wear than others, which will very much tend to increase the friction and other inequalities and resistances; and will greatly diminish the efficiency of the machine. To obviate these difficulties, Mr James White invented a system of pulleys (fig. 57), consisting of two blocks A, B; into which grooves were cut, the radii of those in the upper block being as the numbers 1, 3, 5... and the radii of those in the lower block being as the numbers 2, 4, 6... Now, suppose the lower block to be raised through one inch, then each of its strings will be shortened one inch, and therefore the circumference of the pulley BB1 describes one inch; that of AA1, two inches; that of BB, three inches, and so on; which numbers being proportional to the radii of the respective pulleys, they will all move with the same angular velocity; and, consequently, each block instead of being composed of separate pulleys may consist of one solid piece of wood or metal, containing the grooves before mentioned. The disadvantage of this system is, that if the cord be at all elastic it cannot be kept stretched in every part on account of the tension not being the same throughout, so that the smaller grooves are rather a hindrance to the motion than a help.

HUNTER'S SCREW.

219. We have seen (Art. 215) that the advantage of a screw increases in proportion as the distance between the threads diminishes, and as the length of the lever at which the power acts increases; therefore, by making the threads of the screw sufficiently fine, we may increase the advantage as much as we please; but there is a limit to the fineness of the threads; for as all the weight is borne upon them, if they are too fine they will not be sufficiently strong to bear the load. If we, on the other hand, increase the length of the arm of the lever, with the view of increasing the advantage of the screw, the power will have to describe an inconveniently large circle. To obviate these natural defects, and yet increase the advantage to any degree, Mr Hunter invented the screw in fig. 58; A and B are two common screws, of which A is also a hollow screw to admit

B, which is fastened to the moveable plate D of wood or metal. If D, d be the distances between two threads of the screws A, B respectively; then, while the power describes one circumference, A descends through D, and B ascends in A through d, and the space descended by the plane D is D-d; for when A descends it carries B along with it, though B is at the same time ascending in A. Wherefore, by Art. 217,

Now we can make D and d as nearly equal as we please without diminishing the strength of the machine, and therefore the advantage of this screw admits of indefinite increase.

220. It appears from Art. 209, that the advantage of a wheel and axle is

which might theoretically be augmented ad libitum, either by increasing the radius of the wheel, or by diminishing that of the axle. But by the former means, the power would practically have to describe an inconveniently large space, and the machine would become cumberous; and, in the latter case, it would be too weak to bear the pressure of the weight upon its axle. To remedy these inconveniences, and at the same time to increase the advantage in any requisite degree, the form of fig. 59, has been given to it; where A is the wheel, B and C two axles of unequal radii, firmly fixed to each other, and having the same axis. The cord BDC as P descends is wound upon the axle B with the larger radius, and is at the same time unwound from the axle C with the smaller radius; it passes under a pulley D, to which the weight W is attached. Let R be the radius of the wheel, rr' those of the axles B, C. Then when the machine turns once round, P descends through 2π R, and the length of the cord wound upon В is 2πr,

and the length unwound at the same time from C is 2πr'; wherefore, upon the whole, the length of cord hanging down from the axles is diminished by

As we can diminish the denominator of this fraction as much as we please, without weakening the materials of the machine, there is no limit to the advantage of it, except what arises from the very great length of cord that must be used in raising W through a very small space.

THE GENOU.

221. This instrument is represented in its simplest form in fig. 60, where AF is the profile of a frame in which the rods AB, BC work. AB is moveable about a fixed axis passing through A; it is connected with BC by a compass joint at B; and the other end C of BC, by means of a pin passing through it, is compelled to move in the vertical groove EF. The power is applied at G, a point in AB, in the plane of the rods ABC. It causes B to come nearer to AF; and, consequently, C presses downwards upon any obstacle opposed to it. It is obvious this machine is only applicable in those cases in which C is required to descend through a small space, as in printing, where it presses the paper upon the type.

Let W the reaction at C, P the power applied horizontally at G, 0= the angle BAF, a = AB, b = BC, c = AG,

Also AF = a cos 0 + b cos BCA, therefore the virtual ve

locity of W

Wherefore, by Art. 114, the advantage of the machine

An

222. A combination of wheels and axles may be used instead of the machine in Art. 220, when that is inconvenient and great advantage is required. Fig. 61 represents a combination of three of these mechanical powers. endless strap passes over the axle a and the wheel B, and another strap passes over the axle b and the wheel C. If two successive wheels are required to turn in opposite

directions, the strap must be crossed as between A and B in the figure; when the wheels are to turn in the same direction, the strap must not be crossed. B and C are turned by the friction of the straps upon their surfaces; and hence it is manifest, that if the force to be overcome by any wheel be greater than the friction of its strap, the strap will slip round without carrying the wheel with it, and the action of the machine will cease. Wherefore, in order to make the friction upon the surfaces of the wheels and axles as great as possible, they are covered with leather, which is nailed or glued on them; and both this leather and the concave side of the straps are suffered to be in a rough state; the friction is also increased by crossing the straps.

To calculate the advantage of this combination, denote the tension of the strings d and e by T, T'; then since P balances the tension T on the axle a, we have, by Art. 209,

and, therefore, by multiplying these equations together, we have

W product of radii of all the wheels

=

P product of radii of all the axles

TOOTHED WHEELS.

223. By far the most general modification under which wheels and axles are used in practical Mechanics, is that of toothed wheels.

Let A, a (fig. 62) be the centres of two wheels BC, bc; upon the circumferences of which let teeth or cogs D, E, F ; d, e, f; of any proposed form, be raised at equal distances