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piece BM to become moveable, being prevented from revolv ing, as CD was, by the intervention of a groove and guide, then might the instrument be applied to overcome any given resistance R opposed to the motion of this piece CD by the constant pressure of its pivot upon that piece. The screw is applied under these circumstances in the

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common screw press. The piece A, fixed to the solid frame of the machine, contains a female screw whose thread corresponds to that of the male screw; this screw, when made to turn by means of a handle fixed across it, presses by the intervention of a pivot B, at its extremity, upon the surface of a solid piece EF moveable vertically, but prevented from turning with the screw by grooves receiving two vertical pieces, which serve it as guides, and form parts of the frame of the machine.

The formulæ determined in Art. 251. for the preceding case of the application of the screw, obtain also in this case, if we assume =0. The loss of power due to the friction of the piece EF upon its guides will, however, in this calculation, be neglected; that expenditure is in all cases exceedingly small, the pressure upon the guides, whence their friction results, being itself but the result of the friction of the pivot B upon its bearings; and the former friction being therefore, in all cases, a quantity of two dimensions in respect to the coefficient of friction.

If, instead of the lamina A (p. 326.) being fixed upon the convex surface of a solid cylinder, and B upon the concave surface of a hollow cylinder, the order be reversed, A being fixed upon the hollow and B on the solid cylinder, it is evident that the conditions of the equilibrium will remain the same, the male instead of the female screw being in this case made to progress in the direction of its length. If, however, the longitudinal motion of the male screw B (p. 326.) be, under these circumstances, arrested, and that screw thus become fixed, whilst the obstacle opposed to the longitudinal motion of the female screw A is removed, and that screw thus becomes free to revolve upon the male screw, and also to traverse it longitudinally, except in as far as the latter

motion is opposed by a certain resistance R, which the screw is intended, under these circumstances, to overcome; then will the combination assume the well known form of the screw and nut.

To adapt the formulæ of Art. 251. to this case,, must be made = 0, and instead of assuming the friction upon the extremity of the screw (equation 311) to be that of a solid pivot, we must consider it as that of a hollow pivot, applying to it (by exactly the same process as in Art. (251.), the formula of Art. (177.) instead of Art. (176.).

THE DIFFERENTIAL SCREW.

253. In the combination of three inclined planes discussed in Art. 245., let the plane B be conceived of much greater width than is given to it in the figure (p. 319.), and let it then be conceived to be wrapped upon a convex cylindrical surface. Its two edges ab and cd will thus become the helices of two screws, having their threads of different inclinations wound round different portions of the same cylinder,

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as represented in the accompanying figure, where the thread of one screw is seen winding upon the surface of a solid cylinder from A to C, and the thread of another, having a different inclination, from D to B.

Let, moreover, the planes A and C (p. 319.) be imagined to be wrapped round two hollow cylindrical surfaces, of equal diameters with the above-mentioned solid cylinder, and contained within the solid pieces E and F, through which hollow cylinders AB passes. Two female screws will thus be generated within the pieces E and F, the helix of

the one adapting itself to that of the male screw exter.ding from A to C, and the helix of the other to that upon the male screw extending from D to B. If, then, the piece E be conceived to be fixed, and the piece F moveable in the direction of the length of the screw, but prevented from turning with it by the intervention of a guide, and if a pressure P, be applied at A to turn the screw AB, the action of this combination will be precisely analogous to that of the system of inclined planes discussed in Art. 245., and the conditions of the equilibrium precisely the same; so that the relation between the pressure P, applied to turn the screw (when estimated at the circumference of the thread) and that P1, which it may be made to overcome, are determined by equation (301), and its modulus by equation (302).

The invention of the differential screw has been claimed by M. Prony, and by Mr. White of Manchester. A comparatively small pressure may be made by means of it to yield a pressure enormously greater in magnitude.* It admits of numerous applications, and, among the rest, of that suggested in the preceding engraving.

HUNTER'S SCREW.

254. If we conceive the plane B (p. 319.) to be divided by a horizontal line, and the upper part to be wrapped upon the inner or concave surface of a hollow cylinder, whilst the lower part is wrapped upon the outer or convex circumference of the same cylinder, thus generating the thread of a female screw within the cylinder, and a male screw without it; and if the plane C be then wrapped upon the convex surface of a solid cylinder just fitting the inside or concave surface of the above-mentioned hollow cylin

*It will be seen by reference to equation (301), that the working pressure P, depends for its amount, not upon the actual inclinations t t of the threads, but on the difference of their inclinations; so that its amount may be enor mously increased by making the threads nearly of the same inclination. Thus, neglecting friction, we have, by equation (301), P2=P1· ; sin. (4-4) expression becomes exceedingly great when, nearly equals 12.

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der, and the plane A upon a concave cylindrical surface just capable of receiving and adapting itself to the outside or convex surface of that cylinder, the male screw thus generated adapting itself to the thread of the screw within the hollow cylinder, and the female screw to the thread of that without it; if, moreover, the female screw last mentioned be fixed, and the solid male screw be free to traverse in the direction of its length, but be prevented turning upon its axis by the intervention of a guide; if, lastly, a moving pressure or power be applied to turn the hollow screw, and a resistance be opposed to the longitudinal motion of the solid screw which is received into it; then the combination will be obtained, which is represented in the preceding engraving, and which is well known as Mr. Hunter's screw, having been first described by that gentleman in the seventeenth volume of the Philosophical Transactions.

The theory of this screw is identical with that of the preceding, the relation of its driving and working pressures is determined by equation (301), and its modulus by equation (302).

THE THEORY OF THE SCREW WITH A SQUARE THREAD IN REFERENCE TO THE VARIABLE INCLINATION OF THE THREAD AT DIFFERENT DISTANCES FROM THE AXIS.

255. In the preceding investigation, the inclined plane which, being wound upon the cylinder, generates the thread of the screw, has been imagined to be an exceedingly thin sheet, on which hypothesis every point in the thread may be conceived to be situated at the same distance from the axis of the screw; and it is on this supposition that the relation between the driving and working pressure in the screw and its modulus have been determined.

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Let us now consider the actual case in which the thread

of the screw is of finite thickness, and different elements of it situated at different distance from its axis.

Let mb represent a portion of the square thread of a screw, in which form of thread a line be, drawn from any point b on the outer edge of the thread perpendicular to the axis ef, touches the thread throughout its whole depth bd. Let AC represent a plane perpendicular to its axis, and of the projection of be upon this plane. Take p any point in bd, and let q be the projection of p. 9 Let ep=r, mean radius of thread R, inclination of that helix of the thread whose radius is R*=I, inclination of the helix passing through p=t, whole depth of thread =2D, distance between threads (or pitch) of screw L. Now, since the helix passing through p may be considered to be generated by the enwrapping of an inclined plane whose inclination is upon a cylinder whose radius is r, the base of which inclined plane will then become the arc tq, we have pq=tq. tan. . But, if the angle Afa be increased to 2, pq will become equal to the common distance L between the threads of the screw, and tq will become a complete circle, whose radius is r; therefore L=2r, tan., and this being true for all values of r, therefore L=2-R tan. I. Equating the second members of these equations, and solving in respect to tan. 4,

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From which expression it appears, that the inclination of the thread of a square screw increases rapidly as we recede froin its edge and approach its axis, and would become a right angle if the thread penetrated as far as the axis. Considering, therefore, the thread of the screw as made up of an infinite number of helices, the modulus of each one of which is determined by equation (312), in terms of its corresponding inclination, it becomes a question of much practical importance to determine, if the screw act upon the resistance at one point only of its thread, at what distance from its axis that point should be situated, and if its pressure be applied at all the different points of the depth of its thread, as is commonly the case, to determine how far the conditions of its action are influenced by the different inclinations of the thread at these different depths.

This may be called the mean helix of the thread. The term helix is here taken to represent any spiral line drawn upon the surface of the thread; the distance of every point in which, from the axis of the screw, is the same.

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