by this consideration, but holds even in cases where in certain of fluid which leaves the parallelepiped across one of the faces Equating this to the expression (a), previously found, we have and coa 195. Several references have been made in preceding Freedom sections to the number of independent variables in a dis- straint. placement, or to the degrees of freedom or constraint under which the displacement takes place. It may be well, therefore, to take a general view of this part of the subject by itself. Of a point. Of a rigid body. Freedom and con rigid body. 196. A free point has three degrees of freedom, inasmuch as the most general displacement which it can take is resolvable into three, parallel respectively to any three directions, and independent of each other. It is generally convenient to choose these three directions of resolution at right angles to one another. If the point be constrained to remain always on a given surface, one degree of constraint is introduced, or there are left but two degrees of freedom. For we may take the normal to the surface as one of three rectangular directions of resolution. No displacement can be effected parallel to it: and the other two displacements, at right angles to each other, in the tangent plane to the surface, are independent. If the point be constrained to remain on each of two surfaces, it loses two degrees of freedom, and there is left but In fact, it is constrained to remain on the curve which is common to both surfaces, and along a curve there is at each point but one direction of displacement. 197. Taking next the case of a free rigid body, we have evidently six degrees of freedom to consider-three independent translations in rectangular directions as a point has, and three independent rotations about three mutually rectangular axes. If it have one point fixed, it loses three degrees of freedom; straint of a in fact, it has now only the rotations just mentioned. If a second point be fixed, the body loses two more degrees of freedom, and keeps only one freedom to rotate about the line joining the two fixed points. If a third point, not in a line with the other two, be fixed, the body is fixed. 198. If a rigid body is forced to touch a smooth surface, one degree of freedom is lost; there remain five, two displacements parallel to the tangent plane to the surface, and three rotations. As a degree of freedom is lost by a constraint of the body to touch a smooth surface, six such conditions completely determine the position of the body. Thus if six points on the barrel and stock of a rifle rest on six convex and con rigid body. portions of the surface of a fixed rigid body, the rifle may be Freedom placed, and replaced any number of times, in precisely the straint of a same position, and always left quite free to recoil when fired, for the purpose of testing its accuracy. A fixed V under the barrel near the muzzle, and another under the swell of the stock close in front of the trigger-guard, give four of the contacts, bearing the weight of the rifle. A fifth (the one to be broken by the recoil) is supplied by a nearly vertical fixed plane close behind the second V, to be touched by the trigger-guard, the rifle being pressed forward in its V's as far as this obstruction allows it to go. This contact may be dispensed with and nothing sensible of accuracy lost, by having a mark on the second V, and a corresponding mark on barrel or stock, and sliding the barrel backwards or forwards in the V's till the two marks are, as nearly as can be judged by eye, in the same plane perpendicular to the barrel's axis. The sixth contact may be dispensed with by adjusting two marks on the heel and toe of the butt to be as nearly as need be in one vertical plane judged by aid of a plummet. This method requires less of costly apparatus, and is no doubt more accurate and trustworthy, and more quickly and easily executed, than the ordinary method of clamping the rifle in a massive metal cradle set on a heavy mechanical slide. clamp. A geometrical clamp is a means of applying and main- Geometrical taining six mutual pressures between two bodies touching one another at six points. slide. A "geometrical slide" is any arrangement to apply five Geometrical degrees of constraint, and leave one degree of freedom, to the relative motion of two rigid bodies by keeping them pressed together at just five points of their surfaces. geometrical Ex. 1. The transit instrument would be an instance if Examples of one end of one pivot, made slightly convex, were pressed slide. against a fixed vertical end-plate, by a spring pushing at the other end of the axis. The other four guiding points are. the points, or small areas, of contact of the pivots on the Y's. Ex. 2. Let two rounded ends of legs of a three-legged stool rest in a straight, smooth, V-shaped canal, and the third geometrical on a smooth horizontal plane*. Gravity maintains positive determinate pressures on the five bearing points; and there is a determinate distribution and amount of friction to be overcome, to produce the rectilineal translational motion thus accurately provided for. Example of Ex. 3. Let only one of the feet rest in a V canal, and let clamp. another rest in a trihedral hollowt in line with the canal, the third still resting on a horizontal plane. There are thus six bearing points, one on the horizontal plane, two on the sides of the canal, and three on the sides of the trihedral hollow: and the stool is fixed in a determinate position as long as all these six contacts are unbroken. Substitute for gravity a spring, or a screw and nut (of not infinitely rigid material), binding the stool to the rigid body to which these six planes belong. Thus we have a "geometrical clamp," which clamps two bodies together with perfect firmness in a perfectly definite position, * Thomson's reprint of Electrostatics and Magnetism, § 346. + A conical hollow is more easily made (as it can be bored out at once by an ordinary drill), and fulfils nearly enough for most practical applications the geometrical principle. A conical, or otherwise rounded, hollow is touched at three points by knobs or ribs projecting from a round foot resting in it, and thus again the geometrical principle is rigorously fulfilled. The virtue of the geometrical principle is well illustrated by its possible violation in this very case. Suppose the hollow to have been drilled out not quite "true," and instead of being a circular cone to have slightly elliptic horizontal sections:-A hemispherical foot will not rest steadily in it, but will be liable to a slight horizontal displacement in the direction parallel to the major axes of the elliptic sections, besides the legitimate rotation round any axis through the centre of the hemispherical surface: in fact, on this supposition there are just two points of contact of the foot in the hollow instead of three. When the foot and hollow are large enough in any particular case to allow the possibility of this defect to be of moment, it is to be obviated, not by any vain attempt to turn the hollow and the foot each perfectly "true:"-even if this could be done the desired result would be lost by the smallest particle of matter such as a chip of wood, or a fragment of paper, or a hair, getting into the hollow when, at any time in the use of the instrument, the foot is taken out and put in again. On the contrary, the true geometrical method, (of which the general principle was taught to one of us by the late Professor Willis thirty years ago,) is to alter one or other of the two surfaces so as to render it manifestly not a figure of revolution, thus:-Roughly file three round notches in the hollow so as to render it something between a trihedral pyramid and a circular cone, leaving the foot approximately round; or else roughly file at three places of the rounded foot so that horizontal sections through and a little above and below the points of contact may be (roughly) equilateral triangles with rounded corners. geometrical without the aid of friction (except in the screw, if a screw Example of is used); and in various practical applications gives very clamp. readily and conveniently a more securely firm connexion by one screw slightly pressed, than a clamp such as those commonly made hitherto by mechanicians can give with three strong screws forced to the utmost. geometrical Do away with the canal and let two feet (instead of only one) Example of rest on the plane, the other still resting in the conical hollow, slide. The number of contacts is thus reduced to five (three in the hollow and two on the plane), and instead of a "clamp" we have again a slide. This form of slide,-a three-legged stool with two feet resting on a plane and one in a hollow,—will be found very useful in a large variety of applications, in which motion about an axis is desired when a material axis is not conveniently attainable. Its first application was to the "azimuth mirror," an instrument placed on the glass cover of a mariner's compass and used for taking azimuths of sun or stars to correct the compass, or of landmarks or other terrestrial objects to find the ship's position. It has also been applied to the "Deflector," an adjustible magnet laid on the glass of the compass bowl and used, according to a principle first we believe given by Sir Edward Sabine, to discover the "semicircular" error produced by the ship's iron. The movement may be made very frictionless when the plane is horizontal, by weighting the moveable body so that its centre of gravity is very nearly over the foot that rests in the hollow. One or two guard feet, not to touch the plane except in case of accident, ought to be added to give a broad enough base for safety. The geometrical slide and the geometrical clamp have both been found very useful in electrometers, in the "siphon recorder," and in an instrument recently brought into use for automatic signalling through submarine cables. An infinite variety of forms may be given to the geometrical slide to suit varieties of application of the general principle on which its definition is founded. An old form of the geometrical clamp, with the six pressures produced by gravity, is the three V grooves on a stone slab bearing the three legs of an astronomical or magnetic instru |