body unstrained, but rotated through some angle about some axis, We shall have, later, most important and interesting applications to fluid motion, which will be proved to be instantaneously, or differentially, irrotational; but which may result in leaving a whole fluid mass merely turned round from its primitive position, as if it had been a rigid body. [The following elementary geometrical investigation, though not bringing out a thoroughly comprehensive view of the subject, affords a rigorous demonstration of the proposition, by proving it for a particular case. Let us consider, as above ($ 150), a simple shearing motion. A point O being held fixed, suppose the matter of the body in a plane, cutting that of the diagram perpendicularly in CD, to move in this plane from right to left parallel to CD; and in other planes parallel to it let there be motions proportional to their distances from 0. Consider first a shear from P to P,; then from P, on to P,; and let O be taken in a line through Pi, perpendicular to CD. During the shear from P to P, _P2 QiPi a pointe moves of course to l through a distance el, = PP Choose e midway be tween P and P, so that Pl= QP = {P.P. Now, as we have seen above ($ 152), the line of the body, which is the principal axis of contraction in the shear from l to lı, is 0A, bisecting the angle QOE at the beginning, and OA,, bisecting QOE at the end, of the whole motion considered. The angle between these two lines is half the angle QOQ, that is to say, is equal to P,OQ. Hence, if the plane CD is rotated through an angle equal to P,00, in the plane of the diagram, in the same way as the hands of a watch, during the shear from to Qı, or, which is the same thing, the shear from P to P1, this shear will be effected without final rotation of its principal axes. (Imagine the diagram turned round till OA, lies along OA. The actual and the newly imagined position of CD will show how this plane of the body has moved during such nonrotational shear.) Now, let the second step, P, to P, be made so as to complete the whole shear, P to Pn, which we have proposed to consider. Such second partial shear may be made by the common shearing process parallel to the new position imagined in the preceding parenthesis) of CD, and to make itself also non-rotational, as its predecessor has been made, we must turn further round, in the same direction, through an angle equal to QOP, Thus in these two steps, each made non-rotational, we have turned the plane CD round through an angle equal to QOQ. But now, we have a whole shear PP,; and to make this as one non-rotational shear, we must turn CD through an angle P,OP only, which is less than QOQ by the excess of P,OQ above QOP. Hence the resultant of the two shears, PP, P P2, each separately deprived of rotation, is a single shear PP, and a rotation of its principal axes, in the direction of the hands of a watch through an angle equal to QOP,-POQ. 161. Make the two partial shears each non-rotationally. Return from their resultant in a single non-rotational shear: we conclude with the body unstrained, but turned through the angle QOP,-POQ, in the same direction as the hands of a watch.] 162. As there can be neither annihilation nor generation of matter in any natural motion or action, the whole quantity of a fluid within any space at any time must be equal to the quantity originally in that space, increased by the whole quantity that has entered it, and diminished by the whole quantity that has left it. This idea, when expressed in a perfectly comprehensive manner for every portion of a fluid in motion, constitutes what is commonly called the equation of continuity. 163. Two ways of proceeding to express this idea present themselves, each affording instructive views regarding the properties of fluids. In one we consider a definite portion of the fluid; follow it in its motions; and declare that the average density of the substance varies inversely as its volume. We thus obtain the equation of continuity in an integral form. The form under which the equation of continuity is most commonly given, or the differential equation of continuity, as we may call it, expresses that the rate of diminution of the density bears to the density, at any instant, the same ratio as the rate of increase of the volume of an infinitely small portion bears to the volume of this portion at the same instant. 164. To find the differential equation of continuity, imagine a space fixed in the interior of a fluid, and consider the fluid which flows into this space, and the fluid which flows out of it, across different parts of its bounding surface, in any time. If the fluid is of the same density and incompressible, the whole quantity of matter in the space in question must remain constant at all times, and therefore the quantity flowing in must be equal to the quantity flowing out in any time. If, on the contrary, during any period of motion, more fluid enters than leaves the fixed space, there will be condensation of matter in that space; or if more fluid leaves than enters, there will be dilatation. The rate of augmentation of the average density 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 freedom; 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 limitation 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. 169. If one point be constrained to remain in a curve, there remain four degrees of freedom. If two points be constrained to remain in given curves, there are four degrees of constraint, and we have left two degrees of freedom. One of these may be regarded as being a simple rotation about the line joining the constrained parts, a motion which, it is clear, the body is free to receive. It may be shown that the other possible motion is of the most general character for one degree of freedom; that is to say, translation and rotation in any fixed proportions, as of the nut of a screw. If one line of a rigid system be constrained to remain parallel to itself, as for instance, if the body be a three-legged stool standing on a perfectly smooth board fixed to a common window, sliding in its frame 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 perpendicular to that line (two degrees freedom); and translation in any direction perpendicular to the same line (two degrees freedom). These last four, with the one degree of freedom to screw, constitute the five degrees of freedom, which, with one degree of constraint, 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; 2nd, 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 motion 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. CHAPTER II. 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 heretofore 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 causes 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 dynamics 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 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 twice the original quantity of matter, air for example, be forced into a vessel of given capacity, the density will be doubled, and so on. But it also |