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of the fluid, per unit of time, in the fixed space in question, bears to the actual density, at any instant, the same ratio that the rate of acquisition of matter into that space bears to the whole matter in that space.
165. Several references have been made in preceding sections to the number of independent variables in a displacement, or to the degrees of freedom or constraint under which the displacement takes place. It may be well, therefore, to take a general (but cursory) view of this part of the subject itself.
166. 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 one.
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.
167. Taking next the case of a free rigid system, we have evidently six degrees of freedom to consider three independent displacements or translations in rectangular directions as a point has, and three independent rotations about three mutually rectangular axes.
If it have one point fixed, the system loses three degrees of free dom; in fact, it has now only the rotations above mentioned.
This fixed point may be, and in general is, a point of a continuous surface of the
body in contact with a continuous fixed surface. These surfaces may be supposed 'perfectly rough, so that sliding may be impossible.
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,
168. If one point of the rigid system is forced to remain on a smooth surface, one degree of freedom is lost; there remain five,. two displacements in the tangent plane to the surface, and three rotations. As an additional degree of freedom is lost by each successive limita tion of a point in the body to a smooth surface, six such conditions completely determine the position of the body. Thus if six points
properly chosen on the barrel and stock of a rifle be made to rest on six convex portions of the surface of a fixed rigid body, the rifle may be replaced any number of times in precisely the same position, 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 back. wards 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 ordi. nary method of clamping the rifle in a massive metal cradle set on a heavy.mechanical slide.
A geometrical clamp is a means of applying and maintaining six mutual pressures between two bodies touching one another at six points.
A geometrical slide' is any arrangement to apply five 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.
Ex. I. The transit instrument would be an instance if one end of one pivot, made slightly convex, were pressed 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 con. cact 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 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.
Ex. 3. Let only one of the feet rest in a V canal, and let another rest in a trihedral hollow 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, without the aid of friction (except in the screw, If a screw is used); and in various practical applications gives very 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.
Do away with the canal and let two feet (instead of only one) rest on the plane, the other still resting in the conical hollow. 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 move able 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 instrument. It is not generally however so well-conditioned' as the trihedral hole, the V groove, and the horizontal plane contact, described above.
There is much room for improvement by the introduction of geometrical slides and geometrical clamps, in the mechanism of mathematical, optical, geodetic, and astronomical instruments : which as made at present are remarkable for disregard of geome. trical and dynamical principles in their slides, micrometer screws, and clamps. Good workmanship cannot compensate for bad design, whether in the safety-valve of an ironclad, or the movements and adjustments of a theodolite.
169. If one point be constrained to remain in a curve, there reinain four degrees of freedom.
If two points be constrained to remain in given curves, there are our degrees of constraint, and we have left two degrees of freedom. One of these may be regarded as being a simple rotation about the ine joining the constrained parts, a motion which, it is clear, the body is free to receive. It may be shown that the other possible notion is of the most general character for one degree of freedom; that is to say, translation and rotation in any fixed proportions, as of he nut of a screw.
If one line of a rigid system be constrained to remain parallel to .tself, as for instance, if the body be a three-legged stool standing on
perfectly smooth board fixed to a common window, sliding in its rame with perfect freedom, there remain three displacements and one rotation.
But, we need not farther pursue this subject, as the number of combinations that might be considered is almost endless; and those already given suffice to show how simple is the determination of the degrees of freedom or constraint in any case that may present itself.
170. One degree of constraint of the most general character, is not producible by constraining one point of the body to a curve surface; but it consists in stopping one line of the body from longitudinal motion, except accompanied by rotation round this line, in fixed proportion to the longitudinal motion. Every other motion being left unimpeded, there remains free rotation about any axis perpen. dicular to that line (two degrees of freedom); and translation in any direction perpendicular to the same line (two degrees of freedom). These last four, with the one degree of freedom to screw, con. stitute the five degrees of freedom, which, with one degree of con. straint, make up the six elements. This condition is realized in the following mechanical arrangement, which seems the simplest that can be imagined for the purpose :
Let a screw be cut on one shaft of a Hooke's joint, and let the other shaft be joined to a fixed shaft by a second Hooke's joint. A nut turning on that screw-shaft has the most general kind of motion admitted when there is one degree of constraint. Or it is subjected to just one degree of constraint of the most general character. It has five degrees of freedom ; for it may move, ist, by screwing on its shaft, the two Hooke's joints being at rest; and, it may rotate about either axis of the first Hooke's joint, or any axis in their plane (two more degrees of freedom : being freedom to rotate about two axes through one point); 3rd, it may, by the two Hooke's joints, each bending, have translation without rotation in any direction perpendicular to the link or shaft between the two Hooke's joints (two more degrees of freedom). But it cannot have a notion of translation parallel to the line of the link without a definite proportion of rotation round this line; nor can it have rotation round this line without a definite proportion of translation parallel to it.
DYNAMICAL LAWS AND PRINCIPLES.
171. In the preceding chapter we considered as a subject of pure geometry the motion of points, lines, surfaces, and volumes, whether taking place with or without change of dimensions and form; and the results we there arrived at are of course altogether independent of the idea of matter, and of the forces which matter exerts. We have here. tofore assumed the existence merely of motion, distortion, etc.; we now come to the consideration, not of how we might consider such motion, etc., to be produced, but of the actual cause's which in the material world do produce them. The axioms of the present chapter must therefore be considered to be due to actual experience, in the shape either of observation or experiment. How such experience is to be conducted will form the subject of a subsequent chapter.
172. We cannot do better, at all events in commencing, than follow Newton somewhat closely. Indeed the introduction to the Principia contains'in a most lucid form the general foundations of dynamics. The Definitiones and Axiomata, sive Leges Motús, there laid down, require only a few amplifications and additional illustrations, suggested by subsequent developments, to suit them to the present state of science, and to make a much better introduction to dynanics than we find in even some of the best modern treatises.
173. We cannot, of course, give a definition of Matter which will satisfy the metaphysician; but the naturalist may be content to know matter as that which can be perceived by the senses, or as that which can be acted upon by, or can exert, force. The latter, and indeed the former also, of these definitions involves the idea of Force, which, in point of fact, is a direct object of sense; probably of all our senses, and certainly of the muscular sense.' To our chapter on Properties of Matter we must refer for further discussion of the question, What is matter 9
174. The Quantity of Matter in a body, or, as we now call it, the. Mass of a body, is proportional, according to Newton, to the Volume and the Density conjointly. In reality, the definition gives us the meaning of density rather than of mass; for it shows us that if tivice the original quantity of matter, air for example, be sorced into a vessel