418 FOURTH REPORT — 1834. 



surface, and the square of the velocity, plus a small fraction of 

 the velocity, and is in the inverse ratio of the section *. 



In the preceding determinations we have supposed the water 

 to have an unifonn motion ; that is to say, that each section of 

 the fluid mass presents the same expenditure and velocity, 

 and consequently the same depth of water. M. B^langer has 

 taken into consideration a motion in canals of a different kind ; 

 it is that in which the fluid mass gives the same expenditure 

 through its sections, but has not the same depth throughout, nor 

 with its surface parallel to the bottom. There are examples in 

 canals where the length is insufficient to produce uniformity of 

 motion at the commencement and extremity ; likewise, where 

 the breadth and inclinations are unequal : but it is essential to 

 the theory of M. B^langer that these variations should take 

 place insensibly. By admitting the hypothesis of the paral- 

 lelism of filaments being perpendicular to the canal, this engi- 



* Formula of Motion. 



Let ^p be the force of acceleration, 



A and B the two constant coefficients, 

 A' the constant multiplier, 



B V the friction of the velocity assumed, 

 c the wetted perimeter of the section, 



the mean radius or relation of the area to the wetted perimeter 



* of the section ; 



then A' — {v^ + B v) will be the expression of the resistance. 



Then, adopting the principle of the resistance to be equal to the force of ac- 

 celeration gj), we shall have the following equation, viz. 



gp = A'—iv' + Bv), orp = A ^(v^' + Bv), 



A' • ... 



by making — = A for a portion of the canal taken when the motion is uniform, 



of which I = the length. P' being the absolute inclination, we shall have 



p =z-^and P'= A (v^ + Bv); or, taking the whole extent of the canal, L 



being the length, P the total inclination, (from which inclination must be de- 

 ducted the height due to the velocity v of the uniform motion,) the equation 



will then be P — = A — '- (t^ 4- Bv). We must now determine the two 



2g S ^ ^ ' 



constant coefficients A and B. 



Tlien, taking Eytelwein's results from 91 experiments made on different 

 canals and rivers, in which the velocity varies from 0-124 metre to 2-42 metres, 

 and the fluid section from 0-014 metre to 2'604 metres, it follows that 



A' = 00035855, or 



A = 0-00036554, 



B = 0-0G638. 

 So that, putting g for the numerical value, the fundamental equation of the mo- 



