REPORT ON HYDRAULICS.— PART II. 453 



are as the square roots of their depths. We have^ therefore, 

 formulae for calculating the different cases (two of the data 

 being given,) of the breadth, depth, mean radius, velocity, and 

 inclination, derived from a table of experiments on trapeziums 

 and rectangular canals, on the canal of Jard and the river 

 Hayne. In order to facilitate the use of these tables the late 

 Professor Robison reduced them for his Mechanical Philoso- 

 phy/* : they have since been greatly enlarged by Mr. Laurie of 

 Glasgow, but are now in a great measure superseded by the 

 more accurate researches of Eytelwein. 



For the curves and salient angles of rivers, and during permanent 

 and periodical floods, the author endeavours to establish theories 

 which have no relation to the actual state of things ; but it results 

 from his observations that an inclination of loioo ^^^Y produces 

 on account of bends a velocity due to axioo- I" applying his 

 formula of uniform motion to the course of rivers, he compared 

 the velocities of the Seine and Loire in their mean state : he 

 found that the mean inclination of the Seine was 1. metre for 

 100 toises, or j-J-^-^ ; that its mean depth was 3 feet 7 inches, and 

 its mean velocity 25 inches per second ; and as the theoretical 

 velocity of an inclination of y^^^^ gave 26 inches 10 lines instead 

 of 25 inches, the excess was occasioned by friction and the bend 

 of the river. In the Loire the inclination was 2 metres per 100 

 toises, or ^^Vo j ^^^ "^^an depth 34 inches ; but the velocity due to 

 the depth was 38 inches per second, consequently y'j- was lost 

 by friction and adhesion ; the actual velocity being 35 in 6 

 lines. 



In regard to the velocity, Dubuat may be said to have dis- 

 covered the following laws : 



1st, In small velocities, the velocity in the axis is less than 

 that at the bottom ; 



2nd, This ratio diminishes as the velocity increases, and in 

 very great velocities approaches to the ratio of equality ; 



3rd, Neither the magnitude of the channel nor its'slope has 

 any influence in changing this proportion while the mean velo- 

 city remains the same, whatever be the nature of the bed ; 



4th, When the velocity in the axis is constant, the velocity 

 at the bottom is also constant, and is not affected by the bottom 

 of the river or the magnitude of the stream. 



In some experiments the depth was thrice the width, and in 

 others the reverse, without any change in the ratio of the velo- 

 cities. Another most important fact discovered by him is, that 



* See an excellent article on Hydrodynamics in Brewster's Eil'mbiirnh En- 

 cyclopeedia, . 



