parallelogram will be on the same base AC', and between the same parallels AC and CA', and their other sides will be equally inclined on the two sides of a perpendicular to them. Hence, irrespectively of any rotation, or other absolute motion of the body not involving change of form or dimensions, the strain under consideration may be produced by holding fast and unaltered the plane of the body through AC', perpendicular to the plane of the diagram, and making every plane parallel to it slide, keeping the same distance, through a space proportional to this distance (i. e. different planes parallel to the fixed one slide through spaces proportional to their distances). 150. This kind of strain is called a simple shear. The plane of a shear is a plane perpendicular to the undistorted planes, and parallel to the lines of the relative motion. It has (1) the property that one set of parallel planes remain each unaltered in itself; (2) that another set of parallel planes remain each unaltered in itself. This other set is got when the first set and the degree or K amount of shear are given, thus :-Let N CC, be the motion of one point of one plane, relative to a plane KL held fixedthe diagram being in a plane of the shear, Bisect CC, in N. Draw NA perpendicular to it. A plane perpendicular to the plane of the diagram, initially through AC, and finally through AC, remains unaltered in its dimensions. 151. One set of parallel undistorted planes and the amount of their relative parallel shifting having been given, we have just seen how to find the other set. The shear may be otherwise viewed, and considered as a shifting of this second set of parallel planes, relative to any one of them. The amount of this relative shifting is of course equal to that of the first set, relatively to one of them. 152. The principal axes of a shear are the lines of maximum elongation and of maximum contraction respectively. They may be found from the preceding construction ($ 150), thus :-In the plane of the shear bisect the obtuse and acute angles between the planes destined not to become deformed. The former bisecting line is the principal axis of elongation, and the latter is the principal axis of contraction, in their initial positions. The former angle (obtuse) becomes equal to the latter, its supplement (acute), in the altered condition of the body, and the lines bisecting the altered angles are the D. N. E principal axes of the strain in the altered body. Otherwise, taking a plane of shear for the plane of the diagram, let AB be a B line in which it is cut by one of either set of parallel planes of no distortion. On any portion AB of this as diameter, describe a semicircle. Through C, its middle point, draw, by the preceding construction, CD the initial, and CE the final, position of an unstretched line. Join DA, DB, EA, EB. DA, DB are the initial, and EA, EB the final, positions of the principal axes. 153. The ratio of a shear is the ratio of elongation and contraction of its principal axes. Thus if one principal axis is elongated in the ratio 1 : a, and the other therefore (§ 148) contracted in the ratio a:1, a is called the ratio of the shear. It will be convenient generally to reckon this as the ratio of elongation; that is to say, to make its numerical measure greater than unity. In the diagram of $ 152, the ratio of DB to EB, or of EA to DA, is the ratio of the shear. 154. The amount of a shear is the amount of relative motion per unit distance between planes of no distortion. It is easily proved that this is equal to the excess of the ratio of the shear above its reciprocal. 155. The planes of no distortion in a simple shear are clearly the circular sections of the strain ellipsoid. In the ellipsoid of this case, be it remembered, the mean axis remains unaltered, and is a mean proportional between the greatest and the least axis. 156. If we now suppose all lines perpendicular to the plane of the shear to be elongated or contracted in any proportion, without altering lengths or angles in the plane of the shear, and if, lastly, we suppose every line in the body to be elongated or contracted in some other fixed ratio, we have clearly (§ 142) the most general possible kind of strain. 157. Hence any strain whatever may be viewed as compounded of a uniform dilatation in all directions, superimposed on a simple elongation in the direction of one principal axis superimposed on a simple shear in the plane of the two other principal axes. 158. It is clear that these three elementary component strains may be applied in any other order as well as that stated. Thus, if the simple elongation is made first, the body thus altered must get just the same shear in planes perpendicular to the line of elongation as the originally unaltered body gets when the order first stated is followed. Or the dilatation may be first, then the elongation, and finally the shear, and so on. 159. When the axes of the ellipsoid are lines of the body whose direction does not change, the strain is said to be pure, or unaccompanied by rotation. The strains we have already considered were pure strains accompanied by rotations. 160. If a body experience a succession of strains, each unaccompanied by rotation, its resulting condition will generally be producible by a strain and a rotation. From this follows the remarkable corollary that three pure strains produced one after another, in any piece of matter, each without rotation, may be so adjusted as to leave the Q1 PP 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 Py; then from P, on to P,; and let O be taken in a line through P1, perpendicular to CD. During the shear from P to P Р. -D a point l moves of course to l through a distance la -E Choose ( midway be tween P and P, so that PQ= 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 Q to Q, is 04, 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 gol, that is to say, is equal to P,00. 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 li, 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 0A. 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 Pą, 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 Q OQ by the excess of POQ above QOP. Hence the resultant of the two shears, PP, P,P,, 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: w 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 E 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 |