Draw any line intersecting the directions of two of the forces and not parallel to the third force, and take this line for the axis of x. Then the first two forces have no moment round the axis of ; therefore the equation L = 0 requires that the third force should have no moment round the axis of x; that is, the direction of the third force must pass through the axis of x. Since then any line, which meets the directions of two of the forces, and is not parallel to the direction of the third meets that direction, the three forces must lie in one plane. Combining this proposition with that in Art. (58), we see that if three forces keep a body in equilibrium, they must all lie in the same plane and must meet in a point or be parallel. 85. If the axes of coordinates be oblique, suppose l, m, n to denote the sines of the angles between the axes of y and z, z and x, x and y, respectively; then we may shew, as in Art. (72), that any system of forces can be reduced to ΣX, ΣY, ΣZ, acting at the origin along the axes of x, y, z respectively, and three couples in the three coordinate planes, having their moments equal to IL, mM, nN respectively, where, as before, L = (Zy – Yz), &c. Σ equilibrium, we must have, as before, Also for ΣΧ = 0, ΣΥ = 0, ΣΖ = 0; L= 0, M=0, N = 0. That the forces may admit of a single resultant we must have, as before, ΙΣΧ + ΜΣΥ + NEZ = 0, and ΣΧ, ΣΥ, ΣΖ not all vanishing. EXAMPLES. 1. Four parallel forces act at the angles of a plane quadrilateral and are inversely proportional to the segments of its diagonals nearest to them; shew that the point of application of their resultant lies at the intersection of the diagonals. 2. A cone whose vertical angle is 30°, and whose weight is W is placed with its vertex on a smooth horizontal plane; shew that it may be kept with its slant side in a vertical position by a couple whose arm is equal to the length of 3 W the slant side of the cone, and each force 16 3. A cube is acted on by four forces; one in a diagonal, the others in edges no two of which are in the same plane and which do not meet the diagonal; find the condition that they may have a single resultant. 4. A uniform heavy triangle is supported in a horizontal position by three parallel strings attached to the three sides respectively; shew that there is an infinite number of ways in which the strings may be relatively disposed so that their tensions may be equal, but that the situation of one being given, that of each of the other two is determinate. 5. A sphere of given weight rests upon three planes whose equations are la+my+ nz = 0, 1,x + m1y + n1z = 0, lx + my + n ̧z = 0, the axis of z being vertical; shew that the pressures upon the planes are respectively proportional to Im1 - 1m, lm - 1m, and 1m-lm, and find each pressure. 6. A heavy triangle ABC is suspended from a point by three strings, mutually at right angles, attached to the angular points of the triangle; if O be the inclination of the triangle to the horizon in its position of equilibrium, then 3 cos = √(1+sec A sec B sec C) 7. An equilateral triangle without weight has three unequal particles placed at its angular points; the system is suspended from a fixed point by three equal strings at right angles to each other fastened to the corners of the triangle; find the inclination of the plane of the triangle to the horizon. 8. A uniform bent lever whose arms are at right angles to each other is capable of being enclosed in the interior of a smooth spherical surface; determine the position of equilibrium. 9. Four smooth equal spheres are placed in a hemispherical bowl. The centres of three of them are in the same horizontal plane, and that of the other is above it. If the radius of each sphere be one-third that of the bowl, shew that the mutual pressures of the spheres are all equal. 10. Three equal spheres hang in contact from a fixed point by three equal strings; find the heaviest sphere of given radius that may be placed upon them without causing them to separate. CHAPTER VI. EQUILIBRIUM OF A CONSTRAINED BODY. 86. PROP. To find the conditions of equilibrium of forces acting upon a rigid body when one point is fixed. Let the fixed point be taken as the origin of coordinates. The action of the forces on the body will produce a pressure on the fixed point; let X', Y, Z be the resolved parts of this pressure parallel to the axes. Then the fixed point will exert forces - X', -Y', - Z', against the body; and if we take these forces in connexion with the given forces, we may suppose the body to be free, and the equations of equilibrium are ΣΧ - Χ = 0, ΣΥ - Υ = 0, ΣΖ - Ζ = 0, The first three equations give the resolved parts of the pressure on the fixed point; and the last three are the only conditions to be satisfied by the given forces. If all the forces are parallel, we may take the axis of ≈ passing through the fixed point parallel to the forces. The above equations then reduce to ΣΖ - Ζ = 0, ΣZy=0, ΣΖ = 0; the first determines the pressure on the fixed point, and the other two are conditions which must be satisfied by the given forces. If all the forces act in one plane passing through the fixed point, and we take this plane for that of (x, y), the above equations reduce to EX-X'=0, ΣY- Y' = 0, (Yx - Xy) = 0; the first two determine the pressure on the fixed point, and the third is the only condition which the forces must satisfy. From the equations X' = ΣX, Y' = Σ Y, Z' = ΣZ, it follows that the pressure on the fixed point is equal to the resultant of all the forces of the system moved parallel to themselves up to the fixed point. 87. PROP. To find the conditions of equilibrium of a body which has two points in it fixed. Let the axis of z pass through the two fixed points; and let the distances of the points from the origin be z' and z". Also let X', Y', Z'; X", Y", Z", be the resolved parts of the pressures on these points. Then, as in Art. (86), the equations of equilibrium will be N = 0. The first, second, fourth, and fifth of these equations will determine X', X", Y', Y"; the third equation gives Z' + Z", shewing that the pressures on the fixed points in the direction of the line joining them are indeterminate, being connected by one equation only. The last is the only condition of equilibrium, namely N=0. 88. The indeterminateness which occurs as to the values of Z' and Z" might have been expected; for if two forces, -Z' and -Z", act upon a rigid body in the same straight line, their effect will be the same at whatever point in their - |