needle moving over the graduated circle ef. The support g, which
carries the apparatus, also receives in a tubular guide the end of the
torsion rod gr and a set screw for fixing the upper end of the torsion
rod when necessary. The impulse of the stream of water is received
on a circular disk x, in the plane of the torsion rod and the frame
abcd. To raise and lower the a paratus easily, it is not fixed directly
to the rod mn, but to a tube k/Qsliding on mn.
Suppose the apparatus arranged so that the disk x is at that level
the stream where the velocity is to be determined. The plane

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abcd is placed parallel to the direction of motion of the water. Then
the disk x (acting as a rudder) will place itself parallel to the stream
on the down stream side of the frame. The torsion rod will be unstained,
and the needle will be at zero on the graduated circle.
If then the instrument is turned by pressing the needle till the lane Boileau ex eriments on
v, ~ » » P

abcd of the disk and the zero of the graduated circle is at right angles to the stream, the torsion rod will be twisted through an anglle which measures the normal impulse of the stream on the disk x. T at angle forces in the torsion rod is

will be given by the distance of the needle from zero. Observation shows that the velocity of the water at a given point is not constant. It varies between limits more or less wide. /Vhen the apparatus is nearly in its right position, the set screw at g is made to clamp the torsion spring. Then the needle is fixed, and the apparatus carrying the graduated circle oscillates. It is not, then, difficult to note the mean angle marked by the needle.

Let r be the radius of the torsion, - rod, l its length from the needle I over ef to r, and a the observed

torsion angle. Then the moment

of the couple due to the molecular

- "1

UI

f

£1,

M =E, Ia/l; J

where E, is the modulus of elasticity for torsion, and I the polar

moment of inertia of the section of the rod. If the rod is of circular section, I=%7|'T4. Let R be the

radius of the disk, and b its

leverage, or the distance of its

centre from the axis of the torsion rod. The moment of the pressure

of the water on the disk is

Fb = kb(G/2g)1rR21J2,

where G is the heaviness of water and k an experimental coefficient. Eila/Z = kb(G/zg)-/rR2v2.

For any given instrument, 5

v =c/ a,

where c is a constant coefficient for the instrument.

The instrument as constructed had three disksvwhich could be used at will. Their radii and leverages were in feet R = 1, =

i

l

l

- ° ' '.£ .z;;';;L

Then

"

W' >w F 2399 1.;§ s/$'.¥§ l“'“'5»Z%§ °> FIG. 147.

1st disk . . o-052 o-x6

2nd, , o-105 0-32

3rd, , . O'2IO o-66

For a thin circular plate, the coefficient k=I'I2J In the actual instrument the torsion rod was a brass wire o-06 in. diameter and 6% ft. long. Supposing a. measured in degrees, we get by calculation v=o~335/a; o-1154 a; o-o42/a.

Very careful experiments were made with the instrument. It was fixed to a wooden turning bridgexrevolving over a circular channel of 2 ft. width, and about Z6 ft. 'circumferential length. An allowance was made for the slight current produced in the channel. These experiments gave for the coefficient c, in the formulav =rx/ a, 1st disk, c=o~3126 for velocities of 3 to 16 ft. 2nd H 0'1177 vv H Ii to rv

3rd n 0°O349 u rv less than 1% n

The instrument is preferable to the current meter in giving the velocity in terms of a single observed quantity, the angle of torsion, while the current meter involves the observation of two quantities, the number of rotations and the time. The current meter, except in some improved forms, must be withdrawn from the water to read the result of each experiment, and the law connecting the velocity and number of rotations of a current meter is less well-determined than that connecting the pressure on a disk and the torsion of the wire of a hydro dynamo meter.

The Pitot tube, like the hydro dynamo meter, does not require a time observation. But, where the velocity is a varying one, and consequently the columns of water in the Pitot tube are oscillating, there is room for doubt as to Whether, at any given moment of closing the cock, the difference of level exactly measures the impulse of the stream at the moment. The Pitot tube also fails to give measurable indications of very low velocities. PROCESSES FOR GAUGING STREAMS

§ 146. Gauging by Observation of the Maximum Surface Velocity.-The method of gauging which involves the least trouble is to determine the surface velocity at the thread of the stream, and to degluce from it the mean velocity of the whole cross section. The maximum surface velocity may be determined by floats or by a current meter. Unfortunately the ratio of the maximum surface' to the mean velo» city is extremely variable. Thus putting U., for the surface velocity at the thread of the stream, and Um for the mean velocity of the whole cross section, 'vm/vp has been found to have the following values:- v, ,, /110

De Prony, experiments on small wooden channels 0-8164 Experiments on the Seine ....... 0-62 Destrem and De Prony, experiments on the Neva 0-78 D Canals ...... o-82

Baumgartner, experiments on the Garonne . . 0-80 Brunings (mean) ........ . 0-85

Cunningham, Solaniaqueduct. . . . . 0-823