ings of tan placement Analytically thus :-Let x, y, z be the co-ordinates of any Two reckonpoint, P, in the undisplaced curve; x1, 1, 21, those of P, the gential dis point to which the same point of the curve is displaced. Let compared. dr, dy, dz be the increments of the three co-ordinates corresponding to any infinitely small arc, ds of the first; so that ds=(dx2+dy+dz2)1, and let corresponding notation apply to the corresponding element of the displaced curve. Let denote the angle between the line PP, and the tangent to the undisplaced curve through P; so that we have D= {(x,−x)2+(y, − y)2 + (z, −2)}} D cos 0,ds,(x,-x)dx,+(y,-y)dy,+(z,-z)dz, and therefore or D cos 0,ds,-D cos Ods=(x,—x)d(x,−x)+(y1—y)d(y,—y) +(2,−2)d(≈,−2) D cos 0,ds,-D cos 0ds=d(D). To find the difference of the tangential displacements reckoned the two ways, we have only to integrate this expression. Thus we obtain fDcos, ds,-D cos 0ds=(D-D'2) = }(D'' + D'') (D'' —D') where D and D' denote the displacements of the two ends. 188. The entire tangential displacement of a closed curve Tangential is the same whether reckoned along the undisplaced or the ment of a displaced curve. 189. The entire tangential displacement from one to another of two conterminous ares, is the same reckoned along either as along the other. displace closed curve. a rigid closed 190. The entire tangential displacement of a rigid closed Rotation of Curve when rotated through any angle about any axis, is equal curve. to twice the area of its projection on a plane perpendicular to the axis, multiplied by the sine of the angle. (a) Prop.-The entire tangential displacement round a closed Tangential curve of a homogeneously strained solid, is equal to displacement in a solid, in terms of components of strain. Tangential displacement in a solid, in terms of components of strain. where P, Q, R denote, for its initial position, the areas of its projections on the planes YOZ, ZOX, XOY respectively, and , P, σ are as follows: Hence, according to the previously investigated expression, we have, for the tangential displacement, reckoned along the undisplaced curve, {(x,-x)dx+(y1 —y)dy + (z, —z)dz} 1 =S[}d{ (A — 1)x2+(B − 1)y2+(C− 1)z2+2(ayz+bzx+cxy)} +(ydz-zdy)+p(zdx-xdz)+σ(xdy — ydx)]. The first part, d{}, vanishes for a closed curve. The remainder of the expression is af(yđã —zdy)+Pf(zd — xdz)+of(xdy—yde) which, according to the formulæ for projection of areas, is equal double the area of the orthogonal projection of the curve on that plane; and similarly for the other integrals. (b) From this and § 190, it follows that if the body is rigid. and therefore only rotationally displaced, if at all, [Zy] — [Y:] is equal to twice the sine of the angle of rotation multiplied by the cosine of the inclination of the axis of rotation to the line of reference OX. (c) And in general [Zy] — [Yz] measures the entire tangential displacement, divided by the area on ZOY, of any closed curve given, if a plane curve, in the plane YOZ, or, if a tortuous curve. given so as to have zero area projections on ZOX and XOY. The entire tangential displacement of any closed curve given in a plane, A, perpendicular to a line whose direction-cosines are proportional to, p, o, is equal to twice its area multiplied by displace √(=2+p2+o3). And the entire tangential displacement of any Tangential closed curve whatever is equal to twice the area of its projection ment in a on A, multiplied by √(~2+p2+σ3). solid, in terms of of strain. In the transformation of co-ordinates, ≈, p, σ transform by the components elementary cosine law, and of course 2+p2+σ2 is an invariant; that is to say, its value is unchanged by transformation from one set of rectangular axes to another. (d) In non-rotational homogeneous strain, the entire tangential displacement along any curve from the fixed point to (x, y, z), reckoned along the undisplaced curve, is equal to } { {( A − 1 ) x2 + ( B — 1)y2 + ( C — 1 ):2+2(ayz+bzx+cxy)}. Reckoned along displaced curve, it is, from this and § 187, (A-1)x+(B-1)y2+(C-1)z2+2(ayz+bzx+cxy)} + }} {[(A−1)x+cy+bz]2+[cx+(B−1)y+az]3 +[bx+ay+(C− 1)z]2 }. And the entire tangential displacement from one point along any curve to another point, is independent of the curve, i..., is the same along any number of conterminous curves, this of course whether reckoned in each case along the undisplaced or along the displaced curve. geneous (e) Given the absolute displacement of every point to find the Heterostrain. Let a, ẞ, y be the components, relative to fixed axes, strain. OX, OY, OZ, of the displacement of a particle, P, initially in the position x, y, z. That is to say, let x+a, y+ẞ, z+y be the co-ordinates, in the strained body, of the point of it which was initially at x, y, z. Consider the matter all round this point in its first and second positions. Taking this point P as moveable origin, let έ, 7, be the initial co-ordinates of any other point near it, and §1, 71, Ši the final co-ordinates of the same. The initial and final co-ordinates of the last-mentioned point, with reference to the fixed axes OX, OY, OZ, will be are the components of the displacement of the point which had η, the higher powers and products of §, 7, § being neglected. Com paring these expressions with (1) of § 181, we see that they express the changes in the co-ordinates of any displaced point of a body relatively to three rectangular axes in fixed directions through one point of it, when all other points of it are displaced relatively to this one, in any manner subject only to the condition of giving a homogeneous strain. Hence we perceive that at distances all round any point, so small that the first terms only of the expressions by Taylor's theorem for the differences of displacement are sensible, the strain is sensibly homogeneous and we conclude that the directions of the principal axes of the strain at any point (x, y, z), and the amounts of the elongations of the matter along them, and the tangential displacements in closed curves, are to be found according to the general methods described above, by taking Homo geneous strain. Infinitely small strain Most general motion of matter. dz If each of these nine quantities is constant (i.e., the same for all values of x, y, z), the strain is homogeneous: not unless. (ƒ) The condition that the strain may be infinitely small is that da da da (g) These formulæ apply to the most general possible motion of any substance, and they may be considered as the fundamental equations of kinematics. If we introduce time as independent variable, we have for component velocities u, v, w, parallel to the fixed axes OX, OY, OZ, the following expressions; x, y, z, t Most general being independent variables, and a, ß, y functions of them : motion of matter. (h) If we introduce the condition that no line of the body experiences any elongation, we have the general equations for the kinematics of a rigid body, of which, however, we have had Change of enough already. The equations of condition to express this a rigid body da will be six in number, among the nine quantities , etc., which (g) are. in this case, each constant relatively to x, y, z. There are left three independent arbitrary elements to express any angular motion of a rigid body. position of tional strain (1) If the disturbed condition is so related to the initial con- Non-rotadition that every portion of the body can pass from its initial to its disturbed position and strain, by a translation and a strain without rotation; i.e., if the three principal axes of the strain at any point are lines of the substance which retain their parallelism, we must have, § 183 (18), and if these equations are fulfilled, the strain is non-rotational, as is the differential of a function of three independent variables. the principal axes of the strain at every point are lines of the substance which have retained their parallelism. The displacement back from (x1, y1, z1) to (x, y, z) fulfils the same condition, and therefore we must have partial differential co-efficients with reference to this system of variables. The relation between F and F is clearly |