454 FOURTH REPORT 1834. 



the mean velocity in any pipe or open stream is an arithmetical 

 mean between the velocity in the axis and the velocity at the 

 sides of the pipe or bottom of the open channels. 



Let V be the mean velocity, v the velocity at the axis, u the 

 velocity of the bottom : 



u = a/v — 1 and V = — - — ; 



also V = (^V- i + \Y and v = {\^u + l)'^ 



V= {Vu - \f + i) and V= {Vu + \f+i 



u={\^v-\f B.x^Au = {^/N-\-\f 



Also V — u=2 \/V — ^ and ti — V = V — m= »/y — \; 

 that is, the difference between these velocities increases in the 

 ratio of the square roots of the mean velocities diminished by a 

 small constant quantity. The place of the mean velocity in mo- 

 derate velocities is about ^th or ^th of the depth from the bottom ; 

 in very great velocities it is higher. (See Dubuat's Table of Velo- 

 cities, also Robison's Mechanical Philosophy and Theory of 

 Rivers.) There are, however, anomalies in these principles which 

 render their application extremely doubtful. It is unnecessary 

 to enter into detail of Dubuat's method of rendering rivers na- 

 vigable by increasing their breadth or by diminishing their incli- 

 nations, nor of the different cases of the motions of canals for 

 irrigation or drainage, and the effect of obstruction, such as 

 bridges, sluices, dams, &c. ; they have been investigated very 

 fully by Professor Robison and by M. Le Creulx*. 



Such is a brief outline of Dubuat's workt, ingenious in many 

 respects and abounding with new views and valuable suggestions; 

 but whoever has had occasion to investigate the uncertain mo- 

 tions of rivers will find that the analogies attempted to be derived 

 from the motions of water in pipes and artificial channels are 

 extremely vague. His formula of the uniform motion of water, 

 modified as it is by contraction and resistances, approximates 

 very nearly to reality. In all cases his theory of the effects of 

 curves is quite contrary to nature, and this he acknowledges in 

 reference to several experiments in the Seine and Marne rivers. 

 His application of his theorem of the expenditure, velocity, and 

 inclination of the surface of a river being known, to determine 

 the dimensions of the bed, is necessarily incorrect. 



• Examen Critique Je lOuvrage de M. Duhuat sur les Priiicipes de I' Hydi au- 

 lique : pav M. Le Creulx. Paris 1809. 



f Elements of Mechanical Philosophy, vol. ii., edited by Dr. Brewster ; and 

 Theory of Rivers. 



