REPORT ON HYDRAULICS. — PART II. 419 



neer* arrives at an equation which expresses the relation of the 

 expenditure, the length, the section, and the depth of the water 

 in the canal under consideration ; by a series of very com- 

 plicated calculations, M. Poncelet has arrived at similar re- 

 sults. But uniformity of motion cannot take place when 

 the surface of the water is not of the same inclination as the 

 bottom of the canal, as, for example, when the bottom is hori- 

 zontal, or the declivity is contrary to the current : it was there- 

 fore very important to establish the distinction between the two 

 kinds of regimen, and only to regard the uniform regimen as 

 a modification of the permanent regimen ; that is, it was neces- 

 sary to find a general formula which should represent all the 



tion of water in canals becomesp = 0"00036554 - (vi + 0*0664t)) ; that is, if 



Q * 



v= -^ , Q being the expenditure, 



p S3 = 0-00036554 c (Q^ + 0-0664 QS); 



for the expression of the velocity, v= — 0-0332 + ^2736^ + 0-011; 



for the expenditure, Q = S (— 0-0332+ 'V'' 2736^-1+ 0-0011) ; 



c 



or, what is sufficiently near, Q = S (^2736^ — 0-0332). 



In great velocities, where the resistance is simply proportional to their 

 squares, we have 



v = 5]Vp1 andQ = 51sV^ 

 c c 



These formulae might be illustrated in practice in the method adopted by 

 Messrs. Prony, Girard, and d'Aubuisson. See Traite d' Hydraulique, 8vo, 

 Paris, 1834, wherein several cases, such as the breadth, height and form of the 

 channel which gives the greatest expenditure, are determined ; — the circle, the 

 semicircle, and segment of a circle ; afterwards the regular half-polygons, the 

 regular semi-hexagon, the semi-pentagon, and the semi-square. 



But as many of these figures are inadmissible in practice for canals, we must 

 adopt a trapezium, with its smallest base for the bottom, and having its sides 

 inclined to angles of about 34", but which form will be, however, obliterated in 

 the angles in time by deposits, and present a concave bottom. 

 By preserving the slope at 2 to 1, or n, 

 the mean velocity, l-\-nh=^2h; I being the length, 



h = depth or height ; 

 and consequently s = 2 A2 and c=z2h — nh-\-2h')/n-\-\-=z n'h, 



by making 2 — n -\- 2 V ns -\- 1 = n'. These values will form the equation 

 pks= 0-00004569 n' (Q Vi + 0-133 Q h^), from whence we can deduce k. 

 In the case of rectangular canals moving through an aqueduct or rock, the 



2 O 

 breadth ought to be double the depth, and consequently V 



• Essai sur la Solution Numerique de quelques Problemes relatifs au Mouve- 

 ment permanent des Eaux Courantes .- par M. B^langer. 

 2 e2 



