Case II. Figure 3.—When the two given forces tend to two different points. Let A B and 13 c be the two given forces, let the tendency of A B be from II towards A, and that of II c from c towards II ; produce either, as A B, to E: make 13 E equal to A B, and complete the parallelogram Dona; draw the diagonal B D, and it will be equivalent to n c and II a, or because II E is equal to B A, and both in the same straight line : and since both forces tend to the same point, A, the force a a is equivalent to II A and n c.
It is evident, that though the angles and directions given were the same in both eases, yet the tendency and quantity would he different in each.
Proposition 111.—To resolve any force into two others, in any given directions, which shall act against any point of the line of direction of the given force. Figure 4.
Let B a be the line of direction of the given force, B the given point, from which the required intensities are to act, and n F, n o their directions. Nake. n D equal to the intensity of the given force ; complete the parallelogram Anon; then 13 A is the force acting in the line n F; B 0 that in the line 13 and their tendencies are contrary to'the middle force.
Proposition I V.—If any two forces keep a third in equi librio, the direction of the third has the same point of con course, and is in the same plane with the other two, and all the three forces are to each other as the sides and diagonal of a parallelogram formed on their lines of direction, Figure 5.
Let n c, be the two fin-ces ; complete the parallelogram A 11 c n ; then the force n B is equivalent to A 13 and B e; but if any three be in equilibrio with D n, it must be equal and opposite ; therefore, make II a equal and opposite, and the two forces n v and B a are in equilibrio : take away the force n II, and let its equivalent fbrces A n and B c counteract 13 E, then the three 11)rces A D, c n, and B a, are also in equilibrio: because n D and 13 E are hi a straight line, the direction of E II passes through the point n, and is in the same plane with A n and n c ; for D D is in that plane : and because n a is equal to B D, the three forces A C 13, E B, are expressed by the two sides A n and u c, and the diagonal D n of the parallelogram A B c D, formed on their lines of direction.
Corollary 1.—Hence, if ally three forces be in equilibrio with each other, they are as the sides of a triangle drawn parallel to their directions.
Corollary 2.—If the directions of any three forces, acting against the same point, keeping it in equilibria, be given, and one of the intensities; the intensities of the other two may be found.
Proposition V.—The lines of direction of three tbrees keeping each other in equilibria or a solid, and the inten sity and tendency of one of them being given ; to lied the intensity and tendency of the other two.
Case 1. Figure 4.—When two of the angles formed by the three lines of direction are less than two right angles. Let the three directions be B F, B E, 13 G, and let the given intensity be in the line B E, and let its tendency be from E towards n. Make B D equal to the given intensity, and complete the parallelogram A 13 C D. A B is the intensity in its own line of direction B F, its tendency being from B towards F; and B C is the intensity of the force in its line of direction B a, its tendency being from B towards G: for produce E B to n, since the force acting in the line E B presses the point B, then, by Axiom, 12, it is the same thing. whether the force in the line E B press the point II, or an equal force on the other side of B in E it draw the point 13, and instead of the force pressing the point B by a three at E, let the point B be drawn by a force at u; thus the point B will be drawn by three forces, which are in equilibrio by the last Proposition. Or if the point tit had been drawn by a force acting at E, the two forces acting in the lines B F and B G would have pressed these lines, and conse quently three farces acting at F, 0, would be all pressing the point B ; it therefore appears, when three forces keep each other in equilibria and their lines of direction make two angles less than two right angles, that the force acting in the inter mediate line will be contrary to those in the two extreme lines.
Though this example only shows how to find the two extreme forces when the intermediate force is given ; yet the intermediate force and one of the extreme forces may as readily be found by having the other extreme force given : because when one of the angles of a parallelogram is given, and the position of a diagonal passing through that angle, it may be described as readily by having either of the sides as the diagonal.