R B fig. 10. m/ A forces parallel to three rectangular axes ox, oy, oz, fig. 10, which would represent the action of the forces mA, mB, &c., estimated in the direction of the axes; or, which is the same thing, each of the forces mA, mB, &c. acting on m, may be resolved into three other forces parallel to the axes. 33. It is evident that when the partial forces act in the same direction, their sum is the force in that axis; and when some act in one direction, and others in an opposite direction, it is their difference that is to be estimated. 34. Thus any number of forces of any kind are capable of being resolved into other forces, in the direction of two or of three rectangular axes, according as the forces act in the same or in different planes. 35. If a particle of matter remain in a state of equilibrium, though acted upon by any number of forces, and free to move in every direction, the resulting force must be zero. 36. If the material point be in equilibrio on a curved surface, or on a curved line, the resulting force must be perpendicular to the line or surface, otherwise the particle would slide. The line or surface resists the resulting force with an equal and contrary pressure. 37. Let oA=X, oB=Y, oC=Z, fig. 10, be three rectangular component forces, of which om=F is their resulting force. Then, if mA, mB, mC be joined, om=F will be the hypothenuse common to three rectangular triangles, oAm, oBm, and oCm. Let the angles moA a, moB=b, moC=c; then X=F cos a, Y=F cos b, Z=F cos c. Thus the partial forces are proportional to the cosines of the angles which their directions make with their resultant. But BQ being a rectangular parallelopiped (1). FX + Y + Z3. (2). Hence x2+Y2+Z cosa+cos b+cos c = 1. When the component forces are known, equation (2) will give a value of the resulting force, and equations (1) will determine its direction by the angles a, b, and c; but if the resulting force be given, its resolution into the three component forces X, Y, Z, making with it the angles a, b, c, will be given by (1). If one of the component forces as Z be zero, then c = 90°, F = √X2 + Y1, X= F cos a, Y = F cos b. 38. Velocity and force being each represented by the same space, whatever has been explained with regard to the resolution and composition of the one applies equally to the other. The general Principles of Equilibrium. 39. The general principles of equilibrium may be expressed analytically, by supposing a to be the origin of a force F, acting on a particle of matter at m, fig. 11, in the direction om. If o' be the origin of the coordinates; a, b, c, the co-ordinates of o, and x, y, z those of m;. the diagonal om, which may be represented by r, will be B m fig.11. But F, the whole force in om, is to its component force in hence the component force parallel to the axis or is F (x − a) r In the same manner it may be shown, that are the component forces parallel to oy and oz. Now the equation Again, if F' be another force acting on the particle at m in another direction r', its component forces parallel to the co-ordinates will be, And any number of forces acting on the particle m may be resolved in the same manner, whatever their directions may be. If Σ be employed to denote the sum of any number of finite quantities, represented by the same general symbol ; Σ.Γ or ; are the sums of the is the sum of the partial forces urging the particle parallel to the axis ox. Likewise Σ.F. partial forces that urge the particle parallel to the axis oy and oz. Now if F, be the resulting force of all the forces F, F', F", &c. that act on the particle m, and if u be the straight line drawn from the origin of the resulting force to m, by what precedes are the expressions of the resulting force F, resolved in directions parallel to the three co-ordinates; hence or if the sums of the component forces parallel to the axis x, y, z, represented by X, Y, Z, we shall have be If the first of these be multiplied by dr, the second by dy, and the third by dz, their sum will be Fsu = Xô + Yông + Zôz. 40. If the intensity of the force can be expressed in terms of the distance of its point of application from its origin, X, Y, and Z may be eliminated from this equation, and the resulting force will then be given in functions of the distance only. All the forces in nature are functions of the distance, gravity for example, which varies inversely as the square of the distance of its origin from the point of its application. Were that not the case, the preceding equation could be of no use. 41. When the particle is in equilibrio, the resulting force is zero; consequently Xộc + Yôg + Z = 0 (3), which is the general equation of the equilibrium of a free particle. 42. Thus, when a particle of matter urged by any forces whatever remains in equilibrio, the sum of the products of each force by the element of its direction is zero. As the equation is true, whatever be the values of dx, dy, dz, it is equivalent to the three partial equations in the direction of the axes of the co-ordinates, that is to X=0, Y = 0, Z =0, for it is evident that if the resulting force be zero, its component forces must also be zero. On Pressure. 43. A pressure is a force opposed by another force, so that no motion takes place. 44. Equal and proportionate pressures are such as are produced by forces which would generate equal and proportionate motions in equal times. 45. Two contrary pressures will balance each other, when the motions which the forces would separately produce in contrary directions are equal; and one pressure will counterbalance two others, when it would produce a motion equal and contrary to the resultant of the motions which would be produced by the other forces. 46. It results from the comparison of motions, that if a body remain at rest, by means of three pressures, they must have the same ratio to one another, as the sides of a triangle parallel to the directions. ture, in which no two of its elements are in the same plane, then, mN = √(x − a)2 + (y − b)2 + (≈ — c)2 x, y, z being the co-ordinates of m, and a, b, c those of N, The centre of curvature N, which is the intersection of two consecutive normals mN, m'N, never varies in the circle and sphere, because the curvature is every where the same; but in all other curves and surfaces the position of N changes with every point in the curve or surface, and a, b, c, are only constant from one point to another. By this property, the equation of the radius of curvature is formed from the equation of the curve, or surface. If r be the radius of curvature, it is evident, that though it may vary from one point to another, it is constant for any one point m where dr = 0. Equilibrium of a Particle on a curved Surface. 48. The equation (3) is sufficient for the equilibrium of a particle of matter, if it be free to move in any direction; but if it be constrained to remain on a curved surface, the resulting force of all the forces acting upon it must be perpendicular to the surface, otherwise it would slide along it; but as by experience it is found that re-action is equal and contrary to action, the perpendicular force will be resisted by the re-action of the surface, so that the re-action is equal, and contrary to the force destroyed; hence if R, be the resistance of the surface, the equation of equilibrium will be Sr, Sy, Szare arbitrary; these variations may therefore be assumed to take place in the direction of the curved surface on which the particle moves: then by the property of the normal, dr = 0; which reduces the preceding equation to Xô? + Tây + Z) = 0. But this equation is no longer equivalent to three equations, but to two only, since one of the elements Sx, dy, Sz, must be eliminated by the equation of the surface. 49. The same result may be obtained in another way. For if u = 0 be the equation of the surface, then du = 0; but as the equation of the normal is derived from that of the surface, the equation Sr 0 is connected with the preceding, so that dr = Ndu. But r = √(x−a)2 + (y−b)2 + (z−c)2 whence |