"^20 FOURTH REPORT — 1834. 



circumstances of the permanent motion of running streams. 

 This M. Bdlanger has done, in adopting the following hypo- 

 theses. 



He first supposes the current, when in a state of permanent 

 regimen, to be divided into plates perpendicular to the fluid 

 filaments, and the velocity of the particles of the water to be 

 constantly the same in the same and every plate : 



Secondly, That each particle of water moves in a right line, 

 so that the centrifugal motion generated by the curvilinear mo- 

 tion (if there be any) may be neglected : 



Thirdly, That, although the analysis which follows cannot ap- 

 ply but to cases where the dimensions of the section vary in very 

 small quantities in proportion to the length, the velocity of each 

 particle may be considered to be perpendicular to the plate it 

 traverses, neglecting the transversal velocities which exist as 

 soon as the section varies from one plate to the other. 



Of these hypotheses, the first, namely, the motion of the cur- 

 rent by parallel plates, is never realized in natui'e, because the 

 resistance which the periphery of the canal opposes to the mo- 

 tion of the current is transmitted to the adjacent filaments, and 

 so on to the central filament, which moves the quickest ; but in 

 general the velocities of the different filaments differ very little, 

 and may be safely represented by a mean velocity, which is the 

 quotient, or the volume of water expended in a given time, di- 

 vided by the area of the section, as before stated. As soon as 

 this compensation is admitted, each fluid filament may be con- 

 sidered as retarded by a force equivalent to friction. M. Gi- 

 rard* has applied this idea to the formation of an equation to 

 represent the motion of the current, but with the supposition 

 that the two powers of the velocity are represented by the same 

 coefficients ; a supposition which cannot apply to every case. 

 M. Prony has shown that the function which expressed the re- 

 tarding force may be represented by 



g — {av -\- b v^), in which the metre is taken for unity and 



the second for the time : 

 g' = 9™ 8088 represents the accelerating force of gravity : 

 w = the area of the transverse section to which belongs the 



particle under consideration : 

 X = the length of the wetted perimeter of the section : 

 V = the mean velocity supposed to be common to ail the par- 

 ticles which traverse the section : 



• Rapport sur le Projet general du Canal de I'Ourcq : par P. S. Girard. 

 Paris, 1803. 



