46 BULLETIN 852, U. S. DEPARTMENT OF AGRICULTURE. 



In Plate VI the loss of head per thousand feet of pipe, H, for each 

 observation on concrete pipes, is platted on logarithmic paper as an 

 ordinate against the corresponding value of the mean velocity, V, as 

 an abscissa. For a given series of observations the resulting curve 

 represents equation 12 in its logarithmatic form 



log H = logm + z log V (16) 



which is now recognized as the equation of a straight line where m 

 is the intercept on the axis of H (which is the line V— 1) and z is the 

 tangent of the angle between the curve and the axis of V (indicated 

 by a on PL VI). 



A study of this plate, in connection with the descriptions of the 

 various pipes, shows that pipes of the same structural characteristics 

 follow a rather definite order of position on the plot. If all of the 

 curves were of the same inclination to the axis V, then this order of 

 position would be definitely and fully disclosed by a diagram in which 

 the values of m are platted on logarithmic paper as ordinates and the 

 diameters of the pipes, in inches, are platted as abscissas. The 

 experiments upon some of the pipes, of necessity, covered such a 

 short, range of velocities that the curves indicate a slope quite at 

 variance with that which would have probably resulted for more 

 complete series. For this reason the writer has not projected the 

 curves to an intersection with the axis of H, in order to determine the 

 relative positions of the values of m. Obviously some system of 

 weighting should be assigned to the various series and, in the study 

 of experiments upon wood-stave pipe, an arbitrary weighting method 

 was employed; but because there was some criticism of this pro- 

 cedure the writer hesitates to repeat it. 



If all the pipes were made in the same manner, had the same 

 interior surfaces, and were subject to the same hydraulic conditions, 

 a law derived by the method of least squares should be the best and 

 most accurate. This method of handling experimental data, however, 

 ascribes all variation from a given law to errors, according to the 

 probability of errors, whereas most of the variation from an average 

 law of data on commercially made concrete pipes is due to inherent 

 differences in the pipes. 



For any given series of experiments upon the same pipe the method 

 of least squares is applicable, in its simplest form; that is, by the 

 ccntcr-of-gravity method. 1 This method was used in computing 

 the individual formulas given in column 10, Table 4. The curve is 

 represented graphically in Plate VI, where the center of gravity of 

 all the points in any one series is shown as two circles around a 

 center which is typical of the observation points for that particular 

 series. That is, if the observation points arc given as open circles, 



1 Described in Bui. 376, U. S. Dept. Agr., p. 50, and in Amer. Civil Engineers l'ocketbook, 3d ed.,New 

 York, 1916, p. 847. 



