COR. 3. When the two forces act at an angle, the equivalent is less than the sum, and greater than the difference; and it is more nearly equal to the sum the smaller the angle, so that when the angle vanishes it will become equal to the sum.
This is evident from the properties of a triangle.
Cox. 4. The simple forces are inversely as the per pendiculars dropt on their directions front any point in the direction of the equivalent.
1st, If dropt from D, (Fig. 3.) the ACD=AA DB, hence Therefore AC : AB: : DN : DM.
2d, If dropt from any other point 0, it is easy to see by similar triangles, that DN : DM : : GE : OF; hence AC : Ali : : GE : OF.
Cask: 2. Any numb fr of (Ojai& a! a point.— The equivalent will be found by I onipeunding two ac cording to last case, then compounding the equivalent with a third, and so on.
If all the forces lic in one plane, it is evident that the most expeditious w.ay of finding the equivalent will be by drawing successively lines equal and paralltl to tiio•e representing the forces, as in Figure (Fig. 4.); and then the line drawn in completing the polygon will be the e inivalent.
The equivalent of three forces is the diagonal of a pa- • ralielopiped, of which the three forces arc the adjacent lineal edges.
All, AC, AD (Fig. 5.) being the three forces applied
at A, the diagonal AE will he the equivalent.
For joinity.; I1E, AG, the equivalent of !SC, Al) is AG, because DC is a parallelogram, and since the plane in which the parallels AB, EG are, is cut by parallel planes, the lines of common section BE, AG, wilt be parallel, (Encl. 16. I L) hence the Figure BG is a pa rallelogram, there lore AE is the equivalent of A13, AG, that is of All. AC, Al).
It is easy from the Figure to perceive, by help of (Eucl. 47. I.) that if the parallelopiped is rectangular, and if a, 6, c denote the three simple forces, and d the equivalent, then CAsE 3. Two parallel forces applied to different points of a body in the same direction.—Since the two parallels may be considered as coming from a point infinitely dis tant, or as making an infinitely small angle with one an other, their equivalent will be equal and parallel to their sum by Cor. G. to Case I. and it will divide the straight line joining the points of application in the inverse ratio of the forces.
For by Cor. 4. the forces are inversely as the perpen diculars on their directions, which, in this case, are as the segments of the joining line. The Figure (Fig. 6.) is sufficient to show this.