the section of the river at the second string must be ascertained by taking various depths, as before. If this section be the same as the former, it may be taken for the mean section: if not, add both together, and take half the sum for the mean section. Then the area of the mean section in square feet being multiplied by the distance between the strings in feet, will give the contents of the water in solid feet, which passed from one string to the other during the time of observation; and this by the rule of three may be adapted to any other portion of time. Suppose, for example, the time were 12", and the hourly expenditure of water were required, the proportion would be, as 12′′: 3600" :: the number of cubic feet between the two strings: the hourly expenditure in cubic feet. If the mere velocity be required with reference to any fixed interval of time, a similar proportion will give it, only observing to take, instead of the solid content or capacity in the third term, the distance between the two strings.

The operation may often be greatly abridged by taking notice of the arrival of the floating body opposite two stations on the shore, especially when it is not convenient to stretch a string across. An arch of a bridge is a good station for an expe riment of this kind, because it affords a very regular section and two fixed points of observation: and in some instances the sea practice of heaving the log may be advantageous. Where a time-piece is not at hand, the observer may easily construct a half-seconds or quarter-seconds pendulum: the former may be made by suspending a small round (not flat) button, or other spherical weight, by a thread looped over a pin of such a length that the distance from the point of suspension to the centre of the weight shall be 9.8 inches: the quarter-seconds pendulum must be a fourth of this length. If, by observations at several stations above and below any particular point of the river, the velocity does not appear to vary, the section of the river in all that space may be considered as uniform; and it will not be necessary to determine more than one section by actual measure

ment.

47. The effect of undershot and overshot wheels has been very variously stated by different authors; the most valuable and correct observations are those of Mr. Smeaton, an abstract of which was given in Chap. 4. Book IV. vol. I. The numerous practical remarks and experiments related in that chapter and the second chapter of the same book, will render it unnecessary for us now to dwell longer upon the effects of water as a mover of machinery.

48. AIR is the next natural mover we propose to consider. And this, like water, may be regarded either as at rest, or in motion. The pressure of the atmosphere in a medium state is,

equivalent to the weight of 14 or 15 lbs. averdupois on a square inch, and this pressure will support, and, by means of a sucking pump, raise water to the height of about 33 feet; it supports mercury in the barometer at the height of 28 to 32

inches.

The density of air is, at a medium, about 833 times less than that of water: if we take round numbers and reckon 800 to 1 for the ratio of the densities, and put s2 for the surface on which the wind strikes, v for the velocity with which it moves, and 1 for the angle of incidence, then the force of the wind will be equal to the weight of a volume of water expressed by . sin 21 = .0012144 v2s2 sin 21 lbs. averdupois.

r

2g

This formula, however, is only an approximation, and would lead to considerable errors when the velocities are great: on this subject we have treated pretty fully in art. 554, &c. Book V. vol. I., where the tables of Dr. Hutton, Mr. Rouse, &c. are exhibited: the following is Mr. Rouse's table of velocity and corresponding force in the form it was originally given by Mr. Smeaton; but the form in which it is thrown in art. 554 is more useful.

49. As it is not easy to observe the true velocity of the wind, and thence determine its force, several philosophers have invented instruments called Anemometers or wind gages, by which the force of the wind may be ascertained independent of its velocity. M. Bouguer contrived a very simple instrument for this purpose: it is a hollow tube AABB (fig. 5. pl. I.) in which a spiral spring CD is fixed, that may be more or less compressed by a rod FSD passing through a hole within the tube at AA. Having observed to what degree different forces or given weights are capable of compressing the spiral, put divisions upon the rod in such a manner that the mark observed at s in all positions of that rod shall indicate the weight requisite to force the spring into the corresponding position CD. Afterwards join perpendicularly to this rod at F a plane surface EFE of a given area, either greater or less, as may be judged proper : then nothing more is necessary than to oppose this instrument to the wind, in order that it may strike the surface in the directions VE, VE, parallel to that of the rod; and the mark at s will shew the weight to which the wind is equivalent. It will then be easy to reduce any observed force to a volume of water equivalent to it in energy; and so in all cases ascertain the magnitude of the force which the wind exerts.

50. The most usual method of applying wind as a mover of machinery is in the construction of windmills for different purposes, in which the wind produces its effect by impulse upon the sails. In these machines, therefore, whatever varieties there may be in the internal structure, there are certain rules with regard to the position, shape, and magnitude of the sails, which will bring them into the best state for the action of the wind, and the production of useful effect. These particulars have been considered much at large by Mr. Smeaton: for this purpose he constructed a machine of which a particular description is given in the Philosophical Transactions, vol. 51, or in the quarto collection of his "Miscellaneous Papers," p. 55. By means of a determinate weight it carried round an axis with an horizontal arm, upon which were four small moveable sails. Thus the sails met with a constant and equable blast of air; and as they moved round, a string with a weight affixed to it was wound about their axis, and thus showed what kind of size or construction of sails answered the purpose best. With this machine a great number of experiments were made; the results of which were as follow:

(1.) The sails set at the angle with the axis, proposed as the best by M. Parent and others, viz. 55°, was found to be the worst proportion of any that was tried.

(2.) When the angle of the sails with the axis was increased from 72° to 75°, the power was augmented in the proportion

of 31 to 45; and this is the angle most commonly in use when the sails are planes. See art. 547. vol. i.

(3.) Were nothing more requisite than to cause the sails to acquire a certain degree of velocity by the wind, the position recommended by M. Parent would be the best. But if the sails are intended with given dimensions to produce the greatest effects possible in a given time, we must, if planes are made use of, confine our angle within the limits of 72 and 75 degrees.

(4.) The variation of a degree or two, when the angle is near the best, is but of little consequence.

(5.) When the wind falls upon concave sails it is an advantage to the power of the whole, though each part separately taken should not be disposed of to the best advantage.

(6.) From several experiments on a large scale, Mr. Smeaton has found the following angles to answer as well as any. The radius is supposed to be divided into six parts; and th, reckoning from the centre, is called 1, the extremity being denoted 6. Angle with the plane of motion. 18°

No.

Angle with
that axis.

1

72°

(7.) Having thus obtained the best method of weathering the sails, i. e. the most advantageous manner in which they can be placed, our author's next care was to try what advantage could be derived from an increase of surface upon the same radius. The result was, that a broader sail requires a larger angle; and when the sail is broader at the extremity than near the centre, the figure is more advantageous than that of a parallelogram. The figure and proportion of enlarged sails, which our author determines to be most advantageous on a large scale, is that where the extreme bar is one-third of the radius or whip (as the workmen call it), and is divided by the whip in the proportion of 3 to 5. The triangular or loading sail is covered with board from the point downward of its height, the rest as usual with cloth. The angles above mentioned are likewise the most proper for enlarged sails; it being found in practice, that the sails should rather be too little than too much exposed to the direct action of the wind.

Some have imagined, that the more sail the greater would be the power of the windmill, and have therefore proposed to fill up the whole area; and by making each sail a sector of an ellip

sis, according to M. Parent's method, to intercept the whole cylinder of wind, in order to produce the greatest effect possible. From our author's experiments, however, it appeared, that when the surface of all the sails exceeded seven-eighths of the area, the effect was rather diminished than augmented. Hence he concludes, that when the whole cylinder of wind is intercepted, it cannot then produce the greatest effect for want of proper interstices to escape.

It is certainly desirable (says Mr. Smeaton), that the sails of windmills should be as short as possible; but it is equally desirable, that the quantity of cloth should be the least that may be, to avoid damage by sudden squalls of wind. The best structure, therefore, for large mills, is that where the quantity of cloth is the greatest in a given circle that can be: on this condition, that the effect holds out in proportion to the quantity of cloth; for otherwise the effect can be augmented in a given degree by a lesser increase of cloth upon a larger radius than would be required if the cloth was increased upon the same radius."

(8.) The ratios between the velocities of windmill sails unloaded, and when loaded to their maximum, turned out very different in different experiments; but the most common proportion was as 3 to 2. In general it happened that where the power was greatest, whether by an enlargement of the surface of the sails or an increased velocity of the wind, the second term of the ratio was diminished.

(9.) The ratios between the least load that would stop the sails and the maximum with which they would turn, were confined betwixt that of 10 to 8 and 10 to 9; being at a medium about 10 to 8.3, and 10 to 9, or about 6 to 5; though on the whole it appeared, that where the angle of the sails or quantity of cloth was greatest, the second term of the ratio was less.

(10.) The velocity of windmill sails, whether unloaded or loaded, so as to produce a maximum, is nearly as the velocity of the wind, their shape and position being the same. On this subject Mr. Ferguson remarks, that it is almost incredible to think with what velocity the tips of the sails move when acted upon by a moderate wind. He has several times counted the number of revolutions made by the sails in 10 or 15 minutes; and, from the length of the arms from tip to tip, has computed, that if an hoop of the same size were to run upon plain ground with an equal velocity, it would go upwards of 30 miles in an hour.

(11.) The load at the maximum is nearly, but somewhat less than, as the square of the velocity of the wind; the shape and position of the sails being the same.