452 FOURTH REPORT — 1834. 



and from which results a resistance proportional to the square 

 of the velocity with small velocities, and diminishing to nothing 

 in high velocities, the relation between the velocity and inclina- 



tion being expressed by V = .-= =^ 



6thly, That the resistance which the whole mass experiences 

 from the friction of a part of it against the bed, is in the direct 

 ratio of the bed, and inversely as the section ; 



7thly, That each molecule experiences a resistance in pro- 

 portion to its distance from the bed ; 



8thly, That these velocities taken conjointly produce a mean 

 velocity, which leads to the following general expression : 



^ ^ 297(Vr---01) _o.3(^,_.oi). 

 -v/A - L^/^+ 1-6 ^ ' 



M. Dubuat considers that the amount of friction being pro- 

 portional to the extent of surface, and the circle containing the 

 least perimeter, that figure is preferable for pipes on account of 

 presenting less friction, but that rectangular figures are preferable 

 for aqueducts, and trapeziums for rivers, from the nature of the 

 channel and the velocity in all cases being sensibly proportional 

 to the square root of the mean radius of the bed : it follows that 

 a trapezium in which the breadth at the bottom is f of the 

 height of the water, and the slope of the sides | of the depth, will 

 give the least i-esistance. 



The following ai-e the results of his experiments : 



Inches. 



Fine gravel 4 per second. 



Middling ditto 7 ditto. 



Large ditto 12 ditto. 



Gravel of the size of an egg ... 36 ditto. 



Hence the reason why in the channels of rivers there is necessarily 

 a relation between the tenacity of the soil and the velocity of their 

 currents ; and in general, if we call q the relation to the breadth 



and depth of a channel, we shall have r = and r = -—^ ; 



^ ' q + 2 q + 2' 



or if the depth be undetermined and the breadth be finite, we 



shall have r = — ; and, vice versa, if the depth be finite and the 



breadth undetermined, we shall have r = h. So that in rivers 

 in which the width is very great in proportion to the depth, we 

 may without any sensible error take the depth for the mean ra- 

 dius, and in this case their mean velocities for equal inclinations 



