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Px(−HI+n × GI)
GI+nXHI

Let now H 1=rad.=1, n=cot. a, as before, and G I➡cot. z; z being the angle at the base of the inclined plane, on which, with the given degree of friction, n, the force forward and backward will have the required ratio f: b. Then dividing by P, we shall have f:b :: cot.a-cot.z+1 cot.a cot.z-1 ::cot.(a-z): cot. (a + z):: sin.2a+sin.2z: sin.2a-sin.2z. Then by com. and div.f+b:f-b:: sin.2a: sin.2z, whence the base G I, or circumference of the screw, and thence its diameter may be easily found.

cot.z-cot.a cot.z+cot.a

By the diameter of a screw is always meant the arithmetical mean between the greatest and least diameters of the bearing surface of the thread. The screw is also supposed to have a square thread, or at least that the section of its bearing surface with a plane passing through the axis, is a right line, and perpendicular to the axis. But when the thread is in the form of a triangular prism, or rather, when the section above mentioned makes an oblique angle with the axis, the pressure, and of course the friction is increased in the ratio of the sine of

n

its sine

that angle to radius, so that must be taken instead of n, through the whole investigation. It is supposed also that there is no pressure except on the threads of the screw; in practice, however, there is almost always a pressure at the other extremity equal to the weight raised, or effect produced, and the increase of power required by the friction resulting from this pressure, is proportional to the mean distance of its action from the axis of the screw. In most of the instruments called vices, or rather wherever the screw acts by tension between the points of action, this pressure is distributed over a circular space surrounding the body of the screw, and the friction thus remote from the axis must absorb a considerable portion of the power; but in the presses, or where it acts by compression between the points of action, it is commonly confined to the extremity of a pivot, and its retarding effect must be comparatively trifling.

18

Degrees of Friction.

Tables showing the results of the preceding investigation, the distance of the same thread after a complete revolution being the unity, CR-100 in., P=100 lbs.

TABLE I.-Showing the circumferences, diameters, radii, and angles of screws, and the proportional powers required for several different degrees of friction.

n Bases and Circumference. Diameters Radii. | Angle a. (> Min. pwr. Least power.

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TABLE II.-Giving the power for several diameters under each degree of friction, and showing that the diameter given by the formula and the preceding table requires the least power.

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TABLE III.-Which, for the several degrees of friction, shows the diameters of the screws, and the angles of the threads necessary to insure a given ratio between the powers requisite to turn the screw forward and backward.

Proportion of Forces Forward and Backward.

Different Degrees of Friction.

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5.285. 3° 27′ 3.152. 5° 46′ 2-073. 8° 41′ 1.709. 10° 33′ 1.525. 11° 47′ 3.74 2.49

5.85 1.95

8.57 1.72

23-953. 4° 36′ 2-345. 7° 44' 1.525. 119 47′ 1-245. 14° 21'1-102. 16° 7' 4-619 1.55

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13-285. 5° 32′ 1.937. 9° 20′ 1.245. 14° 21' 1.004. 17° 35′ 88. 19° 54′

1 3-051. 5° 46' 1-855. 9° 44' 1.188. 15° 0'
1.224: 816

1.732: 577

955. 18° 26' 833. 20° 54' 2.235: 447

3-285. 5° 32′ 1.937. 9° 20′ 1.245. 14° 21′ 1.004. 17° 35' 88. 19° 54′

3.953. 4° 36′ 2.345. 7° 44′ 1.525. 11° 47' 1.245. 14° 21' 1.102. 16° 7' •682: 341

5.285. 3° 27' 3.152. 5° 46′ 2.073. 8° 44'1-709. 10° 33' 1.525. 11° 47' •402 : ·268 .582.186

513:171

6-750. 2° 42′ 4-033. 4° 31'2.668. 6° 48′ 2.209. 8° 12' 1.978. 9° 9'
8264. 2° 12' 4.945. 3° 41'3-28. 3° 33' 2-721. 6° 40′ 2.441. 7° 26'

9.804. 1° 52′ 5.872. 3° 6' 4-91. 4° 40'3.24. 5° 37' 2-91. 6° 15'
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•281: 056

FOR THE JOURNAL OF THE FRANKLIN INSTITUTE.

Wire Ropes on the Inclined Planes of the Allegheny Portage Railroad.

Last year we called the attention of our readers to the successful application of wire ropes upon our State railway, crossing the Allegheny mountains. It gives us pleasure to have occasion to add further evidence of their utility, and of their superiority over hemp ropes, for the service of inclined planes, whether in point of durability, of efficacy, or of economy. With these remarks, we invite attention to the following extract from the report of John Snodgrass, superintendent of the Allegheny Portage Railroad, to the Canal Commissioners of Pennsylvania, dated December 9th, 1844.

Сом. Рив.

"Ropes for Inclined Planes.-There are eight planes on the Portage Railroad on which hemp ropes are in use, and two on which wire ropes are placed."-"The wire rope on plane No. 3, has now been running for more than two seasons. At present it exhibits some evidence of 'wear.' However, this rope labored under very considerable disadvantage when first placed on the planes. It being altogether an experiment, it was not supposed that the machinery could be perfected at once. For the first two months before the introduction of the double-grooved receiving sheeve, and while the small iron sheeves were on the plane the injury to the rope from friction was greater than that sustained ever since; the double grooved receiving sheeve, and small wooden sheeves having been substituted for those formerly in use, the wear and tear' since has been comparatively small. For what length of time it may yet be competent for the service of the plane is somewhat uncertain; yet enough has been already ascertained to establish the fact, that wire ropes are much superior to hemp ropes, owing to their greater durability, and the evident economy of their introduction."

"The wire rope on plane No. 10, was introduced at the commencement of business last spring; it has been in constant use ever since, and at present presents very little injury from wear and tear. inclined to believe that this rope will last for at least three business seasons, and perhaps for a longer period."

"The principal objection heretofore urged against the substitution of wire ropes for hemp ropes, has been the necessary expense to be incurred in rebuilding foundations of stationary engines. As by the directions of the Board the foundations of the engines at planes Nos. 1 and 6, which, at present, are worked with hemp ropes, will be rebuilt during the present winter; and as the foundation of plane No. 2, was rebuilt last winter, it appears to me advisable to procure wire ropes for said planes."

"The following calculation of the relative cost of hemp and wire ropes must prove most conclusively the advantage of the introduction of the latter. I take it for granted that a good wire cable will be all sufficient to do the business of any plane on the road for a term of

three years-probably much longer. Experience has shown that the average durability of hemp ropes is not more than for one season. The hemp rope now on plane No. 6, weighed 2.81 pounds to the foot run, which, at 15 cents per pound, makes the cost 42.15 cents per foot. Now Mr. Robling proposes to furnish a wire rope, made of No. 14 wire. (such as is now on plane No. 10,) at 51 cents per foot; or a rope made of No. 10 wire, at 37 cents per foot."

"Presuming the foregoing data to be correct, it will give the following result:

"For 5,826 feet of hemp rope, per season, at 15 cents per pound, ($2,455.65) or, for three seasons,

"For 5,826 feet of wire rope, (which will most certainly last three seasons,) at 51 cents per pound,

"Saved by the substitution of wire rope on one plane, "Or $1,463.23 per annum.

$7,366.95

2,971.26

$4,395.69

"There are yet eight planes worked with hemp ropes, requiring 41,779 feet of rope. The foregoing statement shows an annual saving, by the introduction of wire ropes, of $1,456.23 for every 5,826 feet; (the length of rope required for plane No. 6,) or 25.14 cents per foot. This will give an entire annual saving, by placing wire ropes on the eight remaining planes, of $10.503.24."

"One objection heretofore urged against the use of wire ropes on planes, has been the difficulty of attaching cars with safety to the main rope, and also the large number of hemp stops worn out in hitching to it. These objections have been entirely overcome by the substitution of an iron stop,' invented by John Little, pattern maker in the Johnstown depot. This 'stop' is a most excellent invention, and its adoption in place of the hemp stops will be a safeguard against any accident in consequence of the hitch' slipping."

There is a decided advantage in passing cars with wire ropes over and above the economy spoken of, and that is the facility with which cars can be passed in wet weather. The double-grooved receiving sheeves afford sufficient adhesion to draw up any amount of burthen to which the stationary engine power is adequate, while, at certain times, the hemp ropes and single-grooved receiving sheeves are almost useless. My judgment approves of the suggestions made in reference to wire ropes, and I am induced to embrace this opportunity of presenting them to the consideration of the Board of Canal Commissioners.'

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FOR THE JOURNAL OF THE FRANKLIN INSTITUTE.

Modification in the manner of Working Air-Pumps. By O. KEESE. The following is a sketch of a modification in the manner of working air-pumps, which occurred to me a few days since. It may, very possibly, be in use already; but as I am not aware of its being used, and as it appears to me to combine advantages not found in the com

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