126 tiydraulic Investigations* 



to the area of the section, and the surface producing the 

 friction to its circumference or diameter, the relative mag- 

 nitude of the friction must also he inversely as the diameter, 

 or nearly 50, as Dubuat has justly observed. We shall then 



find that a must be -boooool A30 i£?2 - l44 ° - 18 ° ") > 



and c = -0000001 ( ^°° dd +_L / ]0 50 + ^ + - 9 )Y 



Hence it is easy to calculate the velocity for any given pipe 

 or river, and with any given head of water. For the height 

 required for producing the velocity, independently of friC- 

 tion, is, according to Dubuat,- _, or rather, as tt appears 

 from almost all the experiments which I have compared, 



: and the whole heisrht h is therefore equal to /' -f , 

 550 ?.'• 1 J 55Q 



or A =r / -j \i--h --v ; and man in 2; b = 



\d 550/ d ' ° al:d+ 00182' 



7 •/ 



and e = - , y* -f 2ey == M, whence v = ^/ (£/z 4- e 1 )— c* 

 a 



In order to adapt this formula to the case of rivers, we 



must make I infinite,* then b becomes and hh = — « 



al a I 



d* .... ' 



= — -, $ being the sine of the inclination, and d four timer, 



the hvdraulic mean depth ; and since e is here = — , v =» 



a 



?— 1 1—1 x and in most rivers, v becomes nearly 



y {20000 ds). f 



In order to show the agreement of these formula? with 

 the result of observation, I have extracted, as indiscrimi- 

 nately and impartially as possible, forty of the experiments 

 made and collected by Dubuat ; I have added to these some 

 of Gerstner's, with a few of my own • and I have compared 

 the results of these experiments with Dubuat's calculations, 

 and with my own formula?, in separate columns. There are 

 six of Dubuat's experiments, which he has rejected as irre- 

 gular, apparently without any very sufficient reason, since 

 he has accidentally mentioned, that some of them were made 

 with great care; I have therefore calculated the velocities for 

 thc*e experiments in both ways, and compared the results 

 in a separate table. Tabular 



