being in one right line. The vessels were about 10 inches deep, and the branches FG, fg of the syphon were about 5 feet long. They were then set on two tables of equal height, and (the hole e being stopped) the vessel ABCD, and the whole syphon, were filled with water, which was also poured into the vessel abcd till it stood at a certain height LM. The syphon was then turned into a horizontal position, and the plug drawn out of e, and the time carefully noted which the water employed in rising to the level HKkh in both vessels. The whole apparatus was now inclined so that the water ran back into ABCD. The syphon was now put in a vertical position, and the experiment repeated: no sensible or regular difference was observed in the time; yet in this experiment the pressure on the part Gg of the syphon was more than six times greater than before. As it was thought that the friction on this small part (only 6 inches) was too small a portion of the whole resistance, various additional obstructions were put into this part of the syphon, and it was even lengthened to 9 feet; but still no remarkable difference was observed. It was even thought that the times were less when the syphon was vertical; nor has any variation ever been observed in the friction of water in these different positions when the surface was glass, lead, iron, wood, &c. (Principes d'Hydraulique, tome i., §§ 34 and 36, Dubuat.)

Second Law. The resistance is, at any one velocity, proportional to the surface exposed to the action of the flowing water. In order to obtain an expression for this law, we may remark, in the first place, that in any channel or pipe the resistance arising from the surface is shared by all the particles in the volume of water flowing down, those nearest the sides being most retarded, and each in succession less and less influenced. This is

proved by the result of observations shown in the engraving, Fig. 70, which represents the transverse section of a trapezoidal channel, with lines of equal velocity plotted upon it, as given in the recent work of M. Darcy and M. Bazin. The width of this experimental chan

nel at the water surface was 2 metres, qp, and its depth 0.540 metre, with side slopes about 45°.

The measured

second (= 44.5

discharge was 1.236 cubic metre per cubic feet), and the mean velocity 1.497 metres (= 5 ft. nearly) per second; obtained by dividing the discharge by the area of the transverse section, which was equal to very nearly 0.824 square metre. By improvements on Pitot's tube (p. 162) this instrument was adapted by them to the accurate measurement of the velocity in any

part of the transverse section, and from the observations thus taken the lines of equal velocity were plotted (by a method described further on). The darker line, No. 3, shows the points in the flowing water at which the mean velocity of 1.497 metres per second was found. The line, No. 1, which returns upon itself, shows continuously the points of highest velocity plotted; No. 2 being also greater than the mean, while lines, Nos. 4, 5, and 6, show the successively decreasing velocities below the mean, the least being that nearest the surface of the sides and bottom. It would be easy to interpolate by hand any number of intermediate lines of equal velocity, and thus divide the whole mass of moving water into successive lamina, each suffering less resistance than the previous one as we proceed from the wetted surface of the bottom and sides inwards. The point of maximum velocity was situated on the central dotted line about one-third of the depth from the surface, and was equal to 1.82 metre per second. The greater, then, that surface is, the greater is the resistance. But the greater the volume upon which this retarding action of the surface has to act, the less reduced will be the velocity of the first, and therefore of each successive lamina: and thus we have the resistance directly proportional to the surface and inversely as the volume, i. e. proportional to area of sides and bottom

Now let us suppose the chan

volume of moving water nel, Fig. 70—which is identical in every section throughout its length, and having a uniform flow-to be cut by two parallel planes perpendicular to the axis of the stream; and in the plan, Fig. 71, let aa' and AA' be the horizontal traces of these two planes, and let the base of the section, Fig. 70, be produced on each side until the produced part, CN and C'N', equal the sum of

the sloping sides and short vertical portions, BA and B'A'. If, then, from the extremities N and N' of this line perpendiculars be let fall on the traces, Fig. 71, the rectangle aAA'a' so formed is evidently equal to the wetted surface of the channel between the two planes, that is, to the product of the distance between them, aA and AA', = NN'; also the volume of water between the same planes is equal to the product of aA into the transverse section of the channel. Hence the ratio given.

above is equal to

aA× NN'

aAx transverse section

Striking out from each the length aA of the channel common to both, we have the resistance directly proportional to the border or wetted perimeter, and inversely as the area of the transverse section perpendicular to the axis of the stream. If, then, we put C for the contour of the border, and S for the area of section, C S

we have the resistance proportional to

Third Law. The resistance is proportional to the square of the velocity nearly, the border being constant. For the number of particles drawn in one second from their adhesion to the sides of the channel or pipe is proportional to the number of feet per second with which the water is moving, that is, to the velocity. And the force with which they are drawn is also as the same number of feet per second, or the same velocity: and thus the passive resistance of the wetted border to the flow of the water is proportional to the product of the velocity into the velocity; this part, then, of the expression for the resistance is represented by av2, a being a constant, determined hereafter.

Experimenters have shown that this gives the resistance a very little too high, and that with velocities increased in the ratio 2, 3, 4, &c., it is not represented by a × 4, a × 9, a × 16, &c., but more nearly by adding the simple power of the velocity, thus a (v2+bv), the series of numbers v2 + v not increasing so fast as v3.

Fourth Law.-In gases and elastic fluids we also have the friction proportional to the specific gravity or density.

In order to obtain from these laws a formula for the discharge of water through pipes and channels, we must make use of the well-known principle, that when any body is moving with a uniform velocity, the accelerating are necessarily equal to the retarding forces: for if the accelerating forces be supposed greater than the retarding, the velocity must increase; and if they should become less, then the velocity must, on the other hand, decrease. We must now, as in Chapters I. and II., find a general expression for the mean velocity, for this multiplied into the transverse area gives the discharge with a given inclination and we can thus solve the questions that arise in practice, such as the requisite dimensions of pipe or channel to convey a given quantity of water, &c., &c.

Now in any pipe or channel, whose length is /, and whose height, from the surface of the supply to the point of discharge or extremity of 1, is represented by h, we

h

have the accelerating force expressed by g, or sine of ī

inclination of surface into gravity.

The retarding forces are, from the second and third laws above given, neglecting bv, proportional to