DEVELOPMENT OF FORMULAE FOR FLOW , 173 



The index x, the root n, and the coefficient C depend on the 

 nature of the surface of the channel. For brick sewers in good 

 condition the value of the index x is .61, n is 2, and C is .00746. 

 For cement plaster x is .67, n is 1.74, and C is .004. A dia- 

 gram given at the end of the book on " Sewage Disposal," 

 by Mr. Robinson, allows velocities and discharges to be read 

 directly. 



The other English formula, Santo Crimp's, is given as 



the same meaning being given to the letters, except that S 

 is the fall divided by the length. It would seem that, with 

 no possible variation for the coefficient, this formula could 

 not equal the others in accuracy, though it is suitable for 

 conditions similar to those upon which it was founded, that 

 is in brick sewers. 



With the use of logarithmic plotting, a new interpreta- 

 tion of old data was possible and it remains to refer to two 

 exponential equations based on the old experiments but made 

 possible only by the aid of logarithmic paper. 



Sullivan's formula, presented to the world by Mr. M. E. 

 Sullivan of Denver, Colorado, in a small book called " The New 

 Hydraulics," expresses the relation between the three varia- 

 bles in the form v = CR' 75 S' 5Q in which C varies only as the 

 roughness and as nothing else, and so is absolutely, accord- 

 ing to Mr. Sullivan, a correct index of the roughness. For 

 cast-iron and for terra-cotta pipe, a numerical value of 141 is 

 advised for C, and for brick conduits a value of 120. 



Messrs. Williams and Hazen soon after produced their 

 little book, explaining their slide rule for solving the various 

 problems of water-flow, and deriving their exponential equa- 

 tion which has the form v = CR'* 3 S' 5 *. They say that if expo- 

 nents could be selected agreeing perfectly with the facts, the 

 value of C would depend on the roughness only, and for any 

 degree of roughness C would then be a true constant. They 



