Mution of 

 Water in 

 Pipes and 



Canals. 



530 HYDRODYNAMICS. 



If we divide by U both sides of the equation 

 '=* U+S U*, we obtain ^-~-. <*+/3 U, and put 



a tl 



ting 2 jj = y, we have an equation of the first order 



v=a+/3U, in which all the quantities are linear ex- 

 cept ft, which is an abstract number. 



By calculating as many values of y corresponding to 

 the determined values of U, as we have experiments on 

 the velocity observed in canals, where , *, a, and ;<; 

 have been measured, and by finding and /3 by the 

 methods already mentioned, we obtain an expression of 

 the velocity. 



M. Prony has applied these methods to the twelve 

 experiments, froin which Girard deduced the value of 

 ' coefficient R in his formula, chiefly with the view 

 of comparing the results obtained by Du Buat's for- 

 mula, Girard's Ibrmula, and his own formula. Two of 



these twelve experiments were made by M. Chezy upon 

 the Rigole of Courpalet, and upon the Seine. The 

 other ten are taken from Du Buat's work, and are those 

 which Girard employed. From these experiments, 

 Prony finds * = 0.00093, and 0=0.00266. Hence we 

 obtain 



^2 = 0.00093 U+ 0.00266 U s , 



3687,52 



which, when reduced, gives 



U= 0,174812 + v/ (o.0305592 + 



* ** XT i 



The particulars of the twelve experiments are given in 

 the following Table, in columns 1, 2, 3, 4, 5, 6. Co- 

 lumn 7 is calculated from Du Buat's formula already 

 given, column 8 from Girard's, and column 9 from the 

 preceding formula of Prony's. The four experiments 

 marked with an asterisk are rejected as anomalous. 



TABLE L 



Containing the mean Velocities of Currents of Water deduced from eight Experiments, compared with the 

 Velocities as calculated by the Formula of Du Buat, Girard, and Prony. 



In the preceding Table, the mean velocities in co- 

 lumn 2 were not directly observed, but were deduced 

 from the superficial velocities by a formula of Du Buat, 



vz. 



In this formula, which is reduced to metres, U is the 

 mean velocity, and V the superficial velocity. Girard 

 slso calculated his mean velocities by an equivalent for- 

 mula. 



The relative accuracies of the three formulae will be 

 seen from the following Table of differences. 



Absolute differences between 



the calculated and observed 



mean velocity. 



Positive Negative 



difference. difference. 



Formula of Du Buat ..... 0.0338 0.0391 

 Formula of Girard ..... 0.0238 0.0970 

 Formula of Prony ..... 0.0198 0.0060 



In Du Buat's formula, the errors are between T |,y 

 and iVo- of the observed results; in Girard's they are 



between ^/ s and ^ ; and in Prony's between T g. 

 and T^g-. The great superiority of Prony's formula is 

 therefore manifest. 



As the preceding formula of Prony was drawn on- 

 ly from a few observations, for the purpose of compa- 

 ring it with the other formula, he has deduced more 

 correct values of * and /3 from 31 experiments, including 

 the eight experiments of Du Buat in Table I. The 23 

 new experiments were performed with very great accu- 

 racy upon artificial canals, and have the advantage of 

 giving the mean velocity from direct observation. 



These experiments, which are contained in Table II. 

 give the following values of <*and /3, viz. 



K 0.000436, /3- 0.003034, from which we obtain 



U=-0.07IS5234V(0.00516275 + ' 

 Or more simply, 



(<JOQ<J 5>v 

 0.005+-^- * Y which will be suf- 

 ficiently exact. With the first of these formula;, the 

 numbers in column 8 of Table II. were calculated. 



