Case 11. Figure 5.—When any two angles of direction arc greater than two right angles.
Let A B, E B, C B, be the three directions, whereof any two angles made by these lines are greater than two right angles, and consequently the remaining one less than two right angles. Let the given force act in E B ; produce E B through the opposite angle to D, so as to divide it into two angles; make B D to represent the intensity in E B, then by complet ing the parallelogram A B c D, as before, B A will represent the intensity in B A, and B c in 13 c; and as the forces are supposed to act at the points A, E, c, they are either all drawing the point 13 or all pressing it.
Proposition \I.—Given, the directions of four forces in the same plane, keeping a solid in equilibria, and one of the intensities, to find the intensities of the other three.
Produce any two directions till they meet each other ; also, produce the other two directions till they meet each other ; join the two angular points ; then by means of the given force, find the other two at the same point : then, because two forces acting at each point of concourse in the same right line must be equal, and have opposite tendencies, the force in this line acting at the other point of concourse will now be given : therefore, find the two remaining intensities in the same manner as at the first point of concourse.
Example I. Figure G.—Let E A, F B, G c, a D, be the direction of the forces that support the body A B c D, and let the given force be in E A. Produce E A, F B, till they meet in i ; also produce a c, a D, till they meet in Q. Join n Q, and produce it to P ; then let i K represent the given force, and complete the parallelogram i K L at. Make Q P equal to L, and complete the parallelogram OP QR; then Will i DI represent the intensity in F B, 0 Q in G C, and B Q in II D.
Example 11. Figure 7.—Let A n D be a lever with three arms, A C, it C, D c, revolvable about c, as a fulcrum, sup ported in the direction c o; and let forces act at the extre mities A, B, D, in given directions, A K. D E, E and keep it in equilibrio : it is required to find the proportion of the forces. Produce two of the directions till they meet ; also
produce the other direction, and that of the prop, till they meet; join the twd angular points, and proceed as in Exam ple I., and find the parallelograms n E F G, and K L >t N : then 1, K is the force acting at A, and m K that in the direc tion of the prop, it E, the force acting at 13, and F E that at D. The tendencies of these threes are thus distinguished: let the point /3 be drawn towards E, then the line E B is in a state of tension ; and because the angles u E G and G E F are less than two right angles, the force in the direction E D will also be in a state of tension, and the middle one, E K, in a state of compression. Again, because the angles L K and a K N are less than two right angles, and because E K is in a state of compression, ti A is likewise in a state of compression, and the middle one c K, is in a state of tension ; or the post, c o, on the opposite side, is in a state of compression, acting on the other side of G.
It must be observed, when any force acts upon any point of a solid body, that to draw on one side of the point is the same as to press upon the other side, or to press upon one side is the same as to draw upon the opposite; therefore, as the point c is drawn by the force Dt the prop, c o, is com pressed by the fulcrum at c. The arms c A, c B, c D, are supposed to be void of weight. If the forces acting at A, II, D, be weights, 1', e, R, going over the pulleys s, T, u, all the lines, A 5, B n 11, Will be in a state of tension.
Proposition VII.—Given, the direction of five forces in one plane, keeping a solid in equilibrio, and the intensities and tendencies of two of them, to find the intensities and tendencies of the rest.
Find a force equivalent to the two given forces; than unite this given force with the three remaining ones, and the directions of four forces, with the intensity of one of them, will be given to find the rest, which may be found by the last problem.
Let A B c D F, Figure 8, be a lever, with four arms, F A,