HYDRAULICS. 



given time, must necessarily be as the 

 velocity with which it flows; and if, 

 therefore, the hole D is supposed to be 

 four times as deep below the surface 

 A, as the hole B is, it follo\vs that D 

 will discharge twice the quantity of 

 water, that can flow from B in the same 

 time, because 2 is the square root of 4. 

 So in like manner, if D had been nine 

 times the depth of B, three times the 

 quantity of water would issue from it, 

 3 being the square root of 9. 



From the above law of spouting 

 fluids, if a semicircle cgd be drawn 

 from the central height of the column 

 of fluid as at C, so that c C d may be 

 the perpendicular diameter, and c the 

 top of the fluid, while d is its bottom, 

 any parallel lines drawn from that semi- 

 circle to the diameter, and at right angles 

 to it, as at /B, g C, and eD, will be 

 proportionate to the horizontal distances 

 to which the fluid will spout from holes 

 made at the points BCD where those 

 lines cut the diameter ; and as g C is 

 the longest line that can be drawn, within 

 the semicircle, so we learn that a hole 

 made in the centre of the column at C 

 will project its water to the greatest 

 horizontal distance or range ad, and 

 that range (if in vacuo) would be equal 

 to twice the length of the diameter c d. 

 In like manner two jets of water spout- 

 ing from B and D would be thrown to 

 the same distance and meet in the point 

 b, because the lines /B and e D pro- 

 ceeding from the respective jets are 

 equal to each other. The path of the 

 fluid in so spouting will in every case 

 be a parabola, because it is impelled by 

 two forces, the one being horizontal, 

 while the other (gravitation) is perpen- 

 dicular. The velocity of the jet will 

 not be affected by its direction, because 

 fluids press equally in all directions, and 

 that velocity may be found by multi- 

 plying the square root of the head in 

 feet by 8^-, so that a four-feet head 

 would produce a velocity of discharge 

 of rather more than 16 feet in a second. 

 If the water, instead of flowing out at 

 small holes, as in the figure, had been 

 permitted to run from a long slit, or 

 opening, of equal width throughout, it 

 is evident from the laws above stated, 

 that the discharge from the top and 

 bottom would be very different, but the 

 general velocity of the whole stream 

 will be two-thirds of that at the lowest 

 point. Hence if the head be not kept 

 up to one height by a fresh supply, the 

 initial velocity will soon be lost, and the 



discharge become very languid, which 

 is the reason why canal locks, or reser- 

 voirs, are so long filling, although the 

 process at first proceeds most rapidly. 

 M. De Buat has given the best practical 

 rule for calculating the velocity of rivers 

 when the sectional area and inclination 

 in a certain distance are known ; that 

 is, to suppose the whole quantity of 

 water to be spread on a horizontal sur- 

 face, equal in extent to the bottom and 

 sides of the river, when the height at 

 which the water would so stand is called 

 the hydraulic mean depth. This found, 

 the square of the velocity will be jointly 

 proportional to this depth, and to the 

 fall in a given length. The fall in such 

 length must, therefore, be ascertained, 

 and the square of the velocity in a second 

 will be very nearly equal to the product 

 of this fall multiplied by the hydraulic 

 mean depth: the velocity thus given 

 will, however, be a trifle too great, par- 

 ticularly if the river is very crooked. 

 For practical purposes, the usual pro- 

 cess is to take the sectional area of the 

 stream in superficial feet by soundings, 

 and to measure off ten, twenty, or 

 any number of feet on the banks, and 

 then to ascertain by a stop-watch the 

 mean time that slices of turnip (or any 

 other body of nearly the same weight as 

 the water, and which will therefore float, 

 but not float on the surface) thrown into 

 different parts of the stream, take to 

 pass through this measured distance, 

 from which the number of cubic feet of 

 water flowing through the stream in a 

 given time can be pretty accurately de- 

 termined. 



Pipes must be considered in the same 

 light as small rivers, taking the mean 

 depth as one-fourth of the diameter, 

 and a sufficiently accurate determination 

 of the velocity will be obtained by sup- 

 posing the height of the head of water 

 from its surface to the discharging- 

 orifice to be diminished in the same 

 proportion as the diameter of the pipe 

 would be increased by adding to it one- 

 fiftieth part of its length, and finding the 

 whole velocity corresponding to four- 

 fifths of this height. Thus, if the dia- 

 meter of the pipe was one inch, and its 

 length 100 inches, we must suppose the 

 effective height to be reduced to one- 

 third by friction, and the discharge 

 must be calculated from a height four- 

 fifths as great as this. If the pipe had 

 been two inches, the head would only 

 have been supposed to be reduced to 

 one-half by the friction, and such a pipe 



