250. It is evident from equations 306, 307, 308, that, since, whatever may be the weight of the impinging body or the velocity of the impact, a certain finite amount of work U, is yielded upon the resistances opposed to the motion of the wedge; there is in every such case a certain mean resistance Rovercome through a certain space S, in the direction in which that resistance acts, which resistance and space are such, that If therefore the space S be exceedingly small as compared *This is the form under which the power of the wedge is applied for the expressing of oil. with U,, there will be an exceedingly great resistance R overcome by the impact through that small space, however slight the impact. From this fact the enormous amount of the resistances which the wedge, when struck by the ham mer, is made to overcome, is accounted for. The power of thus subduing enormous resistances by impact is not how ever peculiar to the wedge, it is common to all implements of impact, and belongs to its nature; its effects are rendered permanent in the wedge by the property possessed by that implement of retaining permanently any position into which it is driven between two resisting surfaces, and thereby opposing itself effectually to the tendency of those surfaces, by reason of their elasticity, to recover their original form and position. It is equally true of any the slightest direct impact of the hammer as of its impact applied through the wedge, that it is sufficient to cause any finite resistance opposed to it to yield through a certain finite space, however great that resistance may be. The difference lies in this, that the surface yielding through this exceedingly small but finite space under the blow of the hammer, immediately recovers itself after the blow if the limits of elasticity be not passed; whereas the space which the wedge is, by such an impact, made to traverse, in the direction of its length, becomes a permanent separation. THE SCREW. 251. Let the system of two moveable inclined planes represented in fig. p. 318. be formed of exceedingly thin and pliable laminæ, and conceive one of them, A for instance, to be wound upon a convex cylindrical surface, as shown in the accompanying figure, and the other, B, upon a concave cylindrical surface having an equal diameter, and the same axis with the other; then will the surfaces EF and GH of these planes represent truly the threads or helices of two screws, one of them of the form called the male screw, and the other the female screw. Let the helix EF be continued, so as to form more than one spire or convolution of the thread; if, then, the cylinder which carries this helix be made to revolve upon its axis by the action of a pressure P, applied to its circumference, and the cylinder which carries the helix GH be prevented from re R volving upon its axis by the opposition of an obstacle D, which leaves that cylinder nevertheless free to move in a direction parallel to its axis, it is evident that the helix EF will be made to slide beneath GH, and the cylinder which carries the latter helix to traverse longitudinally; moreover, that the conditions of this mutual action of the helical surfaces EF and GH will be precisely analogous to those of the surfaces of contact of the two moveable inclined planes discussed in Art. 244. So that the conditions of the equilibrium of the pressures P, and P, in the state bordering upon motion, and the modulus of the system, will be the same in the one case as in the other; with this single exception, that the resistance R, of the mass on which the plane A rests (see fig. p. 318.) is not, in the case of the screw, applied only to the thin edge of the base of the lamina A, but to the whole extremity of the solid cylinder on which it is fixed, or to a circular projection from that extremity serving it as a pivot. Now if, in equation 299, we assume =0, we shall obtain that relation of the pressures P, and P, in their state bordering upon motion, which would obtain if there were no friction of the extremity of the cylinder on the mass on which it rests; and observing that the pressure P, is precisely that by which the pivot at the extremity of the cylinder is pressed upon this mass, and therefore the moment (see Art. 176, equation 183) of the resistance to the rotation of the cylinder produced by the friction of this pivot by Pp tan. „, where p represents the radius of the pivot; observing, moreover, that the pressure which must be applied at the circumference of the cylinder to overcome this resistance, above that which would be required to give motion to the screw if there 2 were, no such friction, is represented by P, tan. 92, 7 being P 2 3 r taken to represent the radius of the cylinder, we obtain for the entire value of the pressure P, in the state bordering upon motion P1=P, P. sin. (1 +❤,) cos. 9, + 2P, £ tan. 9, The pressure P, has here been supposed to be applied to turn the screw at its circumference; it is customary, however, to apply it at some distance from its circumference by the intervention of an arm. If a represent the length of such an arm, measuring from the axis of the cylinder, it is evident that the pressure P, applied to the extremity of that arin, would produce at the circumference of the cylinder a pressure a represented by P, which expression being substituted for P, in the preceding equation, and that equation solved in respect to P,, we obtain finally for the relation between P. and P, in their state bordering upon motion, P1 =P, ( ) { 00s. (+++, +93) cos. } + tan. 9,.. (311). 3 If in like manner we assume in the modulus (equation 300) ,=0, and thus determine a relation between the work doneé at the driving point and that yielded at the working point, on the supposition that no work is expended on the friction of the pivot; and if to the value of U, thus obtained we add the work expended upon the resistance of the pivot which is shown (equation 184) to be represented at each revolution by P tan.,, and therefore during n revolutions by 4 29 P1, we shall obtain the following general expression for the modulus; the whole expenditure of work due to the prejudicial resistances being taken into account. U1=U,. 4 3 sin. (+) cos. 9, Representing by λ the common distance between the threads of the screw, i. e. the space which the nut B is made to traverse at each revolution of the screw; and observ 4 4 U. ing that nλP,=U,, so that npP, tan. 9,=*='_tan. 9,= 22. U, tan. 9, in which expression 3λ ጥ 2 obtain finally for the modulus of the screw It is evidently immaterial to the result at what distance from the axis the obstacle D is opposed to the revolution of that cylinder which carries the lamina B; since the amount of that resistance does not enter into the result as expressed in the above formula, but only its direction determined by the angle, which angle depends upon the nature of the resisting surfaces, and not upon the position of the resisting point. APPLICATIONS OF THE SCREW. 252. The accompanying figure represents an application of the screw under the circumstances described in the preceding article, to the well known machine called the VICE. AB is a solid cylinder carrying on its surface the thread of a male screw, and within the piece CD is a hollow cylindrical surface, carrying the corresponding thread of a female screw; this female screw is prevented from revolving with the male screw by a groove in the piece CD, which carries it, and which is received into a corresponding projection EF of the solid frame of the machine, serving it as a guide; which guide nevertheless allows a longitudinal motion to the piece CD. A projection from the frame of the instrument at B, met by a pivot at the extremity of the male screw, opposes itself to the tendency of that screw to traverse in the direction of its length. The pressure P, to be Overcome is applied between the jaws H and K of the vice, and the driving pressure P, to an arm which carries round with it the screw AB. It is evident that, in the state bordering upon motion, the resistance R upon the pivot at the extremity B of the screw AB, resolved in a direction parallel to the length of that Screw, must be equal to the pressure P, (see Art. 16.); so that if we imagine the piece CD to become fixed, and the |