viz., when the direction of the power is parallel to the length of the plane, or to its base. The one method may easily be extended to every other case.

272. In the first place, let the power Q, act in a direction parallel to the length of the plane. In order that the body may begin to slide in the direction G H, the force Q, must be equal to the sum of the relative weight of the body and the friction. But in constructing, as above, the rectangle P A D C, and always calling n, the ratio of the

HI

friction to the pressure, the force, or relative weight, P C= Px GH'

HI

Q=Px +nPx
GH+

GI

nPGH a formula which exhibits the quantity by which the friction enters into the expression of the power Q. Example.-Let the weight P=8000 lbs., the angle of inclination

Q=4000 lbs. +2309.4. The power Q must, therefore, be more than 6309.4, while without friction it would be only 4000 lbs.

273. In the second place, let the power Q act in the direction parallel to the base of the inclined plane. Having resolved, as above, the weight of the body into two forces P C, and P A, the one parallel, the other perpendicular, to the inclined plane, resolve in like manner

the power Q, expressed by the part P O, of its direction, into two other forces P N, and P M, the one parallel, the other perpendicular, to the length of the plane. Then the force,

The whole pressure on the inclined plane being equal to the sum of the two forces P A, P M, if n always represents the ratio of the friction to the pressure, it is plain that the friction,

This being premised, in order that a body may be ready to slide in the direction G H, the force P N must be equal to the sum of the force P C, and the friction, or

GI √3

nP x G H2 GI-nxGIX HI'

or

2

by which the friction increases the power.

=

Example.-Let P-8000, HGI= 30°, or

= 1⁄2, GH

=

2

.866, n. It will be found that Q-almost 9021.7 lbs. Without friction the power would be only about 4618.8 lbs.

Of Friction in the Screw.

Fig. 4.

R

IG

Let the inclined plane G H I, expressing the last value of Q in the above paragraph, as the power in the screw always acts in a direction parallel to G I, or perpendicular to the axis of the cylinder, be wrapped round a cylinder of such radius C p, that G I, the base of the triangle, shall be just equal to the circumference, and, of course, just reach round it; and let G H, the inclined plane itself, be produced in the same direction an indefinite number of turns around the cylinder. Here then obviously G H, forms one thread of a screw, while H I, the height of the inclined plane, is equal to the distance of the threads VOL. IX, 3RD SERIES. No. 4.-MARCH, 1845.

18

parallel to the axis. Then let a nut be perforated, and formed in the interior, so as to correspond to the thread of the screw, applying itself to the surface G H, in its whole length, and perhaps two or three other turns, and let the weight P be laid on the nut, the friction is the same whether it be supported on a small, or large, surface, and let this nut be furnished with a lever C R, perpendicular to the axis, and extending to R, the point where the new power q, is applied, make an equilibrium with the weight P, and its friction in the same manner as Q does at the surface of the screw. On the principle of Let this equation be

Cp

CR'

the lever, q: Q:: Cp: CR, or q=Qx incorporated with that expressing the value of Q in the inclined plane, where the power acts in a direction parallel to the base, and we shall Cpx (HI+nxGI)

have q=Px

CRX (GI-nxH1)'

This is the formula for the

screw as commonly constructed; P=the weight to be raised, q=the power necessary to just not produce motion when acting at the distance C R, and in a plane perpendicular to the axis C, C p=the semi-diameter of the screw, HI=the distance of the threads measured parallel to the axis, G I=the circumference of the screw, n=the ratio of the friction to the pressure.

Now, in order to find the value of C p, when q is a minimum, the other quantities remaining constant, P and C R being omitted, as affecting all the terms equally, for conciseness' sake let Cp-x, HI=h, h x + n x x 2

and
has its least value when its differential,

6.28, &c., and the fraction will become

π X hn

hxxdx+2n«2x2dx—h2ndx—2hn2 xxdx—h¬xdx—n2x2 dx

π X hn 2

which

= 0,

whence 2x2-2hnxx-h2 and x, (the circumference of the screw,) =h×(n+1+n2) from which the diameter may be computed as

usual.

From the denominator of the first fraction in the last paragraph, it appears that when hnx, that is, when the circumference of the screw: distance of the threads:: the friction: the pressure, the denominator becomes 0, and, therefore, the power 9 must become infinite, or the effect 0. When this is the case, if h, or the perpendicular height of the inclined plane, formed by one revolution of the thread, be taken = = 1, or radius, then n=x the circumference of the cylinder, or base of the plane, or cotangent of the angle contained between any thread and a plane perpendicular to the axis. Let a represent this cot.2a-1 angle, then n=cot. a=

2 cot.a

by an :trig: whence cot. 24a-1=

2 n cot. a, and cot. } a=n+1+n2, which is precisely the value of the circumference given by the calculus when the effect is greatest, so that the angle of inclination at which the greatest effect is produced, is exactly half of that angle which, with the same degree of friction, is the limit of all effect.

From the calculus it appears that screws are commonly made much too large in proportion to the distance of the threads, to produce any given effect with the least power, the circumference in hardly any practical case being, for that purpose, greater than twice the distance of the threads, and the diameter much less than that distance. Thus, suppose it be desired to produce a pressure equal to ten tons, with a screw whose threads are one inch asunder, lever eight feet long, and friction one-third the pressure; if the screw be six inches in diameter, the power must exceed 275 lbs., if the diameter be only two inches, any power greater than 122 lbs., will be sufficient. If the friction amount to half the pressure, the dimensions still remaining unaltered, the powers corresponding to the same two diameters, will be 398 and 168 lbs. respectively. The least powers with these two degrees of friction would be 72 lbs. and 98 lbs., and require diameters only 0.4416 in., and 0.515 in. respectively; but no substance is known, which, when reduced to even twice these dimensions, would have strength enough to resist the torsion. It appears, therefore, that a screw, as an engine of force, with any given distance of thread, ought to be made as small as the strength of the material will admit.

In order to ascertain, by computation, the middle course between wasting materials, labor, and power, on one hand, and hazarding the failure of the machine on the other, we may adopt the doctrine "that in resisting torsion the whole lateral cohesion is exerted at 4th the radius of the cylinder from the centre." The notation of the formula for the screw being retained, and S being put for the lateral cohesion of a square unit of the substance used, the area of the section is

π X 2

2

the force of the

cohesion at the end of the lever, which, being substituted for q, in the

are properly substituted, the value of x may be safely adopted as the radius of the cylinder.

But when the exertion of great force, or the weakness of the material, renders a large cylinder indispensable, the distance of the threads should be greater also, because increasing the distance of the threads does not require, by any means, a proportional increase of power. Thus with the six-inch cylinder, above mentioned, with the friction of one-third, increasing the distance of the threads from one to two inches, requires an increase of power less than one-sixth, viz., from 275 lbs. to 319 lbs.; and with a friction of one-half the increase is only one-eighth, or from 398 lbs. to 448 lbs., which may generally be compensated by using a longer lever, while the action will be much more prompt and rapid. And less than double the power will produce the same effect, though the distance of the threads be made

* Edin. Encycl. vol. v, p. 400; art. "Carpentry."

equal to the diameter of the screw, or six times their former distance. The reason of this will be obvious, when it is considered that in this case the whole power except 37 lbs. is employed in overcoming the friction, and that at such a small angle of elevation, the pressure, and, consequently, the friction, is but slightly varied by a variation of that angle. And with the same distance of the threads, if the diameter receive successive increments, the ratios of the corresponding powers will continually approximate to the corresponding ratios of the several diameters, because the effect of friction, which absorbs most of the power, is proportional to its distance from the fulcrum.

But after all it is far from being eligible, in practice, always to give screws the shape above described, especially in the fastening of instruments, or machines, where the object is not to produce but to prevent motion, because if the distance of the threads be greater than n times the circumference, the friction of the screw will not prevent it from running back. On this account it may not be improper to investigate the figure of a screw so formed, that, without making the angle of ascent of the thread more acute than necessary, yet the friction shall always hold it where the power leaves it. Let it be required, therefore, so to shape a screw, that with any given degree of friction the power necessary to turn it forward shall have a ratio to the power necessary to turn it back equal to the ratio of the two given quantities f: b; f being, of course, larger than b. As the length of the lever to which these powers are applied, cannot effect their ratio to each other, they may be considered as applied at the surface of the cylinder, and this surface may be further considered, as extended on a vertical plane, in which case one revolution of the thread is the length of an inclined plane, the circumference of the cylinder is its base, and the distance of the threads its height; reversing the course of reasoning adopted in the first two sentences of the description of the screw. Bossut, in Art. 273, as translated above, gives the force which, acting parallel to the base, (as it always does in the screw,) just balances the relative gravity, and the friction,

=

Px(HI+nxGI)
GI−nxHI

where P weight, G I=base, H I=height, and n=the ratio of the friction to the pressure. But in order to express the greatest force which, acting in the direction of the relative gravity, would not drag the body down the plane, H I in the preceding equation, must be taken with the contrary sign from what it is in the equation above,

HI
GH'

because the force P C, or P or relative gravity, instead of oppos

ing the power, assists it, and the force P M, or Q x

stead of increasing, now diminishes the pressure which produces the friction, or resisting, force. With this change the greatest force which (acting parallel to the base, and from the summit towards the foot of the plane,) will not drag the body down the plane, will be found,