Hydraulic Investigations. \ 25 



It was therefore necessary for my purpose to substitute, for 

 the formulae of Mr. Dubuat, others of a totally different 

 nature; and I could follow Dubuat in nothing but in his 

 general mode of considering a part of the pressure, or of the 

 height of a given reservoir, as employed in overcoming the 

 friction of the pipe through which the water flows out of it; 

 a principle, which, if not of his original invention, was 

 certainly first reduced by him into a practical form. By 

 comparing the experiments, which he has collected, with 

 some of Gerstner's, and some of my own, I have ultimately 

 discovered a formula, which appears to agree fully as welL 

 as Dubuat's with the experiments from which his rules were 

 deduced, which accords better with Gerstner's experiments, 

 which extends to all the extreme cases with equal accuracy, 

 which seems to represent more simply the actual operation 

 of the forces concerned, and which is direct in its applica- 

 tion to practice, without the necessity of any successive ap- 

 proximations. 



I began by examining the velocities of the water discharged 

 through pipes of a given diameter with different degrees of 

 pressure; and I found, that the friction could not be repre- 

 sented by any single power of the velocity, although it fre- 

 quently approached to the proportion of that power of 

 which the exponent is 1*8 5 but that it appeared to consist 

 of two parts, the one varying simply as the velocity, the 

 other as its square. The proportion of these parts to each 

 other must however be considered as different in pipes of 

 different diameters, the first part being less perceptible in 

 very large pipes, or in rivers, but becoming greater than the 

 second in very minute tubes; while the second also becomes 

 greater for each given portion of the internal surface of the 

 pipe, as the diameter is diminished. 



If we express, in the first place, ail the measures in 

 French inches, calling the height employed in overcoming 

 the frictionyj the velocity in a second v, the diameter of the 



pipe d, and its length /, we may make f~ a-rtf 3 - -f 2 c : .v j 



for it is obvious, that the friction must be directly as the 

 length of the pipe; and since the pressure is proportional 



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