ├───┬───┬───────┬───┬───┬───────┬───────┬───┬───┬───────┬───┬─────┬─────┤ │ 1 │ 2 │ 3 │ 4 │ 5 │ 6 │ 7 │ 8 │ 9 │ 10 │11 │ 12 │ 13 │ ├───┴───┴───────┼───┼───┼───────┼───────┼───┼───┼───────┼───┼─────┴─────┤ │ Depth │_R_│_H_│_H__{ 1}│_d__{ 1}│_V_│_S_│_S__{ 1}│_L_│ Elevation │ │ │ │ │ │ − │ │ │ │ │ │ │ │ │ │ │_H__{ 1}│ │ │ │ │ │ ├───┬───┬───────┼───┼───┼───────┼───────┼───┼───┼───────┼───┼─────┬─────┤ │_D_│_d_│_d__{ 1}│ │ │ │ │ │ │ │ │Sewer│W. L.│ ├───┼───┼───────┼───┼───┼───────┼───────┼───┼───┼───────┼───┼─────┼─────┤ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├───┼───┼───────┼───┼───┼───────┼───────┼───┼───┼───────┼───┼─────┼─────┤ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├───┼───┼───────┼───┼───┼───────┼───────┼───┼───┼───────┼───┼─────┼─────┤ │ │ │ │ │ │ │ │ │ │ │ │ │ │
At the head of the computation sheet should be recorded the diameter of the sewer in feet, the assumed volume of flow, the area of the full cross-section of the sewer, the velocity of the assumed volume flowing through the full bore of the sewer, and the gradient or slope of the invert. In the 1st column enter the assumed depth in decimal parts of the diameter for each cross-section; in the 2nd column enter the same depth in feet; in the 3rd column enter the difference in feet between the successive cross-sections; in the 4th column enter the hydraulic radius corresponding to the depth at each cross-section; in the 8th column enter the velocity, equal to the volume divided by the wetted area, for each cross-section; in the 5th column enter the corresponding velocity head; in the 6th column enter the difference between the velocity heads at successive cross-sections; in the 7th column enter the difference between the quantities in the third and in the sixth columns; in the 9th column enter the hydraulic slope corresponding to the velocity and hydraulic radius of each cross-section; in the 10th column enter the difference between the hydraulic slope and the slope or gradient of the sewer; in the 11th column enter the computed distance between successive cross-sections; in the 12th column enter the elevation of the bottom of the sewer at each cross-section; and in the 13th column enter the corresponding elevation of the surface of the water.