Chap. XVII.
THEORY OF THE MOTION OF FLUIDS.
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extremity of which is a winch, and at the lower a gudgeon; the handle and gudgeon reston square wooden blocks. When the Archimedean screw is applied to emptying water, thelower end is plunged into it by inclining the axle at a certain angle, the limit of whichmust be determined geometrically ; the winch is turned, taking care that its rotation is ina contrary direction to that of the generating point of the helices, which are drawn on theexterior, and serve as guides to the generating lines of the lillets. The barrel is formed ofstaves strongly bound together by iron hoops, the staves sufficiently close to allow of nowater passing, but the atmospheric air penetrates the joints, so that the interior air betweenthe fillets of the screw is always of the same density as the outer: if the thickness of thefillet be disregarded, we may consider it as composed of helices drawn on parallel cylinders,having for their common axis that of the screw; these helices having the same number ofturns, their bases being the circles of the cylinders on which they are drawn, the com-mon axis of these cylinders is inclined to the horizontal plane of the water level; thelimit of this inclination depends on the angle made by the tangents of each helix withthe same plane of. the water level : each helix may thus be considered as the elementof a fillet, or a small hollow channel whose origin is at the orifice at which the water in-troduces itself: having drawn through this orifice a tangent to the helix, we may sup-pose it prolonged to the lower part of the screw as a right tube, through which the watermay introduce itself into the heHcial tube: if the screw be turned on its axis, the righttube will engender an hyperboloid of revolution, the common point of the two tubes, ofwhich one is straight, and the other spiral, and the point placed at the extremity of thestraight tube would describe two circles; the right tube in all its positions would rest ontwo points in these two circles, and it is on the uosition of those two points, with regard to
Fig. 1769.
the level plane, that the introduction of water into the right tube depends ; for it is necessarythat the extremity of this tube should be less elevated than the point in which it touchesthe helicial tube, in order that the water may slide on the tangent to the helix as on aninclined plane; there is, consequently, a necessary relation between the inclination of theaxis of the screw and the tangents to the helices of its threads, which induces the water torise: this relation determines the limit of the inclination of the screw’s axis with regard tothe level of the water. Suppose the orifice of the extreme helix, or the element of thefillet farthest from the axis of the screw, constantly plunged in water, while the screwturns on its axis, the right line, which touches the helix at its origin, will, in turning atthe same time as the screw, generate an hyperboloid of revolution ; a parallel to this rightline, drawn through any point of the axis, would generate a right cone, the asymtote ofthe hyperboloid of revolution ; if through the summit of this right cone, a plane parallel tothe level of the water be drawn, either this plane would not touch the cone, or it wouldcut it at the two edges, or be in a tangent thereto: supposing it a tangent, it would makewith the axis of the screw an angle which will be the limit of inclination of this axis withthe horizontal plane, for if this inclination were increased, there would be no position of theextreme helix in which the water could introduce itself; the middle helices, that is to say,those nearest the newel, whose orifices would turn in the water, could not, a fortiori , receiveany portion of this water, and no useful effect would be obtained from the motion of thewhole screw.
In Holland the screw is constructed without a barrel, and the fillets turn in a fixed por-tion of the hollow cylinder, the lower extremity of which, as well as that of the screwplunges in the water to be elevated, which fills the capacity of the hollow cylinder, and es-capes at its upper extremity.
Hie Dutch screws are generally of large dimensions, the inner diameter of the hollowcylinder being about seventeen decimetres: they are moved by the wind, and the samewindmill turns several.